Step |
Hyp |
Ref |
Expression |
1 |
|
oveq2 |
|- ( x = 0 -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ 0 ) ) |
2 |
|
oveq1 |
|- ( x = 0 -> ( x x. A ) = ( 0 x. A ) ) |
3 |
2
|
fveq2d |
|- ( x = 0 -> ( cos ` ( x x. A ) ) = ( cos ` ( 0 x. A ) ) ) |
4 |
2
|
fveq2d |
|- ( x = 0 -> ( sin ` ( x x. A ) ) = ( sin ` ( 0 x. A ) ) ) |
5 |
4
|
oveq2d |
|- ( x = 0 -> ( _i x. ( sin ` ( x x. A ) ) ) = ( _i x. ( sin ` ( 0 x. A ) ) ) ) |
6 |
3 5
|
oveq12d |
|- ( x = 0 -> ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) = ( ( cos ` ( 0 x. A ) ) + ( _i x. ( sin ` ( 0 x. A ) ) ) ) ) |
7 |
1 6
|
eqeq12d |
|- ( x = 0 -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) <-> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ 0 ) = ( ( cos ` ( 0 x. A ) ) + ( _i x. ( sin ` ( 0 x. A ) ) ) ) ) ) |
8 |
7
|
imbi2d |
|- ( x = 0 -> ( ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) ) <-> ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ 0 ) = ( ( cos ` ( 0 x. A ) ) + ( _i x. ( sin ` ( 0 x. A ) ) ) ) ) ) ) |
9 |
|
oveq2 |
|- ( x = k -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) ) |
10 |
|
oveq1 |
|- ( x = k -> ( x x. A ) = ( k x. A ) ) |
11 |
10
|
fveq2d |
|- ( x = k -> ( cos ` ( x x. A ) ) = ( cos ` ( k x. A ) ) ) |
12 |
10
|
fveq2d |
|- ( x = k -> ( sin ` ( x x. A ) ) = ( sin ` ( k x. A ) ) ) |
13 |
12
|
oveq2d |
|- ( x = k -> ( _i x. ( sin ` ( x x. A ) ) ) = ( _i x. ( sin ` ( k x. A ) ) ) ) |
14 |
11 13
|
oveq12d |
|- ( x = k -> ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) |
15 |
9 14
|
eqeq12d |
|- ( x = k -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) <-> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) |
16 |
15
|
imbi2d |
|- ( x = k -> ( ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) ) <-> ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
17 |
|
oveq2 |
|- ( x = ( k + 1 ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) ) |
18 |
|
oveq1 |
|- ( x = ( k + 1 ) -> ( x x. A ) = ( ( k + 1 ) x. A ) ) |
19 |
18
|
fveq2d |
|- ( x = ( k + 1 ) -> ( cos ` ( x x. A ) ) = ( cos ` ( ( k + 1 ) x. A ) ) ) |
20 |
18
|
fveq2d |
|- ( x = ( k + 1 ) -> ( sin ` ( x x. A ) ) = ( sin ` ( ( k + 1 ) x. A ) ) ) |
21 |
20
|
oveq2d |
|- ( x = ( k + 1 ) -> ( _i x. ( sin ` ( x x. A ) ) ) = ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) |
22 |
19 21
|
oveq12d |
|- ( x = ( k + 1 ) -> ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) |
23 |
17 22
|
eqeq12d |
|- ( x = ( k + 1 ) -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) <-> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) ) |
24 |
23
|
imbi2d |
|- ( x = ( k + 1 ) -> ( ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) ) <-> ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) ) ) |
25 |
|
oveq2 |
|- ( x = N -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) ) |
26 |
|
oveq1 |
|- ( x = N -> ( x x. A ) = ( N x. A ) ) |
27 |
26
|
fveq2d |
|- ( x = N -> ( cos ` ( x x. A ) ) = ( cos ` ( N x. A ) ) ) |
28 |
26
|
fveq2d |
|- ( x = N -> ( sin ` ( x x. A ) ) = ( sin ` ( N x. A ) ) ) |
29 |
28
|
oveq2d |
|- ( x = N -> ( _i x. ( sin ` ( x x. A ) ) ) = ( _i x. ( sin ` ( N x. A ) ) ) ) |
30 |
27 29
|
oveq12d |
|- ( x = N -> ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) ) |
31 |
25 30
|
eqeq12d |
|- ( x = N -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) <-> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) ) ) |
32 |
31
|
imbi2d |
|- ( x = N -> ( ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ x ) = ( ( cos ` ( x x. A ) ) + ( _i x. ( sin ` ( x x. A ) ) ) ) ) <-> ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) ) ) ) |
33 |
|
coscl |
|- ( A e. CC -> ( cos ` A ) e. CC ) |
34 |
|
ax-icn |
|- _i e. CC |
35 |
|
sincl |
|- ( A e. CC -> ( sin ` A ) e. CC ) |
36 |
|
mulcl |
