Step |
Hyp |
Ref |
Expression |
1 |
|
cnelprrecn |
|- CC e. { RR , CC } |
2 |
1
|
a1i |
|- ( T. -> CC e. { RR , CC } ) |
3 |
|
ax-icn |
|- _i e. CC |
4 |
3
|
a1i |
|- ( ( T. /\ x e. CC ) -> _i e. CC ) |
5 |
|
simpr |
|- ( ( T. /\ x e. CC ) -> x e. CC ) |
6 |
4 5
|
mulcld |
|- ( ( T. /\ x e. CC ) -> ( _i x. x ) e. CC ) |
7 |
|
efcl |
|- ( ( _i x. x ) e. CC -> ( exp ` ( _i x. x ) ) e. CC ) |
8 |
6 7
|
syl |
|- ( ( T. /\ x e. CC ) -> ( exp ` ( _i x. x ) ) e. CC ) |
9 |
|
ine0 |
|- _i =/= 0 |
10 |
9
|
a1i |
|- ( ( T. /\ x e. CC ) -> _i =/= 0 ) |
11 |
8 4 10
|
divcld |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) / _i ) e. CC ) |
12 |
|
negicn |
|- -u _i e. CC |
13 |
|
mulcl |
|- ( ( -u _i e. CC /\ x e. CC ) -> ( -u _i x. x ) e. CC ) |
14 |
12 5 13
|
sylancr |
|- ( ( T. /\ x e. CC ) -> ( -u _i x. x ) e. CC ) |
15 |
|
efcl |
|- ( ( -u _i x. x ) e. CC -> ( exp ` ( -u _i x. x ) ) e. CC ) |
16 |
14 15
|
syl |
|- ( ( T. /\ x e. CC ) -> ( exp ` ( -u _i x. x ) ) e. CC ) |
17 |
16 4 10
|
divcld |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) / _i ) e. CC ) |
18 |
17
|
negcld |
|- ( ( T. /\ x e. CC ) -> -u ( ( exp ` ( -u _i x. x ) ) / _i ) e. CC ) |
19 |
11 18
|
addcld |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) e. CC ) |
20 |
8 16
|
addcld |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) e. CC ) |
21 |
8 4
|
mulcld |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) x. _i ) e. CC ) |
22 |
|
efcl |
|- ( y e. CC -> ( exp ` y ) e. CC ) |
23 |
22
|
adantl |
|- ( ( T. /\ y e. CC ) -> ( exp ` y ) e. CC ) |
24 |
|
1cnd |
|- ( ( T. /\ x e. CC ) -> 1 e. CC ) |
25 |
2
|
dvmptid |
|- ( T. -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) ) |
26 |
3
|
a1i |
|- ( T. -> _i e. CC ) |
27 |
2 5 24 25 26
|
dvmptcmul |
|- ( T. -> ( CC _D ( x e. CC |-> ( _i x. x ) ) ) = ( x e. CC |-> ( _i x. 1 ) ) ) |
28 |
3
|
mulid1i |
|- ( _i x. 1 ) = _i |
29 |
28
|
mpteq2i |
|- ( x e. CC |-> ( _i x. 1 ) ) = ( x e. CC |-> _i ) |
30 |
27 29
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. CC |-> ( _i x. x ) ) ) = ( x e. CC |-> _i ) ) |
31 |
|
eff |
|- exp : CC --> CC |
32 |
31
|
a1i |
|- ( T. -> exp : CC --> CC ) |
33 |
32
|
feqmptd |
|- ( T. -> exp = ( y e. CC |-> ( exp ` y ) ) ) |
34 |
33
|
oveq2d |
|- ( T. -> ( CC _D exp ) = ( CC _D ( y e. CC |-> ( exp ` y ) ) ) ) |
35 |
|
dvef |
|- ( CC _D exp ) = exp |
36 |
35 33
|
syl5eq |
|- ( T. -> ( CC _D exp ) = ( y e. CC |-> ( exp ` y ) ) ) |
37 |
34 36
|
eqtr3d |
|- ( T. -> ( CC _D ( y e. CC |-> ( exp ` y ) ) ) = ( y e. CC |-> ( exp ` y ) ) ) |
38 |
|
fveq2 |
|- ( y = ( _i x. x ) -> ( exp ` y ) = ( exp ` ( _i x. x ) ) ) |
39 |
2 2 6 4 23 23 30 37 38 38
|
dvmptco |
|- ( T. -> ( CC _D ( x e. CC |-> ( exp ` ( _i x. x ) ) ) ) = ( x e. CC |-> ( ( exp ` ( _i x. x ) ) x. _i ) ) ) |
40 |
9
|
a1i |
|- ( T. -> _i =/= 0 ) |
41 |
2 8 21 39 26 40
|
dvmptdivc |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( exp ` ( _i x. x ) ) / _i ) ) ) = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) x. _i ) / _i ) ) ) |
42 |
8 4 10
|
divcan4d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) / _i ) = ( exp ` ( _i x. x ) ) ) |
43 |
42
|
mpteq2dva |
|- ( T. -> ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) x. _i ) / _i ) ) = ( x e. CC |-> ( exp ` ( _i x. x ) ) ) ) |
44 |
41 43
|
eqtrd |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( exp ` ( _i x. x ) ) / _i ) ) ) = ( x e. CC |-> ( exp ` ( _i x. x ) ) ) ) |
45 |
|
mulcl |
|- ( ( ( exp ` ( -u _i x. x ) ) e. CC /\ -u _i e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) e. CC ) |
46 |
16 12 45
|
sylancl |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) e. CC ) |
47 |
46 4 10
|
divcld |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) e. CC ) |
48 |
12
|
a1i |
|- ( ( T. /\ x e. CC ) -> -u _i e. CC ) |
49 |
12
|
a1i |
|- ( T. -> -u _i e. CC ) |
50 |
2 5 24 25 49
|
dvmptcmul |
|- ( T. -> ( CC _D ( x e. CC |-> ( -u _i x. x ) ) ) = ( x e. CC |-> ( -u _i x. 1 ) ) ) |
51 |
12
|
mulid1i |
|- ( -u _i x. 1 ) = -u _i |
52 |
51
|
mpteq2i |
|- ( x e. CC |-> ( -u _i x. 1 ) ) = ( x e. CC |-> -u _i ) |
53 |
50 52
|
eqtrdi |
|- ( T. -> ( CC _D ( x e. CC |-> ( -u _i x. x ) ) ) = ( x e. CC |-> -u _i ) ) |
54 |
|
fveq2 |
|- ( y = ( -u _i x. x ) -> ( exp ` y ) = ( exp ` ( -u _i x. x ) ) ) |
