Theorem sqrecd

Description: Square of reciprocal is reciprocal of square. (Contributed by Mario Carneiro, 28-May-2016)

	Ref	Expression
Hypotheses	expcld.1	|- ( ph -> A e. CC )
	sqrecd.1	|- ( ph -> A =/= 0 )
Assertion	sqrecd	|- ( ph -> ( ( 1 / A ) ^ 2 ) = ( 1 / ( A ^ 2 ) ) )

Proof

Step	Hyp	Ref	Expression
1		expcld.1	|- ( ph -> A e. CC )
2		sqrecd.1	|- ( ph -> A =/= 0 )
3		2z	|- 2 e. ZZ
4	3	a1i	|- ( ph -> 2 e. ZZ )
5		exprec	|- ( ( A e. CC /\ A =/= 0 /\ 2 e. ZZ ) -> ( ( 1 / A ) ^ 2 ) = ( 1 / ( A ^ 2 ) ) )
6	1 2 4 5	syl3anc	|- ( ph -> ( ( 1 / A ) ^ 2 ) = ( 1 / ( A ^ 2 ) ) )