Theorem sinhval 14884

Description: Value of the hyperbolic sine of a complex number. (Contributed by Mario Carneiro, 4-Apr-2015.)

Assertion

Ref	Expression
sinhval	|- ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )

Proof of Theorem sinhval

Step	Hyp	Ref	Expression
1		ixi 10656	. . . . . . . . 9 |- ( _i x. _i ) = -u 1
2	1	oveq1i 6660	. . . . . . . 8 |- ( ( _i x. _i ) x. A ) = ( -u 1 x. A )
3		ax-icn 9995	. . . . . . . . 9 |- _i e. CC
4		mulass 10024	. . . . . . . . 9 |- ( ( _i e. CC /\ _i e. CC /\ A e. CC ) -> ( ( _i x. _i ) x. A ) = ( _i x. ( _i x. A ) ) )
5	3, 3, 4	mp3an12 1414	. . . . . . . 8 |- ( A e. CC -> ( ( _i x. _i ) x. A ) = ( _i x. ( _i x. A ) ) )
6		mulm1 10471	. . . . . . . 8 |- ( A e. CC -> ( -u 1 x. A ) = -u A )
7	2, 5, 6	3eqtr3a 2680	. . . . . . 7 |- ( A e. CC -> ( _i x. ( _i x. A ) ) = -u A )
8	7	fveq2d 6195	. . . . . 6 |- ( A e. CC -> ( exp ` ( _i x. ( _i x. A ) ) ) = ( exp ` -u A ) )
9	3, 3	mulneg1i 10476	. . . . . . . . . 10 |- ( -u _i x. _i ) = -u ( _i x. _i )
10	1	negeqi 10274	. . . . . . . . . . 11 |- -u ( _i x. _i ) = -u -u 1
11		negneg1e1 11128	. . . . . . . . . . 11 |- -u -u 1 = 1
12	10, 11	eqtri 2644	. . . . . . . . . 10 |- -u ( _i x. _i ) = 1
13	9, 12	eqtri 2644	. . . . . . . . 9 |- ( -u _i x. _i ) = 1
14	13	oveq1i 6660	. . . . . . . 8 |- ( ( -u _i x. _i ) x. A ) = ( 1 x. A )
15		negicn 10282	. . . . . . . . 9 |- -u _i e. CC
16		mulass 10024	. . . . . . . . 9 |- ( ( -u _i e. CC /\ _i e. CC /\ A e. CC ) -> ( ( -u _i x. _i ) x. A ) = ( -u _i x. ( _i x. A ) ) )
17	15, 3, 16	mp3an12 1414	. . . . . . . 8 |- ( A e. CC -> ( ( -u _i x. _i ) x. A ) = ( -u _i x. ( _i x. A ) ) )
18		mulid2 10038	. . . . . . . 8 |- ( A e. CC -> ( 1 x. A ) = A )
19	14, 17, 18	3eqtr3a 2680	. . . . . . 7 |- ( A e. CC -> ( -u _i x. ( _i x. A ) ) = A )
20	19	fveq2d 6195	. . . . . 6 |- ( A e. CC -> ( exp ` ( -u _i x. ( _i x. A ) ) ) = ( exp ` A ) )
21	8, 20	oveq12d 6668	. . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) = ( ( exp ` -u A ) - ( exp ` A ) ) )
22	21	oveq1d 6665	. . . 4 |- ( A e. CC -> ( ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) / ( 2 x. _i ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) )
23		mulcl 10020	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
24	3, 23	mpan 706	. . . . 5 |- ( A e. CC -> ( _i x. A ) e. CC )
25		sinval 14852	. . . . 5 |- ( ( _i x. A ) e. CC -> ( sin ` ( _i x. A ) ) = ( ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) / ( 2 x. _i ) ) )
26	24, 25	syl 17	. . . 4 |- ( A e. CC -> ( sin ` ( _i x. A ) ) = ( ( ( exp ` ( _i x. ( _i x. A ) ) ) - ( exp ` ( -u _i x. ( _i x. A ) ) ) ) / ( 2 x. _i ) ) )
27		irec 12964	. . . . . . . 8 |- ( 1 / _i ) = -u _i
28	27	negeqi 10274	. . . . . . 7 |- -u ( 1 / _i ) = -u -u _i
29	3	negnegi 10351	. . . . . . 7 |- -u -u _i = _i
30	28, 29	eqtri 2644	. . . . . 6 |- -u ( 1 / _i ) = _i
31	30	oveq1i 6660	. . . . 5 |- ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
32		ine0 10465	. . . . . . . 8 |- _i =/= 0
33	3, 32	reccli 10755	. . . . . . 7 |- ( 1 / _i ) e. CC
34		efcl 14813	. . . . . . . . 9 |- ( A e. CC -> ( exp ` A ) e. CC )
35		negcl 10281	. . . . . . . . . 10 |- ( A e. CC -> -u A e. CC )
36		efcl 14813	. . . . . . . . . 10 |- ( -u A e. CC -> ( exp ` -u A ) e. CC )
37	35, 36	syl 17	. . . . . . . . 9 |- ( A e. CC -> ( exp ` -u A ) e. CC )
38	34, 37	subcld 10392	. . . . . . . 8 |- ( A e. CC -> ( ( exp ` A ) - ( exp ` -u A ) ) e. CC )
39	38	halfcld 11277	. . . . . . 7 |- ( A e. CC -> ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) e. CC )
