Theorem dvsincos 23744

Description: Derivative of the sine and cosine functions. (Contributed by Mario Carneiro, 21-May-2016.)

Assertion

Ref	Expression
dvsincos	|- ( ( CC _D sin ) = cos /\ ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) )

Proof of Theorem dvsincos

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnelprrecn 10029	. . . . . 6 |- CC e. { RR , CC }
2	1	a1i 11	. . . . 5 |- ( T. -> CC e. { RR , CC } )
3		ax-icn 9995	. . . . . . . . . 10 |- _i e. CC
4	3	a1i 11	. . . . . . . . 9 |- ( ( T. /\ x e. CC ) -> _i e. CC )
5		simpr 477	. . . . . . . . 9 |- ( ( T. /\ x e. CC ) -> x e. CC )
6	4, 5	mulcld 10060	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> ( _i x. x ) e. CC )
7		efcl 14813	. . . . . . . 8 |- ( ( _i x. x ) e. CC -> ( exp ` ( _i x. x ) ) e. CC )
8	6, 7	syl 17	. . . . . . 7 |- ( ( T. /\ x e. CC ) -> ( exp ` ( _i x. x ) ) e. CC )
9		ine0 10465	. . . . . . . 8 |- _i =/= 0
10	9	a1i 11	. . . . . . 7 |- ( ( T. /\ x e. CC ) -> _i =/= 0 )
11	8, 4, 10	divcld 10801	. . . . . 6 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) / _i ) e. CC )
12		negicn 10282	. . . . . . . . . 10 |- -u _i e. CC
13		mulcl 10020	. . . . . . . . . 10 |- ( ( -u _i e. CC /\ x e. CC ) -> ( -u _i x. x ) e. CC )
14	12, 5, 13	sylancr 695	. . . . . . . . 9 |- ( ( T. /\ x e. CC ) -> ( -u _i x. x ) e. CC )
15		efcl 14813	. . . . . . . . 9 |- ( ( -u _i x. x ) e. CC -> ( exp ` ( -u _i x. x ) ) e. CC )
16	14, 15	syl 17	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> ( exp ` ( -u _i x. x ) ) e. CC )
17	16, 4, 10	divcld 10801	. . . . . . 7 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) / _i ) e. CC )
18	17	negcld 10379	. . . . . 6 |- ( ( T. /\ x e. CC ) -> -u ( ( exp ` ( -u _i x. x ) ) / _i ) e. CC )
19	11, 18	addcld 10059	. . . . 5 |- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) e. CC )
20	8, 16	addcld 10059	. . . . 5 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) e. CC )
21	8, 4	mulcld 10060	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) x. _i ) e. CC )
22		efcl 14813	. . . . . . . . . 10 |- ( y e. CC -> ( exp ` y ) e. CC )
23	22	adantl 482	. . . . . . . . 9 |- ( ( T. /\ y e. CC ) -> ( exp ` y ) e. CC )
24		1cnd 10056	. . . . . . . . . . 11 |- ( ( T. /\ x e. CC ) -> 1 e. CC )
25	2	dvmptid 23720	. . . . . . . . . . 11 |- ( T. -> ( CC _D ( x e. CC |-> x ) ) = ( x e. CC |-> 1 ) )
26	3	a1i 11	. . . . . . . . . . 11 |- ( T. -> _i e. CC )
27	2, 5, 24, 25, 26	dvmptcmul 23727	. . . . . . . . . 10 |- ( T. -> ( CC _D ( x e. CC |-> ( _i x. x ) ) ) = ( x e. CC |-> ( _i x. 1 ) ) )
28	3	mulid1i 10042	. . . . . . . . . . 11 |- ( _i x. 1 ) = _i
29	28	mpteq2i 4741	. . . . . . . . . 10 |- ( x e. CC |-> ( _i x. 1 ) ) = ( x e. CC |-> _i )
30	27, 29	syl6eq 2672	. . . . . . . . 9 |- ( T. -> ( CC _D ( x e. CC |-> ( _i x. x ) ) ) = ( x e. CC |-> _i ) )
31		eff 14812	. . . . . . . . . . . . 13 |- exp : CC --> CC
32	31	a1i 11	. . . . . . . . . . . 12 |- ( T. -> exp : CC --> CC )
33	32	feqmptd 6249	. . . . . . . . . . 11 |- ( T. -> exp = ( y e. CC |-> ( exp ` y ) ) )
34	33	oveq2d 6666	. . . . . . . . . 10 |- ( T. -> ( CC _D exp ) = ( CC _D ( y e. CC |-> ( exp ` y ) ) ) )
35		dvef 23743	. . . . . . . . . . 11 |- ( CC _D exp ) = exp
36	35, 33	syl5eq 2668	. . . . . . . . . 10 |- ( T. -> ( CC _D exp ) = ( y e. CC |-> ( exp ` y ) ) )
37	34, 36	eqtr3d 2658	. . . . . . . . 9 |- ( T. -> ( CC _D ( y e. CC |-> ( exp ` y ) ) ) = ( y e. CC |-> ( exp ` y ) ) )
38		fveq2 6191	. . . . . . . . 9 |- ( y = ( _i x. x ) -> ( exp ` y ) = ( exp ` ( _i x. x ) ) )
39	2, 2, 6, 4, 23, 23, 30, 37, 38, 38	dvmptco 23735	. . . . . . . 8 |- ( T. -> ( CC _D ( x e. CC |-> ( exp ` ( _i x. x ) ) ) ) = ( x e. CC |-> ( ( exp ` ( _i x. x ) ) x. _i ) ) )
40	9	a1i 11	. . . . . . . 8 |- ( T. -> _i =/= 0 )
41	2, 8, 21, 39, 26, 40	dvmptdivc 23728	. . . . . . 7 |- ( 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 10807	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) x. _i ) / _i ) = ( exp ` ( _i x. x ) ) )