55 |
2 2 14 48 23 23 53 37 54 54
|
dvmptco |
|- ( T. -> ( CC _D ( x e. CC |-> ( exp ` ( -u _i x. x ) ) ) ) = ( x e. CC |-> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) ) |
56 |
2 16 46 55 26 40
|
dvmptdivc |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) = ( x e. CC |-> ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) ) ) |
57 |
2 17 47 56
|
dvmptneg |
|- ( T. -> ( CC _D ( x e. CC |-> -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) = ( x e. CC |-> -u ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) ) ) |
58 |
46 4 10
|
divneg2d |
|- ( ( T. /\ x e. CC ) -> -u ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) = ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / -u _i ) ) |
59 |
3 9
|
negne0i |
|- -u _i =/= 0 |
60 |
59
|
a1i |
|- ( ( T. /\ x e. CC ) -> -u _i =/= 0 ) |
61 |
16 48 60
|
divcan4d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / -u _i ) = ( exp ` ( -u _i x. x ) ) ) |
62 |
58 61
|
eqtrd |
|- ( ( T. /\ x e. CC ) -> -u ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) = ( exp ` ( -u _i x. x ) ) ) |
63 |
62
|
mpteq2dva |
|- ( T. -> ( x e. CC |-> -u ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) ) = ( x e. CC |-> ( exp ` ( -u _i x. x ) ) ) ) |
64 |
57 63
|
eqtrd |
|- ( T. -> ( CC _D ( x e. CC |-> -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) = ( x e. CC |-> ( exp ` ( -u _i x. x ) ) ) ) |
65 |
2 11 8 44 18 16 64
|
dvmptadd |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) ) = ( x e. CC |-> ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) ) ) |
66 |
|
2cnd |
|- ( T. -> 2 e. CC ) |
67 |
|
2ne0 |
|- 2 =/= 0 |
68 |
67
|
a1i |
|- ( T. -> 2 =/= 0 ) |
69 |
2 19 20 65 66 68
|
dvmptdivc |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) ) = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) ) |
70 |
|
df-sin |
|- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) |
71 |
8 16
|
subcld |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) e. CC ) |
72 |
|
2cnd |
|- ( ( T. /\ x e. CC ) -> 2 e. CC ) |
73 |
67
|
a1i |
|- ( ( T. /\ x e. CC ) -> 2 =/= 0 ) |
74 |
71 4 72 10 73
|
divdiv1d |
|- ( ( T. /\ x e. CC ) -> ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( _i x. 2 ) ) ) |
75 |
|
2cn |
|- 2 e. CC |
76 |
3 75
|
mulcomi |
|- ( _i x. 2 ) = ( 2 x. _i ) |
77 |
76
|
oveq2i |
|- ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( _i x. 2 ) ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) |
78 |
74 77
|
eqtrdi |
|- ( ( T. /\ x e. CC ) -> ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) |
79 |
8 16 4 10
|
divsubdird |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) = ( ( ( exp ` ( _i x. x ) ) / _i ) - ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) |
80 |
11 17
|
negsubd |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) = ( ( ( exp ` ( _i x. x ) ) / _i ) - ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) |
81 |
79 80
|
eqtr4d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) = ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) ) |
82 |
81
|
oveq1d |
|- ( ( T. /\ x e. CC ) -> ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) = ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) |
83 |
78 82
|
eqtr3d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) = ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) |
84 |
83
|
mpteq2dva |
|- ( T. -> ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) = ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) ) |
85 |
70 84
|
syl5eq |
|- ( T. -> sin = ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) ) |
86 |
85
|
oveq2d |
|- ( T. -> ( CC _D sin ) = ( CC _D ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) ) ) |
87 |
|
df-cos |
|- cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) |
88 |
87
|
a1i |
|- ( T. -> cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) ) |
89 |
69 86 88
|
3eqtr4d |
|- ( T. -> ( CC _D sin ) = cos ) |
90 |
21 46
|
addcld |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) e. CC ) |
91 |
2 8 21 39 16 46 55
|
dvmptadd |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) ) ) = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) ) ) |
92 |
2 20 90 91 66 68
|
dvmptdivc |
|- ( T. -> ( CC _D ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) ) = ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) / 2 ) ) ) |