40		mulneg12 10468	. . . . . . 7 |- ( ( ( 1 / _i ) e. CC /\ ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) e. CC ) -> ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) )
41	33, 39, 40	sylancr 695	. . . . . 6 |- ( A e. CC -> ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) )
42		2cnd 11093	. . . . . . . . . 10 |- ( A e. CC -> 2 e. CC )
43		2ne0 11113	. . . . . . . . . . 11 |- 2 =/= 0
44	43	a1i 11	. . . . . . . . . 10 |- ( A e. CC -> 2 =/= 0 )
45	38, 42, 44	divnegd 10814	. . . . . . . . 9 |- ( A e. CC -> -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) = ( -u ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
46	34, 37	negsubdi2d 10408	. . . . . . . . . 10 |- ( A e. CC -> -u ( ( exp ` A ) - ( exp ` -u A ) ) = ( ( exp ` -u A ) - ( exp ` A ) ) )
47	46	oveq1d 6665	. . . . . . . . 9 |- ( A e. CC -> ( -u ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) )
48	45, 47	eqtrd 2656	. . . . . . . 8 |- ( A e. CC -> -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) )
49	48	oveq2d 6666	. . . . . . 7 |- ( A e. CC -> ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( 1 / _i ) x. ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) ) )
50	37, 34	subcld 10392	. . . . . . . . 9 |- ( A e. CC -> ( ( exp ` -u A ) - ( exp ` A ) ) e. CC )
51	50	halfcld 11277	. . . . . . . 8 |- ( A e. CC -> ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) e. CC )
52	3	a1i 11	. . . . . . . 8 |- ( A e. CC -> _i e. CC )
53	32	a1i 11	. . . . . . . 8 |- ( A e. CC -> _i =/= 0 )
54	51, 52, 53	divrec2d 10805	. . . . . . 7 |- ( A e. CC -> ( ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) / _i ) = ( ( 1 / _i ) x. ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) ) )
55	50, 42, 52, 44, 53	divdiv1d 10832	. . . . . . 7 |- ( A e. CC -> ( ( ( ( exp ` -u A ) - ( exp ` A ) ) / 2 ) / _i ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) )
56	49, 54, 55	3eqtr2d 2662	. . . . . 6 |- ( A e. CC -> ( ( 1 / _i ) x. -u ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) )
57	41, 56	eqtrd 2656	. . . . 5 |- ( A e. CC -> ( -u ( 1 / _i ) x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) )
58	31, 57	syl5eqr 2670	. . . 4 |- ( A e. CC -> ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) = ( ( ( exp ` -u A ) - ( exp ` A ) ) / ( 2 x. _i ) ) )
59	22, 26, 58	3eqtr4d 2666	. . 3 |- ( A e. CC -> ( sin ` ( _i x. A ) ) = ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) )
60	59	oveq1d 6665	. 2 |- ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) / _i ) )
61	39, 52, 53	divcan3d 10806	. 2 |- ( A e. CC -> ( ( _i x. ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )
62	60, 61	eqtrd 2656	1 |- ( A e. CC -> ( ( sin ` ( _i x. A ) ) / _i ) = ( ( ( exp ` A ) - ( exp ` -u A ) ) / 2 ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   =/= wne 2794   ` cfv 5888  (class class class)co 6650   CCcc 9934   0cc0 9936   1c1 9937   _ici 9938   x. cmul 9941   - cmin 10266   -ucneg 10267   / cdiv 10684   2c2 11070   expce 14792   sincsin 14794

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-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-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  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-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-ico 12181  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  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

This theorem is referenced by:  resinhcl  14886  tanhlt1  14890  sinhpcosh  42481