This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem fltmul

Description: A counterexample to FLT stays valid when scaled. The hypotheses are more general than they need to be for convenience. (There does not seem to be a standard term for Fermat or Pythagorean triples extended to any N e. NN0 , so the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024)

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

Proof

Step Hyp Ref Expression
1 fltmul.s ⊢ ( 𝜑 → 𝑆 ∈ ℂ )
2 fltmul.a ⊢ ( 𝜑 → 𝐴 ∈ ℂ )
3 fltmul.b ⊢ ( 𝜑 → 𝐵 ∈ ℂ )
4 fltmul.c ⊢ ( 𝜑 → 𝐶 ∈ ℂ )
5 fltmul.n ⊢ ( 𝜑 → 𝑁 ∈ ℕ0 )
6 fltmul.1 ⊢ ( 𝜑 → ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) = ( 𝐶 ↑ 𝑁 ) )
7 1 5 expcld ⊢ ( 𝜑 → ( 𝑆 ↑ 𝑁 ) ∈ ℂ )
8 2 5 expcld ⊢ ( 𝜑 → ( 𝐴 ↑ 𝑁 ) ∈ ℂ )
9 3 5 expcld ⊢ ( 𝜑 → ( 𝐵 ↑ 𝑁 ) ∈ ℂ )
10 7 8 9 adddid ⊢ ( 𝜑 → ( ( 𝑆 ↑ 𝑁 ) · ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) ) = ( ( ( 𝑆 ↑ 𝑁 ) · ( 𝐴 ↑ 𝑁 ) ) + ( ( 𝑆 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) )
11 6 oveq2d ⊢ ( 𝜑 → ( ( 𝑆 ↑ 𝑁 ) · ( ( 𝐴 ↑ 𝑁 ) + ( 𝐵 ↑ 𝑁 ) ) ) = ( ( 𝑆 ↑ 𝑁 ) · ( 𝐶 ↑ 𝑁 ) ) )
12 10 11 eqtr3d ⊢ ( 𝜑 → ( ( ( 𝑆 ↑ 𝑁 ) · ( 𝐴 ↑ 𝑁 ) ) + ( ( 𝑆 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) = ( ( 𝑆 ↑ 𝑁 ) · ( 𝐶 ↑ 𝑁 ) ) )
13 1 2 5 mulexpd ⊢ ( 𝜑 → ( ( 𝑆 · 𝐴 ) ↑ 𝑁 ) = ( ( 𝑆 ↑ 𝑁 ) · ( 𝐴 ↑ 𝑁 ) ) )
14 1 3 5 mulexpd ⊢ ( 𝜑 → ( ( 𝑆 · 𝐵 ) ↑ 𝑁 ) = ( ( 𝑆 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) )
15 13 14 oveq12d ⊢ ( 𝜑 → ( ( ( 𝑆 · 𝐴 ) ↑ 𝑁 ) + ( ( 𝑆 · 𝐵 ) ↑ 𝑁 ) ) = ( ( ( 𝑆 ↑ 𝑁 ) · ( 𝐴 ↑ 𝑁 ) ) + ( ( 𝑆 ↑ 𝑁 ) · ( 𝐵 ↑ 𝑁 ) ) ) )
16 1 4 5 mulexpd ⊢ ( 𝜑 → ( ( 𝑆 · 𝐶 ) ↑ 𝑁 ) = ( ( 𝑆 ↑ 𝑁 ) · ( 𝐶 ↑ 𝑁 ) ) )
17 12 15 16 3eqtr4d ⊢ ( 𝜑 → ( ( ( 𝑆 · 𝐴 ) ↑ 𝑁 ) + ( ( 𝑆 · 𝐵 ) ↑ 𝑁 ) ) = ( ( 𝑆 · 𝐶 ) ↑ 𝑁 ) )