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Metamath Proof Explorer

Theorem dffltz

Description: Fermat's Last Theorem (FLT) for nonzero integers is equivalent to the original scope of natural numbers. The backwards direction takes ( a ^ n ) + ( b ^ n ) = ( c ^ n ) , and adds the negative of any negative term to both sides, thus creating the corresponding equation with only positive integers. There are six combinations of negativity, so the proof is particularly long. (Contributed by Steven Nguyen, 27-Feb-2023)

Ref Expression
Assertion dffltz ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )

Proof

Step Hyp Ref Expression
1 oveq1 ⊢ ( 𝑥 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) → ( 𝑥 ↑ 𝑛 ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) )
2 1 oveq1d ⊢ ( 𝑥 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) → ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) )
3 2 eqeq1d ⊢ ( 𝑥 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ) )
4 oveq1 ⊢ ( 𝑦 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) → ( 𝑦 ↑ 𝑛 ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) )
5 4 oveq2d ⊢ ( 𝑦 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) )
6 5 eqeq1d ⊢ ( 𝑦 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) → ( ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ) )
7 oveq1 ⊢ ( 𝑧 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) → ( 𝑧 ↑ 𝑛 ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) )
8 7 eqeq2d ⊢ ( 𝑧 = if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) → ( ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) ) )
9 simp-4r ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
10 eldifi ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → 𝑎 ∈ ℤ )
11 eldifsni ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → 𝑎 ≠ 0 )
12 10 11 jca ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ( 𝑎 ∈ ℤ ∧ 𝑎 ≠ 0 ) )
13 nnabscl ⊢ ( ( 𝑎 ∈ ℤ ∧ 𝑎 ≠ 0 ) → ( abs ‘ 𝑎 ) ∈ ℕ )
14 9 12 13 3syl ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( abs ‘ 𝑎 ) ∈ ℕ )
15 simp-6r ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
16 15 eldifad ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑎 ∈ ℤ )
17 simplr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < 𝑎 )
18 elnnz ⊢ ( 𝑎 ∈ ℕ ↔ ( 𝑎 ∈ ℤ ∧ 0 < 𝑎 ) )
19 16 17 18 sylanbrc ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑎 ∈ ℕ )
20 eldifsni ⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → 𝑏 ≠ 0 )
21 20 ad6antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ≠ 0 )
22 simplr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ¬ 0 < 𝑏 )
23 eldifi ⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → 𝑏 ∈ ℤ )
24 23 ad6antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ℤ )
25 21 22 24 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → - 𝑏 ∈ ℕ )
26 simp-7r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
27 26 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ℤ )
28 simpllr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 0 < 𝑎 )
29 27 28 18 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ℕ )
30 25 29 ifclda ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ∈ ℕ )
31 19 30 ifclda ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ∈ ℕ )
32 11 ad7antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ≠ 0 )
33 simpllr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ¬ 0 < 𝑎 )
34 10 ad7antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ℤ )
35 32 33 34 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → - 𝑎 ∈ ℕ )
36 simp-6r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
37 36 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ℤ )
38 simplr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 0 < 𝑏 )
39 elnnz ⊢ ( 𝑏 ∈ ℕ ↔ ( 𝑏 ∈ ℤ ∧ 0 < 𝑏 ) )
40 37 38 39 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ℕ )
41 35 40 ifclda ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ∈ ℕ )
42 11 ad6antlr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ≠ 0 )
43 simplr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < 𝑎 )
44 10 ad6antlr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ∈ ℤ )
45 42 43 44 negn0nposznnd ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → - 𝑎 ∈ ℕ )
46 41 45 ifclda ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ∈ ℕ )
47 31 46 ifclda ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ∈ ℕ )
48 14 47 ifcld ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ∈ ℕ )
49 simpllr ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
50 23 20 jca ⊢ ( 𝑏 ∈ ( ℤ ∖ { 0 } ) → ( 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0 ) )
51 nnabscl ⊢ ( ( 𝑏 ∈ ℤ ∧ 𝑏 ≠ 0 ) → ( abs ‘ 𝑏 ) ∈ ℕ )
52 49 50 51 3syl ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( abs ‘ 𝑏 ) ∈ ℕ )
53 simp-5r ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
54 53 eldifad ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑏 ∈ ℤ )
55 simpr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < 𝑏 )
56 54 55 39 sylanbrc ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑏 ∈ ℕ )
57 simp-5r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
58 57 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ℤ )
59 simpr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 0 < 𝑐 )
60 elnnz ⊢ ( 𝑐 ∈ ℕ ↔ ( 𝑐 ∈ ℤ ∧ 0 < 𝑐 ) )
61 58 59 60 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ℕ )
62 eldifsni ⊢ ( 𝑐 ∈ ( ℤ ∖ { 0 } ) → 𝑐 ≠ 0 )
63 62 ad5antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ≠ 0 )
64 simpr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ 0 < 𝑐 )
65 eldifi ⊢ ( 𝑐 ∈ ( ℤ ∖ { 0 } ) → 𝑐 ∈ ℤ )
66 65 ad5antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ∈ ℤ )
67 63 64 66 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - 𝑐 ∈ ℕ )
68 61 67 ifclda ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ∈ ℕ )
69 56 68 ifclda ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ 0 < 𝑎 ) → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ∈ ℕ )
70 simp-5r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
71 70 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ℤ )
72 simpr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 0 < 𝑐 )
73 71 72 60 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ℕ )
74 62 ad5antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ≠ 0 )
75 simpr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ 0 < 𝑐 )
76 65 ad5antlr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ∈ ℤ )
77 74 75 76 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - 𝑐 ∈ ℕ )
78 73 77 ifclda ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ∈ ℕ )
