Theorem efival 12518

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 8275	. . . . . 6 |- _i e. CC
2		mulcl 8307	. . . . . 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 12450	. . . . 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 8529	. . . . . 6 |- -u _i e. CC
7		mulcl 8307	. . . . . 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 12450	. . . . 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 8346	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC )
12	5, 10	subcld 8639	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
13		2cn 9378	. . . . 5 |- 2 e. CC
14		2ap0 9400	. . . . 5 |- 2 # 0
15	13, 14	pm3.2i 272	. . . 4 |- ( 2 e. CC /\ 2 # 0 )
16		divdirap 9030	. . . 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 8642	. . . . . 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 6101	. . . . 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 8349	. . . . 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 9553	. . . . 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 6100	. . 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 9031	. . . . 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 12489	. . 3 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
30		2mulicn 9532	. . . . . . 7 |- ( 2 x. _i ) e. CC
31		2muliap0 9534	. . . . . . 7 |- ( 2 x. _i ) # 0
32	30, 31	pm3.2i 272	. . . . . 6 |- ( ( 2 x. _i ) e. CC /\ ( 2 x. _i ) # 0 )
33		div12ap 9027	. . . . . 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 12488	. . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
37	36	oveq2d 6101	. . . 4 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) )
38		divrecap 9021	. . . . . . 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 8330	. . . . . . . 8 |- ( 1 x. _i ) = _i
42	41	oveq1i 6095	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( _i / ( 2 x. _i ) )
43		iap0 9533	. . . . . . . . . . 11 |- _i # 0
44	1, 43	dividapi 9078	. . . . . . . . . 10 |- ( _i / _i ) = 1
45	44	oveq2i 6096	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 / 2 ) x. 1 )
46		ax-1cn 8273	. . . . . . . . . 10 |- 1 e. CC
47	46, 13, 1, 1, 14, 43	divmuldivapi 9105	. . . . . . . . 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 9524	. . . . . . . . 9 |- ( 1 / 2 ) e. CC
50	49	mulridi 8329	. . . . . . . 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 6096	. . . . 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 6103	. 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 4130   ` cfv 5377  (class class class)co 6085   CCcc 8178   0cc0 8180   1c1 8181   _ici 8182   + caddc 8183   x. cmul 8185   - cmin 8499   -ucneg 8500   # cap 8912   / cdiv 9005   2c2 9358   expce 12428   sincsin 12430   cosccos 12431

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8271  ax-resscn 8272  ax-1cn 8273  ax-1re 8274  ax-icn 8275  ax-addcl 8276  ax-addrcl 8277  ax-mulcl 8278  ax-mulrcl 8279  ax-addcom 8280  ax-mulcom 8281  ax-addass 8282  ax-mulass 8283  ax-distr 8284  ax-i2m1 8285  ax-0lt1 8286  ax-1rid 8287  ax-0id 8288  ax-rnegex 8289  ax-precex 8290  ax-cnre 8291  ax-pre-ltirr 8292  ax-pre-ltwlin 8293  ax-pre-lttrn 8294  ax-pre-apti 8295  ax-pre-ltadd 8296  ax-pre-mulgt0 8297  ax-pre-mulext 8298  ax-arch 8299  ax-caucvg 8300

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 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  df-fin 7025  df-pnf 8363  df-mnf 8364  df-xr 8365  df-ltxr 8366  df-le 8367  df-sub 8501  df-neg 8502  df-reap 8906  df-ap 8913  df-div 9006  df-inn 9308  df-2 9366  df-3 9367  df-4 9368  df-n0 9569  df-z 9650  df-uz 9932  df-q 10030  df-rp 10066  df-ico 10307  df-fz 10423  df-fzo 10561  df-seqfrec 10900  df-exp 10991  df-fac 11180  df-ihash 11231  df-cj 11623  df-re 11624  df-im 11625  df-rsqrt 11780  df-abs 11781  df-clim 12064  df-sumdc 12139  df-ef 12434  df-sin 12436  df-cos 12437

This theorem is used by:  efmival  12519  efeul  12520  efieq  12521  sinadd  12522  cosadd  12523  absefi  12555  demoivre  12559  efhalfpi  15992  efipi  15994  ef2pi  15998  efimpi  16012