|- ( ( _i e. CC /\ ( sin ` A ) e. CC ) -> ( _i x. ( sin ` A ) ) e. CC ) |
37 |
34 35 36
|
sylancr |
|- ( A e. CC -> ( _i x. ( sin ` A ) ) e. CC ) |
38 |
|
addcl |
|- ( ( ( cos ` A ) e. CC /\ ( _i x. ( sin ` A ) ) e. CC ) -> ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) e. CC ) |
39 |
33 37 38
|
syl2anc |
|- ( A e. CC -> ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) e. CC ) |
40 |
|
exp0 |
|- ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ 0 ) = 1 ) |
41 |
39 40
|
syl |
|- ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ 0 ) = 1 ) |
42 |
|
mul02 |
|- ( A e. CC -> ( 0 x. A ) = 0 ) |
43 |
42
|
fveq2d |
|- ( A e. CC -> ( cos ` ( 0 x. A ) ) = ( cos ` 0 ) ) |
44 |
|
cos0 |
|- ( cos ` 0 ) = 1 |
45 |
43 44
|
eqtrdi |
|- ( A e. CC -> ( cos ` ( 0 x. A ) ) = 1 ) |
46 |
42
|
fveq2d |
|- ( A e. CC -> ( sin ` ( 0 x. A ) ) = ( sin ` 0 ) ) |
47 |
|
sin0 |
|- ( sin ` 0 ) = 0 |
48 |
46 47
|
eqtrdi |
|- ( A e. CC -> ( sin ` ( 0 x. A ) ) = 0 ) |
49 |
48
|
oveq2d |
|- ( A e. CC -> ( _i x. ( sin ` ( 0 x. A ) ) ) = ( _i x. 0 ) ) |
50 |
34
|
mul01i |
|- ( _i x. 0 ) = 0 |
51 |
49 50
|
eqtrdi |
|- ( A e. CC -> ( _i x. ( sin ` ( 0 x. A ) ) ) = 0 ) |
52 |
45 51
|
oveq12d |
|- ( A e. CC -> ( ( cos ` ( 0 x. A ) ) + ( _i x. ( sin ` ( 0 x. A ) ) ) ) = ( 1 + 0 ) ) |
53 |
|
ax-1cn |
|- 1 e. CC |
54 |
53
|
addid1i |
|- ( 1 + 0 ) = 1 |
55 |
52 54
|
eqtrdi |
|- ( A e. CC -> ( ( cos ` ( 0 x. A ) ) + ( _i x. ( sin ` ( 0 x. A ) ) ) ) = 1 ) |
56 |
41 55
|
eqtr4d |
|- ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ 0 ) = ( ( cos ` ( 0 x. A ) ) + ( _i x. ( sin ` ( 0 x. A ) ) ) ) ) |
57 |
|
expp1 |
|- ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) e. CC /\ k e. NN0 ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
58 |
39 57
|
sylan |
|- ( ( A e. CC /\ k e. NN0 ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
59 |
58
|
ancoms |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
60 |
59
|
adantr |
|- ( ( ( k e. NN0 /\ A e. CC ) /\ ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
61 |
|
oveq1 |
|- ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
62 |
61
|
adantl |
|- ( ( ( k e. NN0 /\ A e. CC ) /\ ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) ) |
63 |
|
nn0cn |
|- ( k e. NN0 -> k e. CC ) |
64 |
|
mulcl |
|- ( ( k e. CC /\ A e. CC ) -> ( k x. A ) e. CC ) |
65 |
63 64
|
sylan |
|- ( ( k e. NN0 /\ A e. CC ) -> ( k x. A ) e. CC ) |
66 |
|
sinadd |
|- ( ( ( k x. A ) e. CC /\ A e. CC ) -> ( sin ` ( ( k x. A ) + A ) ) = ( ( ( sin ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
67 |
65 66
|
sylancom |
|- ( ( k e. NN0 /\ A e. CC ) -> ( sin ` ( ( k x. A ) + A ) ) = ( ( ( sin ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
68 |
33
|
adantl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( cos ` A ) e. CC ) |
69 |
|
sincl |
|- ( ( k x. A ) e. CC -> ( sin ` ( k x. A ) ) e. CC ) |
70 |
65 69
|
syl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( sin ` ( k x. A ) ) e. CC ) |
71 |
|
mulcom |
|- ( ( ( cos ` A ) e. CC /\ ( sin ` ( k x. A ) ) e. CC ) -> ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) = ( ( sin ` ( k x. A ) ) x. ( cos ` A ) ) ) |
72 |
68 70 71
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) = ( ( sin ` ( k x. A ) ) x. ( cos ` A ) ) ) |
73 |
72
|
oveq1d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) + ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) = ( ( ( sin ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
74 |
|
mulcl |
|- ( ( ( cos ` A ) e. CC /\ ( sin ` ( k x. A ) ) e. CC ) -> ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) |
75 |
68 70 74
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) |
76 |
|
coscl |
|- ( ( k x. A ) e. CC -> ( cos ` ( k x. A ) ) e. CC ) |
77 |
65 76
|
syl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( cos ` ( k x. A ) ) e. CC ) |
