Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( j = 0 -> ( A ^ j ) = ( A ^ 0 ) ) |
2 |
1
|
fveq2d |
|- ( j = 0 -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ 0 ) ) ) |
3 |
|
oveq2 |
|- ( j = 0 -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ 0 ) ) |
4 |
2 3
|
eqeq12d |
|- ( j = 0 -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) ) ) |
5 |
|
oveq2 |
|- ( j = k -> ( A ^ j ) = ( A ^ k ) ) |
6 |
5
|
fveq2d |
|- ( j = k -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ k ) ) ) |
7 |
|
oveq2 |
|- ( j = k -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ k ) ) |
8 |
6 7
|
eqeq12d |
|- ( j = k -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) ) |
9 |
|
oveq2 |
|- ( j = ( k + 1 ) -> ( A ^ j ) = ( A ^ ( k + 1 ) ) ) |
10 |
9
|
fveq2d |
|- ( j = ( k + 1 ) -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ ( k + 1 ) ) ) ) |
11 |
|
oveq2 |
|- ( j = ( k + 1 ) -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) |
12 |
10 11
|
eqeq12d |
|- ( j = ( k + 1 ) -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) ) |
13 |
|
oveq2 |
|- ( j = N -> ( A ^ j ) = ( A ^ N ) ) |
14 |
13
|
fveq2d |
|- ( j = N -> ( abs ` ( A ^ j ) ) = ( abs ` ( A ^ N ) ) ) |
15 |
|
oveq2 |
|- ( j = N -> ( ( abs ` A ) ^ j ) = ( ( abs ` A ) ^ N ) ) |
16 |
14 15
|
eqeq12d |
|- ( j = N -> ( ( abs ` ( A ^ j ) ) = ( ( abs ` A ) ^ j ) <-> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) ) |
17 |
|
abs1 |
|- ( abs ` 1 ) = 1 |
18 |
|
exp0 |
|- ( A e. CC -> ( A ^ 0 ) = 1 ) |
19 |
18
|
fveq2d |
|- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( abs ` 1 ) ) |
20 |
|
abscl |
|- ( A e. CC -> ( abs ` A ) e. RR ) |
21 |
20
|
recnd |
|- ( A e. CC -> ( abs ` A ) e. CC ) |
22 |
21
|
exp0d |
|- ( A e. CC -> ( ( abs ` A ) ^ 0 ) = 1 ) |
23 |
17 19 22
|
3eqtr4a |
|- ( A e. CC -> ( abs ` ( A ^ 0 ) ) = ( ( abs ` A ) ^ 0 ) ) |
24 |
|
oveq1 |
|- ( ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) -> ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) ) |
25 |
24
|
adantl |
|- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) ) |
26 |
|
expp1 |
|- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ ( k + 1 ) ) = ( ( A ^ k ) x. A ) ) |
27 |
26
|
fveq2d |
|- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( abs ` ( ( A ^ k ) x. A ) ) ) |
28 |
|
expcl |
|- ( ( A e. CC /\ k e. NN0 ) -> ( A ^ k ) e. CC ) |
29 |
|
simpl |
|- ( ( A e. CC /\ k e. NN0 ) -> A e. CC ) |
30 |
|
absmul |
|- ( ( ( A ^ k ) e. CC /\ A e. CC ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) ) |
31 |
28 29 30
|
syl2anc |
|- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( ( A ^ k ) x. A ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) ) |
32 |
27 31
|
eqtrd |
|- ( ( A e. CC /\ k e. NN0 ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) ) |
33 |
32
|
adantr |
|- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` ( A ^ k ) ) x. ( abs ` A ) ) ) |
34 |
|
expp1 |
|- ( ( ( abs ` A ) e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) ) |
35 |
21 34
|
sylan |
|- ( ( A e. CC /\ k e. NN0 ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) ) |
36 |
35
|
adantr |
|- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( ( abs ` A ) ^ ( k + 1 ) ) = ( ( ( abs ` A ) ^ k ) x. ( abs ` A ) ) ) |
37 |
25 33 36
|
3eqtr4d |
|- ( ( ( A e. CC /\ k e. NN0 ) /\ ( abs ` ( A ^ k ) ) = ( ( abs ` A ) ^ k ) ) -> ( abs ` ( A ^ ( k + 1 ) ) ) = ( ( abs ` A ) ^ ( k + 1 ) ) ) |
38 |
4 8 12 16 23 37
|
nn0indd |
|- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) |