93 |
88
|
oveq2d |
|- ( T. -> ( CC _D cos ) = ( CC _D ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) ) ) |
94 |
71 4 10
|
divcld |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) e. CC ) |
95 |
94 72 73
|
divnegd |
|- ( ( T. /\ x e. CC ) -> -u ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) = ( -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) ) |
96 |
|
sinval |
|- ( x e. CC -> ( sin ` x ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) |
97 |
96
|
adantl |
|- ( ( T. /\ x e. CC ) -> ( sin ` x ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) ) |
98 |
97 78
|
eqtr4d |
|- ( ( T. /\ x e. CC ) -> ( sin ` x ) = ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) ) |
99 |
98
|
negeqd |
|- ( ( T. /\ x e. CC ) -> -u ( sin ` x ) = -u ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) ) |
100 |
3
|
negnegi |
|- -u -u _i = _i |
101 |
100
|
oveq2i |
|- ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u -u _i ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. _i ) |
102 |
|
mulneg2 |
|- ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) e. CC /\ -u _i e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u -u _i ) = -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u _i ) ) |
103 |
71 12 102
|
sylancl |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u -u _i ) = -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u _i ) ) |
104 |
101 103
|
syl5eqr |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. _i ) = -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u _i ) ) |
105 |
|
mulcl |
|- ( ( ( exp ` ( -u _i x. x ) ) e. CC /\ _i e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. _i ) e. CC ) |
106 |
16 3 105
|
sylancl |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. _i ) e. CC ) |
107 |
21 106
|
negsubd |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) + -u ( ( exp ` ( -u _i x. x ) ) x. _i ) ) = ( ( ( exp ` ( _i x. x ) ) x. _i ) - ( ( exp ` ( -u _i x. x ) ) x. _i ) ) ) |
108 |
|
mulneg2 |
|- ( ( ( exp ` ( -u _i x. x ) ) e. CC /\ _i e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) = -u ( ( exp ` ( -u _i x. x ) ) x. _i ) ) |
109 |
16 3 108
|
sylancl |
|- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) = -u ( ( exp ` ( -u _i x. x ) ) x. _i ) ) |
110 |
109
|
oveq2d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) = ( ( ( exp ` ( _i x. x ) ) x. _i ) + -u ( ( exp ` ( -u _i x. x ) ) x. _i ) ) ) |
111 |
8 16 4
|
subdird |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. _i ) = ( ( ( exp ` ( _i x. x ) ) x. _i ) - ( ( exp ` ( -u _i x. x ) ) x. _i ) ) ) |
112 |
107 110 111
|
3eqtr4d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. _i ) ) |
113 |
71 4 10
|
divrecd |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. ( 1 / _i ) ) ) |
114 |
|
irec |
|- ( 1 / _i ) = -u _i |
115 |
114
|
oveq2i |
|- ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. ( 1 / _i ) ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u _i ) |
116 |
113 115
|
eqtrdi |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u _i ) ) |
117 |
116
|
negeqd |
|- ( ( T. /\ x e. CC ) -> -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) = -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) x. -u _i ) ) |
118 |
104 112 117
|
3eqtr4d |
|- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) = -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) ) |
119 |
118
|
oveq1d |
|- ( ( T. /\ x e. CC ) -> ( ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) / 2 ) = ( -u ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) ) |
120 |
95 99 119
|
3eqtr4d |
|- ( ( T. /\ x e. CC ) -> -u ( sin ` x ) = ( ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) / 2 ) ) |
121 |
120
|
mpteq2dva |
|- ( T. -> ( x e. CC |-> -u ( sin ` x ) ) = ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) x. _i ) + ( ( exp ` ( -u _i x. x ) ) x. -u _i ) ) / 2 ) ) ) |
122 |
92 93 121
|
3eqtr4d |
|- ( T. -> ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) ) |
123 |
89 122
|
jca |
|- ( T. -> ( ( CC _D sin ) = cos /\ ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) ) ) |
124 |
123
|
mptru |
|- ( ( CC _D sin ) = cos /\ ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) ) |