Description: Positive integer exponentiation of a product. Proposition 10-4.2(c) of Gleason p. 135, restricted to nonnegative integer exponents. (Contributed by Mario Carneiro, 28-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | expcld.1 | |- ( ph -> A e. CC ) |
|
| mulexpd.2 | |- ( ph -> B e. CC ) |
||
| mulexpd.3 | |- ( ph -> N e. NN0 ) |
||
| Assertion | mulexpd | |- ( ph -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | expcld.1 | |- ( ph -> A e. CC ) |
|
| 2 | mulexpd.2 | |- ( ph -> B e. CC ) |
|
| 3 | mulexpd.3 | |- ( ph -> N e. NN0 ) |
|
| 4 | mulexp | |- ( ( A e. CC /\ B e. CC /\ N e. NN0 ) -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) |
|
| 5 | 1 2 3 4 | syl3anc | |- ( ph -> ( ( A x. B ) ^ N ) = ( ( A ^ N ) x. ( B ^ N ) ) ) |