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Metamath Proof Explorer

Theorem fltdiv

Description: A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (Contributed by SN, 20-Aug-2024)

Ref Expression
Hypotheses fltdiv.s ⊢ ( 𝜑 → 𝑆 ∈ ℂ )
fltdiv.0 ⊢ ( 𝜑 → 𝑆 ≠ 0 )
fltdiv.a ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
fltdiv.b ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
fltdiv.c ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
fltdiv.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
fltdiv.1 ⊢ ( 𝜑 → ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) = ( 𝐶 ↑ 𝑁 ) )
Assertion fltdiv ( 𝜑 → ( ( ( 𝐴 / 𝑆 ) ↑ 𝑁 ) + ( ( 𝐵 / 𝑆 ) ↑ 𝑁 ) ) = ( ( 𝐶 / 𝑆 ) ↑ 𝑁 ) )

Proof

Step Hyp Ref Expression
1 fltdiv.s ⊢ ( 𝜑 → 𝑆 ∈ ℂ )
2 fltdiv.0 ⊢ ( 𝜑 → 𝑆 ≠ 0 )
3 fltdiv.a ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
4 fltdiv.b ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
5 fltdiv.c ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
6 fltdiv.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
7 fltdiv.1 ⊢ ( 𝜑 → ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) = ( 𝐶 ↑ 𝑁 ) )
8 3 6 expcld ⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )
9 4 6 expcld ⊢ ( 𝜑 → ( 𝐵 ↑ 𝑁 ) ∈ ℂ )
10 1 6 expcld ⊢ ( 𝜑 → ( 𝑆 ↑ 𝑁 ) ∈ ℂ )
11 6 nn0zd ⊢ ( 𝜑 → 𝑁 ∈ ℤ )
12 1 2 11 expne0d ⊢ ( 𝜑 → ( 𝑆 ↑ 𝑁 ) ≠ 0 )
13 8 9 10 12 divdird ⊢ ( 𝜑 → ( ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) / ( 𝑆 ↑ 𝑁 ) ) = ( ( ( 𝐴 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) + ( ( 𝐵 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) ) )
14 7 oveq1d ⊢ ( 𝜑 → ( ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) / ( 𝑆 ↑ 𝑁 ) ) = ( ( 𝐶 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) )
15 13 14 eqtr3d ⊢ ( 𝜑 → ( ( ( 𝐴 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) + ( ( 𝐵 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) ) = ( ( 𝐶 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) )
16 3 1 2 6 expdivd ⊢ ( 𝜑 → ( ( 𝐴 / 𝑆 ) ↑ 𝑁 ) = ( ( 𝐴 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) )
17 4 1 2 6 expdivd ⊢ ( 𝜑 → ( ( 𝐵 / 𝑆 ) ↑ 𝑁 ) = ( ( 𝐵 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) )
18 16 17 oveq12d ⊢ ( 𝜑 → ( ( ( 𝐴 / 𝑆 ) ↑ 𝑁 ) + ( ( 𝐵 / 𝑆 ) ↑ 𝑁 ) ) = ( ( ( 𝐴 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) + ( ( 𝐵 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) ) )
19 5 1 2 6 expdivd ⊢ ( 𝜑 → ( ( 𝐶 / 𝑆 ) ↑ 𝑁 ) = ( ( 𝐶 ↑ 𝑁 ) / ( 𝑆 ↑ 𝑁 ) ) )
20 15 18 19 3eqtr4d ⊢ ( 𝜑 → ( ( ( 𝐴 / 𝑆 ) ↑ 𝑁 ) + ( ( 𝐵 / 𝑆 ) ↑ 𝑁 ) ) = ( ( 𝐶 / 𝑆 ) ↑ 𝑁 ) )