78 |
35
|
adantl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( sin ` A ) e. CC ) |
79 |
|
mulcl |
|- ( ( ( cos ` ( k x. A ) ) e. CC /\ ( sin ` A ) e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) e. CC ) |
80 |
77 78 79
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) e. CC ) |
81 |
|
addcom |
|- ( ( ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) e. CC /\ ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) e. CC ) -> ( ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) + ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
82 |
75 80 81
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) + ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
83 |
67 73 82
|
3eqtr2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( sin ` ( ( k x. A ) + A ) ) = ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
84 |
83
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( _i x. ( sin ` ( ( k x. A ) + A ) ) ) = ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
85 |
84
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` ( ( k x. A ) + A ) ) + ( _i x. ( sin ` ( ( k x. A ) + A ) ) ) ) = ( ( cos ` ( ( k x. A ) + A ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
86 |
|
adddir |
|- ( ( k e. CC /\ 1 e. CC /\ A e. CC ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + ( 1 x. A ) ) ) |
87 |
|
mulid2 |
|- ( A e. CC -> ( 1 x. A ) = A ) |
88 |
87
|
oveq2d |
|- ( A e. CC -> ( ( k x. A ) + ( 1 x. A ) ) = ( ( k x. A ) + A ) ) |
89 |
88
|
3ad2ant3 |
|- ( ( k e. CC /\ 1 e. CC /\ A e. CC ) -> ( ( k x. A ) + ( 1 x. A ) ) = ( ( k x. A ) + A ) ) |
90 |
86 89
|
eqtrd |
|- ( ( k e. CC /\ 1 e. CC /\ A e. CC ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + A ) ) |
91 |
63 90
|
syl3an1 |
|- ( ( k e. NN0 /\ 1 e. CC /\ A e. CC ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + A ) ) |
92 |
53 91
|
mp3an2 |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( k + 1 ) x. A ) = ( ( k x. A ) + A ) ) |
93 |
92
|
fveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( cos ` ( ( k + 1 ) x. A ) ) = ( cos ` ( ( k x. A ) + A ) ) ) |
94 |
92
|
fveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( sin ` ( ( k + 1 ) x. A ) ) = ( sin ` ( ( k x. A ) + A ) ) ) |
95 |
94
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) = ( _i x. ( sin ` ( ( k x. A ) + A ) ) ) ) |
96 |
93 95
|
oveq12d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) = ( ( cos ` ( ( k x. A ) + A ) ) + ( _i x. ( sin ` ( ( k x. A ) + A ) ) ) ) ) |
97 |
|
mulcl |
|- ( ( _i e. CC /\ ( sin ` ( k x. A ) ) e. CC ) -> ( _i x. ( sin ` ( k x. A ) ) ) e. CC ) |
98 |
34 97
|
mpan |
|- ( ( sin ` ( k x. A ) ) e. CC -> ( _i x. ( sin ` ( k x. A ) ) ) e. CC ) |
99 |
65 69 98
|
3syl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( _i x. ( sin ` ( k x. A ) ) ) e. CC ) |
100 |
33 37
|
jca |
|- ( A e. CC -> ( ( cos ` A ) e. CC /\ ( _i x. ( sin ` A ) ) e. CC ) ) |
101 |
100
|
adantl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` A ) e. CC /\ ( _i x. ( sin ` A ) ) e. CC ) ) |
102 |
|
muladd |
|- ( ( ( ( cos ` ( k x. A ) ) e. CC /\ ( _i x. ( sin ` ( k x. A ) ) ) e. CC ) /\ ( ( cos ` A ) e. CC /\ ( _i x. ( sin ` A ) ) e. CC ) ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) + ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
103 |
77 99 101 102
|
syl21anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) + ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
104 |
78 34
|
jctil |
|- ( ( k e. NN0 /\ A e. CC ) -> ( _i e. CC /\ ( sin ` A ) e. CC ) ) |
105 |
70 34
|
jctil |
|- ( ( k e. NN0 /\ A e. CC ) -> ( _i e. CC /\ ( sin ` ( k x. A ) ) e. CC ) ) |
106 |
|
mul4 |
|- ( ( ( _i e. CC /\ ( sin ` A ) e. CC ) /\ ( _i e. CC /\ ( sin ` ( k x. A ) ) e. CC ) ) -> ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) = ( ( _i x. _i ) x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
