Theorem cxpmul2zd

Description: Generalize cxpmul2 to negative integers. (Contributed by Mario Carneiro, 30-May-2016)

	Ref	Expression
Hypotheses	cxp0d.1	|- ( ph -> A e. CC )
	cxpefd.2	|- ( ph -> A =/= 0 )
	cxpefd.3	|- ( ph -> B e. CC )
	cxpmul2zd.4	|- ( ph -> C e. ZZ )
Assertion	cxpmul2zd	|- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )

Proof

Step	Hyp	Ref	Expression
1		cxp0d.1	|- ( ph -> A e. CC )
2		cxpefd.2	|- ( ph -> A =/= 0 )
3		cxpefd.3	|- ( ph -> B e. CC )
4		cxpmul2zd.4	|- ( ph -> C e. ZZ )
5		cxpmul2z	|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( B e. CC /\ C e. ZZ ) ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )
6	1 2 3 4 5	syl22anc	|- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )