Description: Product of exponents law for positive integer exponentiation. Proposition 10-4.2(b) 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 ) |
|
| expcld.2 | |- ( ph -> N e. NN0 ) |
||
| expaddd.2 | |- ( ph -> M e. NN0 ) |
||
| Assertion | expmuld | |- ( ph -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | expcld.1 | |- ( ph -> A e. CC ) |
|
| 2 | expcld.2 | |- ( ph -> N e. NN0 ) |
|
| 3 | expaddd.2 | |- ( ph -> M e. NN0 ) |
|
| 4 | expmul | |- ( ( A e. CC /\ M e. NN0 /\ N e. NN0 ) -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |
|
| 5 | 1 3 2 4 | syl3anc | |- ( ph -> ( A ^ ( M x. N ) ) = ( ( A ^ M ) ^ N ) ) |