Description: Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | rpcxpcld.1 | |- ( ph -> A e. RR+ ) |
|
| rpcxpcld.2 | |- ( ph -> B e. RR ) |
||
| cxpmuld.4 | |- ( ph -> C e. CC ) |
||
| Assertion | cxpmuld | |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rpcxpcld.1 | |- ( ph -> A e. RR+ ) |
|
| 2 | rpcxpcld.2 | |- ( ph -> B e. RR ) |
|
| 3 | cxpmuld.4 | |- ( ph -> C e. CC ) |
|
| 4 | cxpmul | |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) ) |
|
| 5 | 1 2 3 4 | syl3anc | |- ( ph -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) ) |