Theorem efival 12482

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 8268	. . . . . 6 |- _i e. CC
2		mulcl 8300	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 428	. . . . 5 |- ( A e. CC -> ( _i x. A ) e. CC )
4		efcl 12414	. . . . 5 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
5	3, 4	syl 14	. . . 4 |- ( A e. CC -> ( exp ` ( _i x. A ) ) e. CC )
6		negicn 8521	. . . . . 6 |- -u _i e. CC
7		mulcl 8300	. . . . . 6 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
8	6, 7	mpan 428	. . . . 5 |- ( A e. CC -> ( -u _i x. A ) e. CC )
9		efcl 12414	. . . . 5 |- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
10	8, 9	syl 14	. . . 4 |- ( A e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
11	5, 10	addcld 8339	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC )
12	5, 10	subcld 8631	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
13		2cn 9358	. . . . 5 |- 2 e. CC
14		2ap0 9380	. . . . 5 |- 2 # 0
15	13, 14	pm3.2i 272	. . . 4 |- ( 2 e. CC /\ 2 # 0 )
16		divdirap 9021	. . . 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 1367	. . 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 415	. 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 8634	. . . . . 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 6095	. . . . 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 8342	. . . . 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 9531	. . . . 5 |- ( A e. CC -> ( 2 x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) + ( exp ` ( _i x. A ) ) ) )
23	20, 21, 22	3eqtr4d 2281	. . . 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 6094	. . 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		divcanap3 9022	. . . . 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 1370	. . . 4 |- ( ( exp ` ( _i x. A ) ) e. CC -> ( ( 2 x. ( exp ` ( _i x. A ) ) ) / 2 ) = ( exp ` ( _i x. A ) ) )
27	5, 26	syl 14	. . 3 |- ( A e. CC -> ( ( 2 x. ( exp ` ( _i x. A ) ) ) / 2 ) = ( exp ` ( _i x. A ) ) )
28	24, 27	eqtr2d 2272	. 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 12453	. . 3 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
30		2mulicn 9510	. . . . . . 7 |- ( 2 x. _i ) e. CC
31		2muliap0 9512	. . . . . . 7 |- ( 2 x. _i ) # 0
32	30, 31	pm3.2i 272	. . . . . 6 |- ( ( 2 x. _i ) e. CC /\ ( 2 x. _i ) # 0 )
33		div12ap 9018	. . . . . 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 1369	. . . . 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 14	. . . 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 12452	. . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
37	36	oveq2d 6095	. . . 4 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) )
38		divrecap 9012	. . . . . . 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 1370	. . . . . 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 14	. . . . 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	mullidi 8323	. . . . . . . 8 |- ( 1 x. _i ) = _i
42	41	oveq1i 6089	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( _i / ( 2 x. _i ) )
43		iap0 9511	. . . . . . . . . . 11 |- _i # 0
44	1, 43	dividapi 9069	. . . . . . . . . 10 |- ( _i / _i ) = 1
45	44	oveq2i 6090	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 / 2 ) x. 1 )
46		ax-1cn 8266	. . . . . . . . . 10 |- 1 e. CC
47	46, 13, 1, 1, 14, 43	divmuldivapi 9096	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 x. _i ) / ( 2 x. _i ) )
48	45, 47	eqtr3i 2261	. . . . . . . 8 |- ( ( 1 / 2 ) x. 1 ) = ( ( 1 x. _i ) / ( 2 x. _i ) )
49		halfcn 9502	. . . . . . . . 9 |- ( 1 / 2 ) e. CC
50	49	mulridi 8322	. . . . . . . 8 |- ( ( 1 / 2 ) x. 1 ) = ( 1 / 2 )
51	48, 50	eqtr3i 2261	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( 1 / 2 )
52	42, 51	eqtr3i 2261	. . . . . 6 |- ( _i / ( 2 x. _i ) ) = ( 1 / 2 )
53	52	oveq2i 6090	. . . . 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	eqtr4di 2289	. . . 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 2281	. . 3 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) )
56	29, 55	oveq12d 6097	. 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 2281	1 |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4128   ` cfv 5375  (class class class)co 6079   CCcc 8171   0cc0 8173   1c1 8174   _ici 8175   + caddc 8176   x. cmul 8178   - cmin 8491   -ucneg 8492   # cap 8903   / cdiv 8996   2c2 9338   expce 12392   sincsin 12394   cosccos 12395

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291  ax-arch 8292  ax-caucvg 8293

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-isom 5384  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-irdg 6635  df-frec 6656  df-1o 6681  df-oadd 6685  df-er 6801  df-en 7017  df-dom 7018  df-fin 7019  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-2 9346  df-3 9347  df-4 9348  df-n0 9547  df-z 9628  df-uz 9905  df-q 10003  df-rp 10038  df-ico 10279  df-fz 10395  df-fzo 10533  df-seqfrec 10868  df-exp 10959  df-fac 11147  df-ihash 11198  df-cj 11590  df-re 11591  df-im 11592  df-rsqrt 11747  df-abs 11748  df-clim 12028  df-sumdc 12103  df-ef 12398  df-sin 12400  df-cos 12401

This theorem is used by:  efmival  12483  efeul  12484  efieq  12485  sinadd  12486  cosadd  12487  absefi  12519  demoivre  12523  efhalfpi  15883  efipi  15885  ef2pi  15889  efimpi  15903