79 20 ad5antlr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ≠ 0 )
80 simpr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < 𝑏 )
81 23 ad5antlr ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ∈ ℤ )
82 79 80 81 negn0nposznnd ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → - 𝑏 ∈ ℕ )
83 78 82 ifclda ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ 0 < 𝑎 ) → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ∈ ℕ )
84 69 83 ifclda ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ∈ ℕ )
85 52 84 ifcld ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ∈ ℕ )
86 simpllr ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
87 86 eldifad ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ℤ )
88 86 62 syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ≠ 0 )
89 nnabscl ⊢ ( ( 𝑐 ∈ ℤ ∧ 𝑐 ≠ 0 ) → ( abs ‘ 𝑐 ) ∈ ℕ )
90 87 88 89 syl2anc ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑐 ) ∈ ℕ )
91 simp-5r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
92 91 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑐 ∈ ℤ )
93 simp-7r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
94 93 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑎 ∈ ℤ )
95 94 zred ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑎 ∈ ℝ )
96 eluz3nn ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → 𝑛 ∈ ℕ )
97 96 ad7antr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑛 ∈ ℕ )
98 97 nnnn0d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑛 ∈ ℕ0 )
99 95 98 reexpcld ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℝ )
100 simp-6r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
101 100 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑏 ∈ ℤ )
102 101 zred ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑏 ∈ ℝ )
103 102 98 reexpcld ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℝ )
104 simplr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < 𝑎 )
105 simpllr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ¬ ( 𝑛 / 2 ) ∈ ℕ )
106 95 97 105 oexpreposd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( 0 < 𝑎 ↔ 0 < ( 𝑎 ↑ 𝑛 ) ) )
107 104 106 mpbid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < ( 𝑎 ↑ 𝑛 ) )
108 simpr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < 𝑏 )
109 102 97 105 oexpreposd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( 0 < 𝑏 ↔ 0 < ( 𝑏 ↑ 𝑛 ) ) )
110 108 109 mpbid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < ( 𝑏 ↑ 𝑛 ) )
111 99 103 107 110 addgt0d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
112 simp-4r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
113 111 112 breqtrd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < ( 𝑐 ↑ 𝑛 ) )
114 92 zred ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑐 ∈ ℝ )
115 114 97 105 oexpreposd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( 0 < 𝑐 ↔ 0 < ( 𝑐 ↑ 𝑛 ) ) )
116 113 115 mpbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 0 < 𝑐 )
117 92 116 60 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → 𝑐 ∈ ℕ )
118 simp-8r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
119 118 eldifad ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ℤ )
120 simpllr ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 0 < 𝑎 )
121 119 120 18 sylanbrc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ℕ )
122 simp-7r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
123 122 20 syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ≠ 0 )
124 simplr ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ 0 < 𝑏 )
125 122 eldifad ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ℤ )
126 123 124 125 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - 𝑏 ∈ ℕ )
127 121 126 ifclda ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ∈ ℕ )
128 117 127 ifclda ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ∈ ℕ )
129 simp-7r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
130 129 eldifad ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ℤ )
131 simplr ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 0 < 𝑏 )
132 130 131 39 sylanbrc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ℕ )
133 simp-8r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
134 133 11 syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ≠ 0 )
135 simpllr ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ 0 < 𝑎 )
136 133 eldifad ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ℤ )
137 134 135 136 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - 𝑎 ∈ ℕ )
138 132 137 ifclda ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ∈ ℕ )
139 simp-5r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
140 139 62 syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑐 ≠ 0 )
141 simp-7r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
142 141 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ∈ ℤ )
143 142 zred ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ∈ ℝ )
144 96 ad7antr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑛 ∈ ℕ )
145 144 nnnn0d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑛 ∈ ℕ0 )
146 143 145 reexpcld ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℝ )
147 simp-6r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
148 147 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ∈ ℤ )
149 148 zred ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ∈ ℝ )
150 149 145 reexpcld ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℝ )
151 146 150 readdcld ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ∈ ℝ )
152 0red ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 0 ∈ ℝ )
153 11 neneqd ⊢ ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ¬ 𝑎 = 0 )
154 141 153 syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 𝑎 = 0 )
155 zcn ⊢ ( 𝑎 ∈ ℤ → 𝑎 ∈ ℂ )
156 141 10 155 3syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ∈ ℂ )
157 expeq0 ⊢ ( ( 𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ) → ( ( 𝑎 ↑ 𝑛 ) = 0 ↔ 𝑎 = 0 ) )
158 156 144 157 syl2anc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) = 0 ↔ 𝑎 = 0 ) )
159 154 158 mtbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ ( 𝑎 ↑ 𝑛 ) = 0 )
160 simplr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < 𝑎 )
161 simpllr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ ( 𝑛 / 2 ) ∈ ℕ )
162 143 144 161 oexpreposd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 0 < 𝑎 ↔ 0 < ( 𝑎 ↑ 𝑛 ) ) )
163 160 162 mtbid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < ( 𝑎 ↑ 𝑛 ) )
164 ioran ⊢ ( ¬ ( ( 𝑎 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑎 ↑ 𝑛 ) ) ↔ ( ¬ ( 𝑎 ↑ 𝑛 ) = 0 ∧ ¬ 0 < ( 𝑎 ↑ 𝑛 ) ) )
165 159 163 164 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ ( ( 𝑎 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑎 ↑ 𝑛 ) ) )
166 146 152 lttrid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) < 0 ↔ ¬ ( ( 𝑎 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑎 ↑ 𝑛 ) ) ) )
167 165 166 mpbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑎 ↑ 𝑛 ) < 0 )
168 zcn ⊢ ( 𝑏 ∈ ℤ → 𝑏 ∈ ℂ )
169 147 23 168 3syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ∈ ℂ )
170 147 20 syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑏 ≠ 0 )
171 eluzelz ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → 𝑛 ∈ ℤ )
172 171 ad7antr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑛 ∈ ℤ )
173 169 170 172 expne0d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑏 ↑ 𝑛 ) ≠ 0 )
174 173 neneqd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ ( 𝑏 ↑ 𝑛 ) = 0 )
175 simpr ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < 𝑏 )
176 149 144 161 oexpreposd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 0 < 𝑏 ↔ 0 < ( 𝑏 ↑ 𝑛 ) ) )
177 175 176 mtbid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < ( 𝑏 ↑ 𝑛 ) )
178 ioran ⊢ ( ¬ ( ( 𝑏 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑏 ↑ 𝑛 ) ) ↔ ( ¬ ( 𝑏 ↑ 𝑛 ) = 0 ∧ ¬ 0 < ( 𝑏 ↑ 𝑛 ) ) )
179 174 177 178 sylanbrc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ ( ( 𝑏 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑏 ↑ 𝑛 ) ) )
180 150 152 lttrid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑏 ↑ 𝑛 ) < 0 ↔ ¬ ( ( 𝑏 ↑ 𝑛 ) = 0 ∨ 0 < ( 𝑏 ↑ 𝑛 ) ) ) )
181 179 180 mpbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑏 ↑ 𝑛 ) < 0 )
182 146 150 152 152 167 181 lt2addd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) < ( 0 + 0 ) )
183 00id ⊢ ( 0 + 0 ) = 0
184 182 183 breqtrdi ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) < 0 )
185 151 152 184 ltnsymd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
186 simp-4r ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
187 186 eqcomd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑐 ↑ 𝑛 ) = ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
188 187 breq2d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 0 < ( 𝑐 ↑ 𝑛 ) ↔ 0 < ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ) )
189 185 188 mtbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < ( 𝑐 ↑ 𝑛 ) )
190 139 eldifad ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑐 ∈ ℤ )
191 190 zred ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑐 ∈ ℝ )
192 191 144 161 oexpreposd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 0 < 𝑐 ↔ 0 < ( 𝑐 ↑ 𝑛 ) ) )
193 189 192 mtbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 0 < 𝑐 )
194 140 193 190 negn0nposznnd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → - 𝑐 ∈ ℕ )
195 138 194 ifclda ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ∈ ℕ )
196 128 195 ifclda ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ∈ ℕ )
197 90 196 ifclda ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ∈ ℕ )
198 simplr ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
199 simp-5r ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
200 199 eldifad ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ℤ )
201 200 zred ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ℝ )
202 absresq ⊢ ( 𝑎 ∈ ℝ → ( ( abs ‘ 𝑎 ) ↑ 2 ) = ( 𝑎 ↑ 2 ) )
203 201 202 syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑎 ) ↑ 2 ) = ( 𝑎 ↑ 2 ) )
204 203 oveq1d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( ( abs ‘ 𝑎 ) ↑ 2 ) ↑ ( 𝑛 / 2 ) ) = ( ( 𝑎 ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
205 199 10 155 3syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑎 ∈ ℂ )
206 205 abscld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑎 ) ∈ ℝ )
207 206 recnd ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑎 ) ∈ ℂ )
208 simpr ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑛 / 2 ) ∈ ℕ )
209 208 nnnn0d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑛 / 2 ) ∈ ℕ0 )
210 2nn0 ⊢ 2 ∈ ℕ0
211 210 a1i ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 2 ∈ ℕ0 )
212 207 209 211 expmuld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑎 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( ( abs ‘ 𝑎 ) ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
213 205 209 211 expmuld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑎 ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( 𝑎 ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
214 204 212 213 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑎 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( 𝑎 ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
215 simp-5l ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑛 ∈ ( ℤ≥ ‘ 3 ) )
216 nncn ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℂ )
217 215 96 216 3syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑛 ∈ ℂ )
218 2cnd ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 2 ∈ ℂ )
219 2ne0 ⊢ 2 ≠ 0
220 219 a1i ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 2 ≠ 0 )
221 217 218 220 divcan2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 2 · ( 𝑛 / 2 ) ) = 𝑛 )
222 221 eqcomd ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑛 = ( 2 · ( 𝑛 / 2 ) ) )
223 222 oveq2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) = ( ( abs ‘ 𝑎 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
224 222 oveq2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑎 ↑ 𝑛 ) = ( 𝑎 ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
225 214 223 224 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
226 simp-4r ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
227 226 eldifad ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ℤ )
228 227 zred ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ℝ )
229 absresq ⊢ ( 𝑏 ∈ ℝ → ( ( abs ‘ 𝑏 ) ↑ 2 ) = ( 𝑏 ↑ 2 ) )
230 228 229 syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑏 ) ↑ 2 ) = ( 𝑏 ↑ 2 ) )
231 230 oveq1d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( ( abs ‘ 𝑏 ) ↑ 2 ) ↑ ( 𝑛 / 2 ) ) = ( ( 𝑏 ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
232 226 23 168 3syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑏 ∈ ℂ )
233 232 abscld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑏 ) ∈ ℝ )
234 233 recnd ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑏 ) ∈ ℂ )
235 234 209 211 expmuld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑏 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( ( abs ‘ 𝑏 ) ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
