Description: Product of exponents law for complex exponentiation. Variation on cxpmul with more general conditions on A and B when C is an integer. (Contributed by Mario Carneiro, 30-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | cxp0d.1 | |- ( ph -> A e. CC ) |
|
| cxpcld.2 | |- ( ph -> B e. CC ) |
||
| cxpmul2d.4 | |- ( ph -> C e. NN0 ) |
||
| Assertion | cxpmul2d | |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | cxp0d.1 | |- ( ph -> A e. CC ) |
|
| 2 | cxpcld.2 | |- ( ph -> B e. CC ) |
|
| 3 | cxpmul2d.4 | |- ( ph -> C e. NN0 ) |
|
| 4 | cxpmul2 | |- ( ( A e. CC /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) |
|
| 5 | 1 2 3 4 | syl3anc | |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) |