43	42	mpteq2dva 4744	. . . . . . 7 |- ( T. -> ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) x. _i ) / _i ) ) = ( x e. CC |-> ( exp ` ( _i x. x ) ) ) )
44	41, 43	eqtrd 2656	. . . . . 6 |- ( T. -> ( CC _D ( x e. CC |-> ( ( exp ` ( _i x. x ) ) / _i ) ) ) = ( x e. CC |-> ( exp ` ( _i x. x ) ) ) )
45		mulcl 10020	. . . . . . . . . 10 |- ( ( ( 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 694	. . . . . . . . 9 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) e. CC )
47	46, 4, 10	divcld 10801	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) e. CC )
48	12	a1i 11	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> -u _i e. CC )
49	12	a1i 11	. . . . . . . . . . . 12 |- ( T. -> -u _i e. CC )
50	2, 5, 24, 25, 49	dvmptcmul 23727	. . . . . . . . . . 11 |- ( T. -> ( CC _D ( x e. CC |-> ( -u _i x. x ) ) ) = ( x e. CC |-> ( -u _i x. 1 ) ) )
51	12	mulid1i 10042	. . . . . . . . . . . 12 |- ( -u _i x. 1 ) = -u _i
52	51	mpteq2i 4741	. . . . . . . . . . 11 |- ( x e. CC |-> ( -u _i x. 1 ) ) = ( x e. CC |-> -u _i )
53	50, 52	syl6eq 2672	. . . . . . . . . 10 |- ( T. -> ( CC _D ( x e. CC |-> ( -u _i x. x ) ) ) = ( x e. CC |-> -u _i ) )
54		fveq2 6191	. . . . . . . . . 10 |- ( y = ( -u _i x. x ) -> ( exp ` y ) = ( exp ` ( -u _i x. x ) ) )
55	2, 2, 14, 48, 23, 23, 53, 37, 54, 54	dvmptco 23735	. . . . . . . . 9 |- ( 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 23728	. . . . . . . 8 |- ( 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 23729	. . . . . . 7 |- ( 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 10815	. . . . . . . . 9 |- ( ( 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 10356	. . . . . . . . . . 11 |- -u _i =/= 0
60	59	a1i 11	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> -u _i =/= 0 )
61	16, 48, 60	divcan4d 10807	. . . . . . . . 9 |- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / -u _i ) = ( exp ` ( -u _i x. x ) ) )
62	58, 61	eqtrd 2656	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> -u ( ( ( exp ` ( -u _i x. x ) ) x. -u _i ) / _i ) = ( exp ` ( -u _i x. x ) ) )
63	62	mpteq2dva 4744	. . . . . . 7 |- ( 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 2656	. . . . . 6 |- ( 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 23723	. . . . 5 |- ( 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 11093	. . . . 5 |- ( T. -> 2 e. CC )
67		2ne0 11113	. . . . . 6 |- 2 =/= 0
68	67	a1i 11	. . . . 5 |- ( T. -> 2 =/= 0 )
69	2, 19, 20, 65, 66, 68	dvmptdivc 23728	. . . 4 |- ( 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 14800	. . . . . 6 |- sin = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
71	8, 16	subcld 10392	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) e. CC )
72		2cnd 11093	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> 2 e. CC )
73	67	a1i 11	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> 2 =/= 0 )
74	71, 4, 72, 10, 73	divdiv1d 10832	. . . . . . . . 9 |- ( ( 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 11091	. . . . . . . . . . 11 |- 2 e. CC
76	3, 75	mulcomi 10046	. . . . . . . . . 10 |- ( _i x. 2 ) = ( 2 x. _i )
77	76	oveq2i 6661	. . . . . . . . 9 |- ( ( ( 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	syl6eq 2672	. . . . . . . 8 |- ( ( 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 10840	. . . . . . . . . 10 |- ( ( 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 10398	. . . . . . . . . 10 |- ( ( 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 2659	. . . . . . . . 9 |- ( ( 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 6665	. . . . . . . 8 |- ( ( 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 2658	. . . . . . 7 |- ( ( 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 4744	. . . . . 6 |- ( 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 2668	. . . . 5 |- ( T. -> sin = ( x e. CC |-> ( ( ( ( exp ` ( _i x. x ) ) / _i ) + -u ( ( exp ` ( -u _i x. x ) ) / _i ) ) / 2 ) ) )
86	85	oveq2d 6666	. . . 4 |- ( 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 14801	. . . . 5 |- cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) )
