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Metamath Proof Explorer

Theorem mulgnn

Description: Group multiple (exponentiation) operation at a positive integer. (Contributed by Mario Carneiro, 11-Dec-2014)

Ref Expression
Hypotheses mulgnn.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
mulgnn.p ⊢ + = ( +g ‘ 𝐺 )
mulgnn.t ⊢ · = ( .g ‘ 𝐺 )
mulgnn.s ⊢ 𝑆 = seq 1 ( + , ( ℕ × { 𝑋 } ) )
Assertion mulgnn ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = ( 𝑆 ‘ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 mulgnn.b ⊢ 𝐵 = ( Base ‘ 𝐺 )
2 mulgnn.p ⊢ + = ( +g ‘ 𝐺 )
3 mulgnn.t ⊢ · = ( .g ‘ 𝐺 )
4 mulgnn.s ⊢ 𝑆 = seq 1 ( + , ( ℕ × { 𝑋 } ) )
5 nnz ⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℤ )
6 eqid ⊢ ( 0g ‘ 𝐺 ) = ( 0g ‘ 𝐺 )
7 eqid ⊢ ( invg ‘ 𝐺 ) = ( invg ‘ 𝐺 )
8 1 2 6 7 3 4 mulgval ⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) )
9 5 8 sylan ⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) )
10 nnne0 ⊢ ( 𝑁 ∈ ℕ → 𝑁 ≠ 0 )
11 10 neneqd ⊢ ( 𝑁 ∈ ℕ → ¬ 𝑁 = 0 )
12 11 iffalsed ⊢ ( 𝑁 ∈ ℕ → if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) = if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) )
13 nngt0 ⊢ ( 𝑁 ∈ ℕ → 0 < 𝑁 )
14 13 iftrued ⊢ ( 𝑁 ∈ ℕ → if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) = ( 𝑆 ‘ 𝑁 ) )
15 12 14 eqtrd ⊢ ( 𝑁 ∈ ℕ → if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) = ( 𝑆 ‘ 𝑁 ) )
16 15 adantr ⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → if ( 𝑁 = 0 , ( 0g ‘ 𝐺 ) , if ( 0 < 𝑁 , ( 𝑆 ‘ 𝑁 ) , ( ( invg ‘ 𝐺 ) ‘ ( 𝑆 ‘ - 𝑁 ) ) ) ) = ( 𝑆 ‘ 𝑁 ) )
17 9 16 eqtrd ⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑋 ∈ 𝐵 ) → ( 𝑁 · 𝑋 ) = ( 𝑆 ‘ 𝑁 ) )