Metamath Proof Explorer


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 ) )