Theorem efival 13764

Description: The exponential function in terms of sine and cosine. (Contributed by NM, 30-Apr-2005.)

Assertion

Ref	Expression
efival	|- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) )

Proof of Theorem efival

Step	Hyp	Ref	Expression
1		ax-icn 9554	. . . . . 6 |- _i e. CC
2		mulcl 9579	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 670	. . . . 5 |- ( A e. CC -> ( _i x. A ) e. CC )
4		efcl 13696	. . . . 5 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
5	3, 4	syl 16	. . . 4 |- ( A e. CC -> ( exp ` ( _i x. A ) ) e. CC )
6		negicn 9826	. . . . . 6 |- -u _i e. CC
7		mulcl 9579	. . . . . 6 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
8	6, 7	mpan 670	. . . . 5 |- ( A e. CC -> ( -u _i x. A ) e. CC )
9		efcl 13696	. . . . 5 |- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
10	8, 9	syl 16	. . . 4 |- ( A e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
11	5, 10	addcld 9618	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC )
12	5, 10	subcld 9936	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
13		2cn 10612	. . . . 5 |- 2 e. CC
14		2ne0 10634	. . . . 5 |- 2 =/= 0
15	13, 14	pm3.2i 455	. . . 4 |- ( 2 e. CC /\ 2 =/= 0 )
16		divdir 10236	. . . 4 |- ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( 2 e. CC /\ 2 =/= 0 ) ) -> ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / 2 ) = ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) + ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) ) )
17	15, 16	mp3an3 1314	. . 3 |- ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC ) -> ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / 2 ) = ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) + ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) ) )
18	11, 12, 17	syl2anc 661	. 2 |- ( A e. CC -> ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / 2 ) = ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) + ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) ) )
19	10, 5	pncan3d 9939	. . . . . 6 |- ( A e. CC -> ( ( exp ` ( -u _i x. A ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) = ( exp ` ( _i x. A ) ) )
20	19	oveq2d 6297	. . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) + ( ( exp ` ( -u _i x. A ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) ) = ( ( exp ` ( _i x. A ) ) + ( exp ` ( _i x. A ) ) ) )
21	5, 10, 12	addassd 9621	. . . . 5 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) = ( ( exp ` ( _i x. A ) ) + ( ( exp ` ( -u _i x. A ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) ) )
22	5	2timesd 10787	. . . . 5 |- ( A e. CC -> ( 2 x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) + ( exp ` ( _i x. A ) ) ) )
23	20, 21, 22	3eqtr4d 2494	. . . 4 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) = ( 2 x. ( exp ` ( _i x. A ) ) ) )
24	23	oveq1d 6296	. . 3 |- ( A e. CC -> ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / 2 ) = ( ( 2 x. ( exp ` ( _i x. A ) ) ) / 2 ) )
25		divcan3 10237	. . . . 5 |- ( ( ( exp ` ( _i x. A ) ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( 2 x. ( exp ` ( _i x. A ) ) ) / 2 ) = ( exp ` ( _i x. A ) ) )
26	13, 14, 25	mp3an23 1317	. . . 4 |- ( ( exp ` ( _i x. A ) ) e. CC -> ( ( 2 x. ( exp ` ( _i x. A ) ) ) / 2 ) = ( exp ` ( _i x. A ) ) )
27	5, 26	syl 16	. . 3 |- ( A e. CC -> ( ( 2 x. ( exp ` ( _i x. A ) ) ) / 2 ) = ( exp ` ( _i x. A ) ) )
28	24, 27	eqtr2d 2485	. 2 |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) + ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) ) / 2 ) )
29		cosval 13735	. . 3 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
30		2mulicn 10768	. . . . . . 7 |- ( 2 x. _i ) e. CC
31		2muline0 10769	. . . . . . 7 |- ( 2 x. _i ) =/= 0
32	30, 31	pm3.2i 455	. . . . . 6 |- ( ( 2 x. _i ) e. CC /\ ( 2 x. _i ) =/= 0 )
33		div12 10235	. . . . . 6 |- ( ( _i e. CC /\ ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( ( 2 x. _i ) e. CC /\ ( 2 x. _i ) =/= 0 ) ) -> ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( _i / ( 2 x. _i ) ) ) )