236 232 209 211 expmuld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑏 ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( 𝑏 ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
237 231 235 236 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑏 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( 𝑏 ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
238 222 oveq2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) = ( ( abs ‘ 𝑏 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
239 222 oveq2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑏 ↑ 𝑛 ) = ( 𝑏 ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
240 237 238 239 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
241 225 240 oveq12d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) + ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
242 87 zred ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ℝ )
243 absresq ⊢ ( 𝑐 ∈ ℝ → ( ( abs ‘ 𝑐 ) ↑ 2 ) = ( 𝑐 ↑ 2 ) )
244 242 243 syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑐 ) ↑ 2 ) = ( 𝑐 ↑ 2 ) )
245 244 oveq1d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( ( abs ‘ 𝑐 ) ↑ 2 ) ↑ ( 𝑛 / 2 ) ) = ( ( 𝑐 ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
246 zcn ⊢ ( 𝑐 ∈ ℤ → 𝑐 ∈ ℂ )
247 86 65 246 3syl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → 𝑐 ∈ ℂ )
248 247 abscld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑐 ) ∈ ℝ )
249 248 recnd ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( abs ‘ 𝑐 ) ∈ ℂ )
250 249 209 211 expmuld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑐 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( ( abs ‘ 𝑐 ) ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
251 247 209 211 expmuld ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑐 ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( ( 𝑐 ↑ 2 ) ↑ ( 𝑛 / 2 ) ) )
252 245 250 251 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑐 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) = ( 𝑐 ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
253 222 oveq2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑐 ) ↑ 𝑛 ) = ( ( abs ‘ 𝑐 ) ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
254 222 oveq2d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( 𝑐 ↑ 𝑛 ) = ( 𝑐 ↑ ( 2 · ( 𝑛 / 2 ) ) ) )
255 252 253 254 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( abs ‘ 𝑐 ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
256 198 241 255 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) + ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) ) = ( ( abs ‘ 𝑐 ) ↑ 𝑛 ) )
257 iftrue ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) = ( abs ‘ 𝑎 ) )
258 257 oveq1d ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) = ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) )
259 iftrue ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) = ( abs ‘ 𝑏 ) )
260 259 oveq1d ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) = ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) )
261 258 260 oveq12d ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) + ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) ) )
262 261 adantl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( ( ( abs ‘ 𝑎 ) ↑ 𝑛 ) + ( ( abs ‘ 𝑏 ) ↑ 𝑛 ) ) )
263 iftrue ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) = ( abs ‘ 𝑐 ) )
264 263 oveq1d ⊢ ( ( 𝑛 / 2 ) ∈ ℕ → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) = ( ( abs ‘ 𝑐 ) ↑ 𝑛 ) )
265 264 adantl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) = ( ( abs ‘ 𝑐 ) ↑ 𝑛 ) )
266 256 262 265 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) )
267 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) = 𝑎 )
268 267 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
269 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) = 𝑏 )
270 269 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
271 268 270 oveq12d ⊢ ( 0 < 𝑏 → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
272 271 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
273 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) = 𝑐 )
274 273 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
275 274 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
276 112 272 275 3eqtr4d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) )
277 simp-7r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
278 277 23 168 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ℂ )
279 simp-8l ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑛 ∈ ( ℤ≥ ‘ 3 ) )
280 279 96 syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑛 ∈ ℕ )
281 simp-4r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ¬ ( 𝑛 / 2 ) ∈ ℕ )
282 2nn ⊢ 2 ∈ ℕ
283 nndivdvds ⊢ ( ( 𝑛 ∈ ℕ ∧ 2 ∈ ℕ ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
284 280 282 283 sylancl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
285 281 284 mtbird ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ¬ 2 ∥ 𝑛 )
286 oexpneg ⊢ ( ( 𝑏 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑏 ↑ 𝑛 ) = - ( 𝑏 ↑ 𝑛 ) )
287 278 280 285 286 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( - 𝑏 ↑ 𝑛 ) = - ( 𝑏 ↑ 𝑛 ) )
288 287 oveq1d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( - 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
289 nnnn0 ⊢ ( 𝑛 ∈ ℕ → 𝑛 ∈ ℕ0 )
290 279 96 289 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑛 ∈ ℕ0 )
291 278 290 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
292 291 negcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → - ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
293 simp-6r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
294 293 65 246 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ℂ )
295 294 290 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑐 ↑ 𝑛 ) ∈ ℂ )
296 292 295 addcomd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) )
297 295 291 negsubd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) )
298 296 297 eqtrd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) )
299 118 10 155 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ℂ )
300 299 290 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℂ )
301 simp-5r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
302 301 eqcomd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑐 ↑ 𝑛 ) = ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
303 300 291 302 mvrraddd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) = ( 𝑎 ↑ 𝑛 ) )
304 288 298 303 3eqtrd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( - 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( 𝑎 ↑ 𝑛 ) )