107 |
|
ixi |
|- ( _i x. _i ) = -u 1 |
108 |
107
|
oveq1i |
|- ( ( _i x. _i ) x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) = ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) |
109 |
106 108
|
eqtrdi |
|- ( ( ( _i e. CC /\ ( sin ` A ) e. CC ) /\ ( _i e. CC /\ ( sin ` ( k x. A ) ) e. CC ) ) -> ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) = ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
110 |
104 105 109
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) = ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
111 |
110
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
112 |
111
|
oveq1d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( ( _i x. ( sin ` A ) ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) + ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) + ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
113 |
|
mul12 |
|- ( ( ( cos ` ( k x. A ) ) e. CC /\ _i e. CC /\ ( sin ` A ) e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) = ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
114 |
34 113
|
mp3an2 |
|- ( ( ( cos ` ( k x. A ) ) e. CC /\ ( sin ` A ) e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) = ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
115 |
77 78 114
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) = ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
116 |
|
mul12 |
|- ( ( ( cos ` A ) e. CC /\ _i e. CC /\ ( sin ` ( k x. A ) ) e. CC ) -> ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) = ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
117 |
34 116
|
mp3an2 |
|- ( ( ( cos ` A ) e. CC /\ ( sin ` ( k x. A ) ) e. CC ) -> ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) = ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
118 |
68 70 117
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) = ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
119 |
115 118
|
oveq12d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) = ( ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) + ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
120 |
|
adddi |
|- ( ( _i e. CC /\ ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) e. CC /\ ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) -> ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) = ( ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) + ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
121 |
34 120
|
mp3an1 |
|- ( ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) e. CC /\ ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) -> ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) = ( ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) + ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
122 |
80 75 121
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) = ( ( _i x. ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) ) + ( _i x. ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
123 |
119 122
|
eqtr4d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) = ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) |
124 |
123
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) + ( ( ( cos ` ( k x. A ) ) x. ( _i x. ( sin ` A ) ) ) + ( ( cos ` A ) x. ( _i x. ( sin ` ( k x. A ) ) ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
125 |
103 112 124
|
3eqtrd |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
126 |
|
mulcl |
|- ( ( ( sin ` A ) e. CC /\ ( sin ` ( k x. A ) ) e. CC ) -> ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) |
127 |
78 70 126
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) |
128 |
|
mulm1 |
|- ( ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) e. CC -> ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) = -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) |
129 |
127 128
|
syl |
|- ( ( k e. NN0 /\ A e. CC ) -> ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) = -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) |
130 |
129
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
131 |
130
|
oveq1d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + ( -u 1 x. ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
132 |
|
mulcl |
|- ( ( ( cos ` ( k x. A ) ) e. CC /\ ( cos ` A ) e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) e. CC ) |
133 |
77 68 132
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) e. CC ) |
134 |
|
negsub |
|- ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) e. CC /\ ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
135 |
133 127 134
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
136 |
135
|
oveq1d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) + -u ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
137 |
125 131 136
|
3eqtrd |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
138 |
|
cosadd |
|- ( ( ( k x. A ) e. CC /\ A e. CC ) -> ( cos ` ( ( k x. A ) + A ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
139 |
65 138
|
sylancom |
|- ( ( k e. NN0 /\ A e. CC ) -> ( cos ` ( ( k x. A ) + A ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` ( k x. A ) ) x. ( sin ` A ) ) ) ) |
140 |
|
mulcom |
|- ( ( ( sin ` ( k x. A ) ) e. CC /\ ( sin ` A ) e. CC ) -> ( ( sin ` ( k x. A ) ) x. ( sin ` A ) ) = ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) |
141 |
70 78 140
|
syl2anc |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( sin ` ( k x. A ) ) x. ( sin ` A ) ) = ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) |
142 |
141
|
oveq2d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` ( k x. A ) ) x. ( sin ` A ) ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
143 |
139 142
|
eqtrd |
|- ( ( k e. NN0 /\ A e. CC ) -> ( cos ` ( ( k x. A ) + A ) ) = ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) ) |
144 |
143
|
oveq1d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( cos ` ( ( k x. A ) + A ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) = ( ( ( ( cos ` ( k x. A ) ) x. ( cos ` A ) ) - ( ( sin ` A ) x. ( sin ` ( k x. A ) ) ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
145 |
137 144
|
eqtr4d |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( cos ` ( ( k x. A ) + A ) ) + ( _i x. ( ( ( cos ` ( k x. A ) ) x. ( sin ` A ) ) + ( ( cos ` A ) x. ( sin ` ( k x. A ) ) ) ) ) ) ) |
146 |
85 96 145
|
3eqtr4rd |
|- ( ( k e. NN0 /\ A e. CC ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) |
147 |
146
|
adantr |
|- ( ( ( k e. NN0 /\ A e. CC ) /\ ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) -> ( ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) x. ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) |
148 |
60 62 147
|
3eqtrd |
|- ( ( ( k e. NN0 /\ A e. CC ) /\ ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) |
149 |
148
|
exp31 |
|- ( k e. NN0 -> ( A e. CC -> ( ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) ) ) |
150 |
149
|
a2d |
|- ( k e. NN0 -> ( ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ k ) = ( ( cos ` ( k x. A ) ) + ( _i x. ( sin ` ( k x. A ) ) ) ) ) -> ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ ( k + 1 ) ) = ( ( cos ` ( ( k + 1 ) x. A ) ) + ( _i x. ( sin ` ( ( k + 1 ) x. A ) ) ) ) ) ) ) |
151 |
8 16 24 32 56 150
|
nn0ind |
|- ( N e. NN0 -> ( A e. CC -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) ) ) |
152 |
151
|
impcom |
|- ( ( A e. CC /\ N e. NN0 ) -> ( ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) ^ N ) = ( ( cos ` ( N x. A ) ) + ( _i x. ( sin ` ( N x. A ) ) ) ) ) |