Description: Complex exponentiation of a product. Proposition 10-4.2(c) of Gleason p. 135. (Contributed by Mario Carneiro, 30-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | recxpcld.1 | |- ( ph -> A e. RR ) |
|
| recxpcld.2 | |- ( ph -> 0 <_ A ) |
||
| recxpcld.3 | |- ( ph -> B e. RR ) |
||
| mulcxpd.4 | |- ( ph -> 0 <_ B ) |
||
| mulcxpd.5 | |- ( ph -> C e. CC ) |
||
| Assertion | mulcxpd | |- ( ph -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | recxpcld.1 | |- ( ph -> A e. RR ) |
|
| 2 | recxpcld.2 | |- ( ph -> 0 <_ A ) |
|
| 3 | recxpcld.3 | |- ( ph -> B e. RR ) |
|
| 4 | mulcxpd.4 | |- ( ph -> 0 <_ B ) |
|
| 5 | mulcxpd.5 | |- ( ph -> C e. CC ) |
|
| 6 | mulcxp | |- ( ( ( A e. RR /\ 0 <_ A ) /\ ( B e. RR /\ 0 <_ B ) /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) ) |
|
| 7 | 1 2 3 4 5 6 | syl221anc | |- ( ph -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) ) |