305 iftrue ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) = - 𝑏 )
306 305 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
307 iftrue ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) = 𝑐 )
308 307 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) = ( 𝑐 ↑ 𝑛 ) )
309 306 308 oveq12d ⊢ ( 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
310 309 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
311 iftrue ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) = 𝑎 )
312 311 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
313 312 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
314 304 310 313 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
315 simp-8r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
316 315 10 155 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ℂ )
317 96 ad8antr ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑛 ∈ ℕ )
318 317 nnnn0d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑛 ∈ ℕ0 )
319 316 318 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℂ )
320 simp-6r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
321 320 65 246 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ∈ ℂ )
322 321 318 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑐 ↑ 𝑛 ) ∈ ℂ )
323 319 322 negsubd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) − ( 𝑐 ↑ 𝑛 ) ) )
324 319 322 subcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) − ( 𝑐 ↑ 𝑛 ) ) ∈ ℂ )
325 122 23 168 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ℂ )
326 325 318 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
327 326 negcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
328 simp-5r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
329 319 326 328 mvlraddd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑎 ↑ 𝑛 ) = ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) )
330 322 319 pncan3d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) + ( ( 𝑎 ↑ 𝑛 ) − ( 𝑐 ↑ 𝑛 ) ) ) = ( 𝑎 ↑ 𝑛 ) )
331 322 326 negsubd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) )
332 329 330 331 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) + ( ( 𝑎 ↑ 𝑛 ) − ( 𝑐 ↑ 𝑛 ) ) ) = ( ( 𝑐 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) )
333 322 324 327 332 addcanad ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) − ( 𝑐 ↑ 𝑛 ) ) = - ( 𝑏 ↑ 𝑛 ) )
334 323 333 eqtrd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) = - ( 𝑏 ↑ 𝑛 ) )
335 simp-4r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ ( 𝑛 / 2 ) ∈ ℕ )
336 317 282 283 sylancl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
337 335 336 mtbird ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ 2 ∥ 𝑛 )
338 oexpneg ⊢ ( ( 𝑐 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑐 ↑ 𝑛 ) = - ( 𝑐 ↑ 𝑛 ) )
339 321 317 337 338 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( - 𝑐 ↑ 𝑛 ) = - ( 𝑐 ↑ 𝑛 ) )
340 339 oveq2d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) )
341 325 317 337 286 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( - 𝑏 ↑ 𝑛 ) = - ( 𝑏 ↑ 𝑛 ) )
342 334 340 341 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) = ( - 𝑏 ↑ 𝑛 ) )
343 iffalse ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) = 𝑎 )
344 343 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) = ( 𝑎 ↑ 𝑛 ) )
345 iffalse ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) = - 𝑐 )
346 345 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
347 344 346 oveq12d ⊢ ( ¬ 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
348 347 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
349 iffalse ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) = - 𝑏 )
350 349 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
351 350 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
352 342 348 351 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
353 314 352 pm2.61dan ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
354 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) = if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) )
355 354 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) )
356 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) = if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) )
357 356 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) )
358 355 357 oveq12d ⊢ ( ¬ 0 < 𝑏 → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) )
359 358 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) )
360 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) = if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) )
361 360 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
362 361 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ↑ 𝑛 ) )
363 353 359 362 3eqtr4d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) )
364 276 363 pm2.61dan ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) → ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) )
365 iftrue ⊢ ( 0 < 𝑎 → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) = if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) )
366 365 oveq1d ⊢ ( 0 < 𝑎 → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) )
367 iftrue ⊢ ( 0 < 𝑎 → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) = if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) )
368 367 oveq1d ⊢ ( 0 < 𝑎 → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) )
369 366 368 oveq12d ⊢ ( 0 < 𝑎 → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) )
370 369 adantl ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) ↑ 𝑛 ) ) )
371 iftrue ⊢ ( 0 < 𝑎 → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) = if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) )
372 371 oveq1d ⊢ ( 0 < 𝑎 → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) )
373 372 adantl ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) ↑ 𝑛 ) )
374 364 370 373 3eqtr4d ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ 0 < 𝑎 ) → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) )
375 simp-8r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ( ℤ ∖ { 0 } ) )
376 375 10 155 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑎 ∈ ℂ )
377 96 ad8antr ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑛 ∈ ℕ )
378 simp-4r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ¬ ( 𝑛 / 2 ) ∈ ℕ )
379 377 282 283 sylancl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
380 378 379 mtbird ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ¬ 2 ∥ 𝑛 )
381 oexpneg ⊢ ( ( 𝑎 ∈ ℂ ∧ 𝑛 ∈ ℕ ∧ ¬ 2 ∥ 𝑛 ) → ( - 𝑎 ↑ 𝑛 ) = - ( 𝑎 ↑ 𝑛 ) )
