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 , hence the label is more about the context in which this theorem is used). (Contributed by SN, 20-Aug-2024)

	Ref	Expression
Hypotheses	fltmul.s	|- ( ph -> S e. CC )
	fltmul.a	|- ( ph -> A e. CC )
	fltmul.b	|- ( ph -> B e. CC )
	fltmul.c	|- ( ph -> C e. CC )
	fltmul.n	|- ( ph -> N e. NN0 )
	fltmul.1	|- ( ph -> ( ( A ^ N ) + ( B ^ N ) ) = ( C ^ N ) )
Assertion	fltmul	|- ( ph -> ( ( ( S x. A ) ^ N ) + ( ( S x. B ) ^ N ) ) = ( ( S x. C ) ^ N ) )

Proof

Step	Hyp	Ref	Expression
1		fltmul.s	|- ( ph -> S e. CC )
2		fltmul.a	|- ( ph -> A e. CC )
3		fltmul.b	|- ( ph -> B e. CC )
4		fltmul.c	|- ( ph -> C e. CC )
5		fltmul.n	|- ( ph -> N e. NN0 )
6		fltmul.1	|- ( ph -> ( ( A ^ N ) + ( B ^ N ) ) = ( C ^ N ) )
7	1 5	expcld	|- ( ph -> ( S ^ N ) e. CC )
8	2 5	expcld	|- ( ph -> ( A ^ N ) e. CC )
9	3 5	expcld	|- ( ph -> ( B ^ N ) e. CC )
10	7 8 9	adddid	|- ( ph -> ( ( S ^ N ) x. ( ( A ^ N ) + ( B ^ N ) ) ) = ( ( ( S ^ N ) x. ( A ^ N ) ) + ( ( S ^ N ) x. ( B ^ N ) ) ) )
11	6	oveq2d	|- ( ph -> ( ( S ^ N ) x. ( ( A ^ N ) + ( B ^ N ) ) ) = ( ( S ^ N ) x. ( C ^ N ) ) )
12	10 11	eqtr3d	|- ( ph -> ( ( ( S ^ N ) x. ( A ^ N ) ) + ( ( S ^ N ) x. ( B ^ N ) ) ) = ( ( S ^ N ) x. ( C ^ N ) ) )
13	1 2 5	mulexpd	|- ( ph -> ( ( S x. A ) ^ N ) = ( ( S ^ N ) x. ( A ^ N ) ) )
14	1 3 5	mulexpd	|- ( ph -> ( ( S x. B ) ^ N ) = ( ( S ^ N ) x. ( B ^ N ) ) )
15	13 14	oveq12d	|- ( ph -> ( ( ( S x. A ) ^ N ) + ( ( S x. B ) ^ N ) ) = ( ( ( S ^ N ) x. ( A ^ N ) ) + ( ( S ^ N ) x. ( B ^ N ) ) ) )
16	1 4 5	mulexpd	|- ( ph -> ( ( S x. C ) ^ N ) = ( ( S ^ N ) x. ( C ^ N ) ) )
17	12 15 16	3eqtr4d	|- ( ph -> ( ( ( S x. A ) ^ N ) + ( ( S x. B ) ^ N ) ) = ( ( S x. C ) ^ N ) )