Step |
Hyp |
Ref |
Expression |
1 |
|
nncn |
|- ( N e. NN -> N e. CC ) |
2 |
1
|
adantl |
|- ( ( A e. CC /\ N e. NN ) -> N e. CC ) |
3 |
|
nnne0 |
|- ( N e. NN -> N =/= 0 ) |
4 |
3
|
adantl |
|- ( ( A e. CC /\ N e. NN ) -> N =/= 0 ) |
5 |
2 4
|
recid2d |
|- ( ( A e. CC /\ N e. NN ) -> ( ( 1 / N ) x. N ) = 1 ) |
6 |
5
|
oveq2d |
|- ( ( A e. CC /\ N e. NN ) -> ( A ^c ( ( 1 / N ) x. N ) ) = ( A ^c 1 ) ) |
7 |
|
simpl |
|- ( ( A e. CC /\ N e. NN ) -> A e. CC ) |
8 |
|
nnrecre |
|- ( N e. NN -> ( 1 / N ) e. RR ) |
9 |
8
|
adantl |
|- ( ( A e. CC /\ N e. NN ) -> ( 1 / N ) e. RR ) |
10 |
9
|
recnd |
|- ( ( A e. CC /\ N e. NN ) -> ( 1 / N ) e. CC ) |
11 |
|
nnnn0 |
|- ( N e. NN -> N e. NN0 ) |
12 |
11
|
adantl |
|- ( ( A e. CC /\ N e. NN ) -> N e. NN0 ) |
13 |
|
cxpmul2 |
|- ( ( A e. CC /\ ( 1 / N ) e. CC /\ N e. NN0 ) -> ( A ^c ( ( 1 / N ) x. N ) ) = ( ( A ^c ( 1 / N ) ) ^ N ) ) |
14 |
7 10 12 13
|
syl3anc |
|- ( ( A e. CC /\ N e. NN ) -> ( A ^c ( ( 1 / N ) x. N ) ) = ( ( A ^c ( 1 / N ) ) ^ N ) ) |
15 |
|
cxp1 |
|- ( A e. CC -> ( A ^c 1 ) = A ) |
16 |
15
|
adantr |
|- ( ( A e. CC /\ N e. NN ) -> ( A ^c 1 ) = A ) |
17 |
6 14 16
|
3eqtr3d |
|- ( ( A e. CC /\ N e. NN ) -> ( ( A ^c ( 1 / N ) ) ^ N ) = A ) |