382 376 377 380 381 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( - 𝑎 ↑ 𝑛 ) = - ( 𝑎 ↑ 𝑛 ) )
383 382 oveq1d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( - 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( - ( 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
384 377 nnnn0d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑛 ∈ ℕ0 )
385 376 384 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℂ )
386 385 negcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → - ( 𝑎 ↑ 𝑛 ) ∈ ℂ )
387 simp-6r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
388 387 65 246 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑐 ∈ ℂ )
389 388 384 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑐 ↑ 𝑛 ) ∈ ℂ )
390 386 389 addcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( - ( 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) ∈ ℂ )
391 129 23 168 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → 𝑏 ∈ ℂ )
392 391 384 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
393 385 negidd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + - ( 𝑎 ↑ 𝑛 ) ) = 0 )
394 393 oveq1d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( ( 𝑎 ↑ 𝑛 ) + - ( 𝑎 ↑ 𝑛 ) ) + ( 𝑐 ↑ 𝑛 ) ) = ( 0 + ( 𝑐 ↑ 𝑛 ) ) )
395 385 386 389 addassd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( ( 𝑎 ↑ 𝑛 ) + - ( 𝑎 ↑ 𝑛 ) ) + ( 𝑐 ↑ 𝑛 ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( - ( 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) ) )
396 389 addlidd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( 0 + ( 𝑐 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
397 394 395 396 3eqtr3d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( - ( 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) ) = ( 𝑐 ↑ 𝑛 ) )
398 simp-5r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
399 397 398 eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( - ( 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) ) = ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
400 385 390 392 399 addcanad ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( - ( 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( 𝑏 ↑ 𝑛 ) )
401 383 400 eqtrd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( - 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( 𝑏 ↑ 𝑛 ) )
402 iftrue ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) = - 𝑎 )
403 402 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
404 403 308 oveq12d ⊢ ( 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
405 404 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( - 𝑎 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
406 iftrue ⊢ ( 0 < 𝑐 → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) = 𝑏 )
407 406 oveq1d ⊢ ( 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
408 407 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
409 401 405 408 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
410 simp-7r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ( ℤ ∖ { 0 } ) )
411 410 23 168 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑏 ∈ ℂ )
412 simp-8l ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑛 ∈ ( ℤ≥ ‘ 3 ) )
413 412 96 289 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑛 ∈ ℕ0 )
414 411 413 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
415 414 negcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
416 simp-6r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ∈ ( ℤ ∖ { 0 } ) )
417 416 65 246 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑐 ∈ ℂ )
418 417 413 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑐 ↑ 𝑛 ) ∈ ℂ )
419 415 418 addcomd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) )
420 418 414 negsubd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) )
421 simp-5r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
422 421 oveq1d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) − ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) )
423 133 10 155 3syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑎 ∈ ℂ )
424 423 413 expcld ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℂ )
425 424 414 pncand ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) − ( 𝑏 ↑ 𝑛 ) ) = ( 𝑎 ↑ 𝑛 ) )
426 422 425 eqtr3d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑐 ↑ 𝑛 ) − ( 𝑏 ↑ 𝑛 ) ) = ( 𝑎 ↑ 𝑛 ) )
427 419 420 426 3eqtrd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( 𝑎 ↑ 𝑛 ) )
428 427 negeqd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = - ( 𝑎 ↑ 𝑛 ) )
429 414 negnegd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - - ( 𝑏 ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
430 429 eqcomd ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 𝑏 ↑ 𝑛 ) = - - ( 𝑏 ↑ 𝑛 ) )
431 430 oveq1d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑏 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) = ( - - ( 𝑏 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) )
432 412 96 syl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → 𝑛 ∈ ℕ )
433 simp-4r ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ ( 𝑛 / 2 ) ∈ ℕ )
434 432 282 283 sylancl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
435 433 434 mtbird ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ¬ 2 ∥ 𝑛 )
436 417 432 435 338 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( - 𝑐 ↑ 𝑛 ) = - ( 𝑐 ↑ 𝑛 ) )
437 436 oveq2d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) = ( ( 𝑏 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) )
438 415 418 negdid ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → - ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) = ( - - ( 𝑏 ↑ 𝑛 ) + - ( 𝑐 ↑ 𝑛 ) ) )
439 431 437 438 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) = - ( - ( 𝑏 ↑ 𝑛 ) + ( 𝑐 ↑ 𝑛 ) ) )
440 423 432 435 381 syl3anc ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( - 𝑎 ↑ 𝑛 ) = - ( 𝑎 ↑ 𝑛 ) )
441 428 439 440 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) = ( - 𝑎 ↑ 𝑛 ) )
442 iffalse ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) = 𝑏 )
443 442 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) = ( 𝑏 ↑ 𝑛 ) )
444 443 346 oveq12d ⊢ ( ¬ 0 < 𝑐 → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
445 444 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( ( 𝑏 ↑ 𝑛 ) + ( - 𝑐 ↑ 𝑛 ) ) )
446 iffalse ⊢ ( ¬ 0 < 𝑐 → if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) = - 𝑎 )
447 446 oveq1d ⊢ ( ¬ 0 < 𝑐 → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
448 447 adantl ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