88	87	a1i 11	. . . 4 |- ( T. -> cos = ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) )
89	69, 86, 88	3eqtr4d 2666	. . 3 |- ( T. -> ( CC _D sin ) = cos )
90	21, 46	addcld 10059	. . . . 5 |- ( ( 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 23723	. . . . 5 |- ( 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 23728	. . . 4 |- ( 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 6666	. . . 4 |- ( T. -> ( CC _D cos ) = ( CC _D ( x e. CC |-> ( ( ( exp ` ( _i x. x ) ) + ( exp ` ( -u _i x. x ) ) ) / 2 ) ) ) )
94	71, 4, 10	divcld 10801	. . . . . . 7 |- ( ( T. /\ x e. CC ) -> ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) e. CC )
95	94, 72, 73	divnegd 10814	. . . . . 6 |- ( ( 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 14852	. . . . . . . . 9 |- ( x e. CC -> ( sin ` x ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
97	96	adantl 482	. . . . . . . 8 |- ( ( T. /\ x e. CC ) -> ( sin ` x ) = ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / ( 2 x. _i ) ) )
98	97, 78	eqtr4d 2659	. . . . . . 7 |- ( ( T. /\ x e. CC ) -> ( sin ` x ) = ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) )
99	98	negeqd 10275	. . . . . 6 |- ( ( T. /\ x e. CC ) -> -u ( sin ` x ) = -u ( ( ( ( exp ` ( _i x. x ) ) - ( exp ` ( -u _i x. x ) ) ) / _i ) / 2 ) )
100	3	negnegi 10351	. . . . . . . . . 10 |- -u -u _i = _i
101	100	oveq2i 6661	. . . . . . . . 9 |- ( ( ( 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 10467	. . . . . . . . . 10 |- ( ( ( ( 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 694	. . . . . . . . 9 |- ( ( 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 2670	. . . . . . . 8 |- ( ( 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 10020	. . . . . . . . . . 11 |- ( ( ( exp ` ( -u _i x. x ) ) e. CC /\ _i e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. _i ) e. CC )
106	16, 3, 105	sylancl 694	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. _i ) e. CC )
107	21, 106	negsubd 10398	. . . . . . . . 9 |- ( ( 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 10467	. . . . . . . . . . 11 |- ( ( ( 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 694	. . . . . . . . . 10 |- ( ( T. /\ x e. CC ) -> ( ( exp ` ( -u _i x. x ) ) x. -u _i ) = -u ( ( exp ` ( -u _i x. x ) ) x. _i ) )
110	109	oveq2d 6666	. . . . . . . . 9 |- ( ( 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 10487	. . . . . . . . 9 |- ( ( 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 2666	. . . . . . . 8 |- ( ( 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 10804	. . . . . . . . . 10 |- ( ( 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 12964	. . . . . . . . . . 11 |- ( 1 / _i ) = -u _i
115	114	oveq2i 6661	. . . . . . . . . 10 |- ( ( ( 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	syl6eq 2672	. . . . . . . . 9 |- ( ( 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 10275	. . . . . . . 8 |- ( ( 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 2666	. . . . . . 7 |- ( ( 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 6665	. . . . . 6 |- ( ( 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 2666	. . . . 5 |- ( ( 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 4744	. . . 4 |- ( 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 2666	. . 3 |- ( T. -> ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) )
123	89, 122	jca 554	. 2 |- ( T. -> ( ( CC _D sin ) = cos /\ ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) ) )
124	123	trud 1493	1 |- ( ( CC _D sin ) = cos /\ ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) ) )

Syntax hints:   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   {cpr 4179   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   _ici 9938   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   expce 14792   sincsin 14794   cosccos 14795   _D cdv 23627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631

This theorem is referenced by:  dvsin  23745  dvcos  23746