Description: Value of a complex number raised to a nonnegative integer power plus one. Part of Definition 10-4.1 of Gleason p. 134. (Contributed by Mario Carneiro, 28-May-2016)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | expcld.1 | |- ( ph -> A e. CC ) |
|
| expcld.2 | |- ( ph -> N e. NN0 ) |
||
| Assertion | expp1d | |- ( ph -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | expcld.1 | |- ( ph -> A e. CC ) |
|
| 2 | expcld.2 | |- ( ph -> N e. NN0 ) |
|
| 3 | expp1 | |- ( ( A e. CC /\ N e. NN0 ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) ) |
|
| 4 | 1 2 3 | syl2anc | |- ( ph -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) ) |