449 441 445 448 3eqtr4d ⊢ ( ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) ∧ ¬ 0 < 𝑐 ) → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
450 409 449 pm2.61dan ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
451 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) = if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) )
452 451 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) )
453 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) = if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) )
454 453 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) )
455 452 454 oveq12d ⊢ ( 0 < 𝑏 → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) )
456 455 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) ↑ 𝑛 ) + ( if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ↑ 𝑛 ) ) )
457 iftrue ⊢ ( 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) = if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) )
458 457 oveq1d ⊢ ( 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
459 458 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) ↑ 𝑛 ) )
460 450 456 459 3eqtr4d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) )
461 186 negeqd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → - ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = - ( 𝑐 ↑ 𝑛 ) )
462 144 282 283 sylancl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 2 ∥ 𝑛 ↔ ( 𝑛 / 2 ) ∈ ℕ ) )
463 161 462 mtbird ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ¬ 2 ∥ 𝑛 )
464 156 144 463 381 syl3anc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( - 𝑎 ↑ 𝑛 ) = - ( 𝑎 ↑ 𝑛 ) )
465 169 144 463 286 syl3anc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( - 𝑏 ↑ 𝑛 ) = - ( 𝑏 ↑ 𝑛 ) )
466 464 465 oveq12d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( - 𝑎 ↑ 𝑛 ) + ( - 𝑏 ↑ 𝑛 ) ) = ( - ( 𝑎 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) )
467 141 11 syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑎 ≠ 0 )
468 156 467 172 expclzd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑎 ↑ 𝑛 ) ∈ ℂ )
469 169 170 172 expclzd ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( 𝑏 ↑ 𝑛 ) ∈ ℂ )
470 468 469 negdid ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → - ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( - ( 𝑎 ↑ 𝑛 ) + - ( 𝑏 ↑ 𝑛 ) ) )
471 466 470 eqtr4d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( - 𝑎 ↑ 𝑛 ) + ( - 𝑏 ↑ 𝑛 ) ) = - ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
472 139 65 246 3syl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → 𝑐 ∈ ℂ )
473 472 144 463 338 syl3anc ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( - 𝑐 ↑ 𝑛 ) = - ( 𝑐 ↑ 𝑛 ) )
474 461 471 473 3eqtr4d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( - 𝑎 ↑ 𝑛 ) + ( - 𝑏 ↑ 𝑛 ) ) = ( - 𝑐 ↑ 𝑛 ) )
475 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) = - 𝑎 )
476 475 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) = ( - 𝑎 ↑ 𝑛 ) )
477 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) = - 𝑏 )
478 477 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) = ( - 𝑏 ↑ 𝑛 ) )
479 476 478 oveq12d ⊢ ( ¬ 0 < 𝑏 → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( ( - 𝑎 ↑ 𝑛 ) + ( - 𝑏 ↑ 𝑛 ) ) )
480 479 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( ( - 𝑎 ↑ 𝑛 ) + ( - 𝑏 ↑ 𝑛 ) ) )
481 iffalse ⊢ ( ¬ 0 < 𝑏 → if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) = - 𝑐 )
482 481 oveq1d ⊢ ( ¬ 0 < 𝑏 → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
483 482 adantl ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) = ( - 𝑐 ↑ 𝑛 ) )
484 474 480 483 3eqtr4d ⊢ ( ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) ∧ ¬ 0 < 𝑏 ) → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) )
485 460 484 pm2.61dan ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) → ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) )
486 iffalse ⊢ ( ¬ 0 < 𝑎 → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) = if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) )
487 486 oveq1d ⊢ ( ¬ 0 < 𝑎 → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) )
488 iffalse ⊢ ( ¬ 0 < 𝑎 → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) = if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) )
489 488 oveq1d ⊢ ( ¬ 0 < 𝑎 → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) )
490 487 489 oveq12d ⊢ ( ¬ 0 < 𝑎 → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) )
491 490 adantl ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ↑ 𝑛 ) + ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ↑ 𝑛 ) ) )
492 iffalse ⊢ ( ¬ 0 < 𝑎 → if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) = if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) )
493 492 oveq1d ⊢ ( ¬ 0 < 𝑎 → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) )
494 493 adantl ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) → ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ↑ 𝑛 ) )
495 485 491 494 3eqtr4d ⊢ ( ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) ∧ ¬ 0 < 𝑎 ) → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) )
496 374 495 pm2.61dan ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) )
497 iffalse ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) = if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) )
498 497 oveq1d ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) )
499 iffalse ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) = if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) )
500 499 oveq1d ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) )
501 498 500 oveq12d ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) )
502 501 adantl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ↑ 𝑛 ) + ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ↑ 𝑛 ) ) )
503 iffalse ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) = if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) )
504 503 oveq1d ⊢ ( ¬ ( 𝑛 / 2 ) ∈ ℕ → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) )
505 504 adantl ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) → ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) = ( if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ↑ 𝑛 ) )
506 496 502 505 3eqtr4d ⊢ ( ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) ∧ ¬ ( 𝑛 / 2 ) ∈ ℕ ) → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) )
