Step |
Hyp |
Ref |
Expression |
1 |
|
1z |
|- 1 e. ZZ |
2 |
|
expaddz |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ ( N e. ZZ /\ 1 e. ZZ ) ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. ( A ^ 1 ) ) ) |
3 |
1 2
|
mpanr2 |
|- ( ( ( A e. CC /\ A =/= 0 ) /\ N e. ZZ ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. ( A ^ 1 ) ) ) |
4 |
3
|
3impa |
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. ( A ^ 1 ) ) ) |
5 |
|
exp1 |
|- ( A e. CC -> ( A ^ 1 ) = A ) |
6 |
5
|
3ad2ant1 |
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ 1 ) = A ) |
7 |
6
|
oveq2d |
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( ( A ^ N ) x. ( A ^ 1 ) ) = ( ( A ^ N ) x. A ) ) |
8 |
4 7
|
eqtrd |
|- ( ( A e. CC /\ A =/= 0 /\ N e. ZZ ) -> ( A ^ ( N + 1 ) ) = ( ( A ^ N ) x. A ) ) |