34	1, 32, 33	mp3an13 1316	. . . . 5 |- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC -> ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( _i / ( 2 x. _i ) ) ) )
35	12, 34	syl 16	. . . 4 |- ( A e. CC -> ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( _i / ( 2 x. _i ) ) ) )
36		sinval 13734	. . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
37	36	oveq2d 6297	. . . 4 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) )
38		divrec 10229	. . . . . . 7 |- ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC /\ 2 e. CC /\ 2 =/= 0 ) -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( 1 / 2 ) ) )
39	13, 14, 38	mp3an23 1317	. . . . . 6 |- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( 1 / 2 ) ) )
40	12, 39	syl 16	. . . . 5 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( 1 / 2 ) ) )
41	1	mulid2i 9602	. . . . . . . 8 |- ( 1 x. _i ) = _i
42	41	oveq1i 6291	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( _i / ( 2 x. _i ) )
43		ine0 9998	. . . . . . . . . . 11 |- _i =/= 0
44	1, 43	dividi 10283	. . . . . . . . . 10 |- ( _i / _i ) = 1
45	44	oveq2i 6292	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 / 2 ) x. 1 )
46		ax-1cn 9553	. . . . . . . . . 10 |- 1 e. CC
47	46, 13, 1, 1, 14, 43	divmuldivi 10310	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 x. _i ) / ( 2 x. _i ) )
48	45, 47	eqtr3i 2474	. . . . . . . 8 |- ( ( 1 / 2 ) x. 1 ) = ( ( 1 x. _i ) / ( 2 x. _i ) )
49		halfcn 10761	. . . . . . . . 9 |- ( 1 / 2 ) e. CC
50	49	mulid1i 9601	. . . . . . . 8 |- ( ( 1 / 2 ) x. 1 ) = ( 1 / 2 )
51	48, 50	eqtr3i 2474	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( 1 / 2 )
52	42, 51	eqtr3i 2474	. . . . . 6 |- ( _i / ( 2 x. _i ) ) = ( 1 / 2 )
53	52	oveq2i 6292	. . . . 5 |- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( _i / ( 2 x. _i ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( 1 / 2 ) )
54	40, 53	syl6eqr 2502	. . . 4 |- ( A e. CC -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) x. ( _i / ( 2 x. _i ) ) ) )
55	35, 37, 54	3eqtr4d 2494	. . 3 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) )
56	29, 55	oveq12d 6299	. 2 |- ( A e. CC -> ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) = ( ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) + ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) ) )
57	18, 28, 56	3eqtr4d 2494	1 |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 369   = wceq 1383   e. wcel 1804   =/= wne 2638   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496   _ici 9497   + caddc 9498   x. cmul 9500   - cmin 9810   -ucneg 9811   / cdiv 10212   2c2 10591   expce 13675   sincsin 13677   cosccos 13678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-inf2 8061  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-pre-sup 9573  ax-addf 9574  ax-mulf 9575

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-fal 1389  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-se 4829  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-isom 5587  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-pm 7425  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-sup 7903  df-oi 7938  df-card 8323  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10213  df-nn 10543  df-2 10600  df-3 10601  df-n0 10802  df-z 10871  df-uz 11091  df-rp 11230  df-ico 11544  df-fz 11682  df-fzo 11804  df-fl 11908  df-seq 12087  df-exp 12146  df-fac 12333  df-hash 12385  df-shft 12879  df-cj 12911  df-re 12912  df-im 12913  df-sqrt 13047  df-abs 13048  df-limsup 13273  df-clim 13290  df-rlim 13291  df-sum 13488  df-ef 13681  df-sin 13683  df-cos 13684

This theorem is referenced by:  efmival  13765  efeul  13774  efieq  13775  sinadd  13776  cosadd  13777  absefi  13808  demoivre  13812  efhalfpi  22736  efipi  22738  ef2pi  22742  efimpi  22756  efif1olem4  22804  1cubrlem  23044  asinsin  23095  atantan  23126