507 266 506 pm2.61dan ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ( ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑎 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑎 , if ( 0 < 𝑐 , - 𝑏 , 𝑎 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , - 𝑎 , 𝑏 ) , - 𝑎 ) ) ) ↑ 𝑛 ) + ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑏 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑐 , - 𝑐 ) , - 𝑏 ) ) ) ↑ 𝑛 ) ) = ( if ( ( 𝑛 / 2 ) ∈ ℕ , ( abs ‘ 𝑐 ) , if ( 0 < 𝑎 , if ( 0 < 𝑏 , 𝑐 , if ( 0 < 𝑐 , 𝑎 , - 𝑏 ) ) , if ( 0 < 𝑏 , if ( 0 < 𝑐 , 𝑏 , - 𝑎 ) , - 𝑐 ) ) ) ↑ 𝑛 ) )
508 3 6 8 48 85 197 507 3rspcedvdw ⊢ ( ( ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑐 ∈ ( ℤ ∖ { 0 } ) ) ∧ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) )
509 508 rexlimdva2 ⊢ ( ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) ∧ 𝑏 ∈ ( ℤ ∖ { 0 } ) ) → ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ) )
510 509 rexlimdva ⊢ ( ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∧ 𝑎 ∈ ( ℤ ∖ { 0 } ) ) → ( ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ) )
511 510 rexlimdva ⊢ ( 𝑛 ∈ ( ℤ≥ ‘ 3 ) → ( ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) → ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ) )
512 511 reximia ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) → ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) )
513 nne ⊢ ( ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) )
514 513 bicomi ⊢ ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
515 514 rexbii ⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
516 rexnal ⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ¬ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
517 515 516 bitri ⊢ ( ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
518 517 rexbii ⊢ ( ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
519 rexnal ⊢ ( ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ¬ ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
520 518 519 bitri ⊢ ( ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
521 520 rexbii ⊢ ( ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ¬ ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
522 rexnal ⊢ ( ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ¬ ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
523 521 522 bitri ⊢ ( ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
524 523 rexbii ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ¬ ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
525 rexnal ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ¬ ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
526 524 525 bitri ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∃ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∃ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( 𝑐 ↑ 𝑛 ) ↔ ¬ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
527 nne ⊢ ( ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) )
528 527 bicomi ⊢ ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
529 528 rexbii ⊢ ( ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑧 ∈ ℕ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
530 rexnal ⊢ ( ∃ 𝑧 ∈ ℕ ¬ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
531 529 530 bitri ⊢ ( ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
532 531 rexbii ⊢ ( ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑦 ∈ ℕ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
533 rexnal ⊢ ( ∃ 𝑦 ∈ ℕ ¬ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
534 532 533 bitri ⊢ ( ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
535 534 rexbii ⊢ ( ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑥 ∈ ℕ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
536 rexnal ⊢ ( ∃ 𝑥 ∈ ℕ ¬ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
537 535 536 bitri ⊢ ( ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
538 537 rexbii ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ¬ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
539 rexnal ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ¬ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
540 538 539 bitri ⊢ ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∃ 𝑥 ∈ ℕ ∃ 𝑦 ∈ ℕ ∃ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) = ( 𝑧 ↑ 𝑛 ) ↔ ¬ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
541 512 526 540 3imtr3i ⊢ ( ¬ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) → ¬ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
542 541 con4i ⊢ ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
543 dfn2 ⊢ ℕ = ( ℕ0 ∖ { 0 } )
544 nn0ssz ⊢ ℕ0 ⊆ ℤ
545 ssdif ⊢ ( ℕ0 ⊆ ℤ → ( ℕ0 ∖ { 0 } ) ⊆ ( ℤ ∖ { 0 } ) )
546 544 545 ax-mp ⊢ ( ℕ0 ∖ { 0 } ) ⊆ ( ℤ ∖ { 0 } )
547 543 546 eqsstri ⊢ ℕ ⊆ ( ℤ ∖ { 0 } )
548 ssel ⊢ ( ℕ ⊆ ( ℤ ∖ { 0 } ) → ( 𝑎 ∈ ℕ → 𝑎 ∈ ( ℤ ∖ { 0 } ) ) )
549 ss2ralv ⊢ ( ℕ ⊆ ( ℤ ∖ { 0 } ) → ( ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) → ∀ 𝑏 ∈ ℕ ∀ 𝑐 ∈ ℕ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
550 548 549 imim12d ⊢ ( ℕ ⊆ ( ℤ ∖ { 0 } ) → ( ( 𝑎 ∈ ( ℤ ∖ { 0 } ) → ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) → ( 𝑎 ∈ ℕ → ∀ 𝑏 ∈ ℕ ∀ 𝑐 ∈ ℕ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) ) )
551 550 ralimdv2 ⊢ ( ℕ ⊆ ( ℤ ∖ { 0 } ) → ( ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) → ∀ 𝑎 ∈ ℕ ∀ 𝑏 ∈ ℕ ∀ 𝑐 ∈ ℕ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
552 547 551 ax-mp ⊢ ( ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) → ∀ 𝑎 ∈ ℕ ∀ 𝑏 ∈ ℕ ∀ 𝑐 ∈ ℕ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )
553 oveq1 ⊢ ( 𝑎 = 𝑥 → ( 𝑎 ↑ 𝑛 ) = ( 𝑥 ↑ 𝑛 ) )
554 553 oveq1d ⊢ ( 𝑎 = 𝑥 → ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) )
555 554 neeq1d ⊢ ( 𝑎 = 𝑥 → ( ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
556 oveq1 ⊢ ( 𝑏 = 𝑦 → ( 𝑏 ↑ 𝑛 ) = ( 𝑦 ↑ 𝑛 ) )
557 556 oveq2d ⊢ ( 𝑏 = 𝑦 → ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) = ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) )
558 557 neeq1d ⊢ ( 𝑏 = 𝑦 → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ) )
559 oveq1 ⊢ ( 𝑐 = 𝑧 → ( 𝑐 ↑ 𝑛 ) = ( 𝑧 ↑ 𝑛 ) )
560 559 neeq2d ⊢ ( 𝑐 = 𝑧 → ( ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ) )
561 555 558 560 cbvral3vw ⊢ ( ∀ 𝑎 ∈ ℕ ∀ 𝑏 ∈ ℕ ∀ 𝑐 ∈ ℕ ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) ↔ ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
562 552 561 sylib ⊢ ( ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) → ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
563 562 ralimi ⊢ ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) → ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) )
564 542 563 impbii ⊢ ( ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑥 ∈ ℕ ∀ 𝑦 ∈ ℕ ∀ 𝑧 ∈ ℕ ( ( 𝑥 ↑ 𝑛 ) + ( 𝑦 ↑ 𝑛 ) ) ≠ ( 𝑧 ↑ 𝑛 ) ↔ ∀ 𝑛 ∈ ( ℤ≥ ‘ 3 ) ∀ 𝑎 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑏 ∈ ( ℤ ∖ { 0 } ) ∀ 𝑐 ∈ ( ℤ ∖ { 0 } ) ( ( 𝑎 ↑ 𝑛 ) + ( 𝑏 ↑ 𝑛 ) ) ≠ ( 𝑐 ↑ 𝑛 ) )