Theorem efival 12292

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 8126	. . . . . 6 |- _i e. CC
2		mulcl 8158	. . . . . 6 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
3	1, 2	mpan 424	. . . . 5 |- ( A e. CC -> ( _i x. A ) e. CC )
4		efcl 12224	. . . . 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 8379	. . . . . 6 |- -u _i e. CC
7		mulcl 8158	. . . . . 6 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
8	6, 7	mpan 424	. . . . 5 |- ( A e. CC -> ( -u _i x. A ) e. CC )
9		efcl 12224	. . . . 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 8198	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) e. CC )
12	5, 10	subcld 8489	. . 3 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
13		2cn 9213	. . . . 5 |- 2 e. CC
14		2ap0 9235	. . . . 5 |- 2 # 0
15	13, 14	pm3.2i 272	. . . 4 |- ( 2 e. CC /\ 2 # 0 )
16		divdirap 8876	. . . 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 1362	. . 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 411	. 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 8492	. . . . . 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 6033	. . . . 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 8201	. . . . 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 9386	. . . . 5 |- ( A e. CC -> ( 2 x. ( exp ` ( _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) + ( exp ` ( _i x. A ) ) ) )
23	20, 21, 22	3eqtr4d 2274	. . . 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 6032	. . 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 8877	. . . . 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 1365	. . . 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 2265	. 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 12263	. . 3 |- ( A e. CC -> ( cos ` A ) = ( ( ( exp ` ( _i x. A ) ) + ( exp ` ( -u _i x. A ) ) ) / 2 ) )
30		2mulicn 9365	. . . . . . 7 |- ( 2 x. _i ) e. CC
31		2muliap0 9367	. . . . . . 7 |- ( 2 x. _i ) # 0
32	30, 31	pm3.2i 272	. . . . . 6 |- ( ( 2 x. _i ) e. CC /\ ( 2 x. _i ) # 0 )
33		div12ap 8873	. . . . . 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 1364	. . . . 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 12262	. . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
37	36	oveq2d 6033	. . . 4 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( _i x. ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) ) )
38		divrecap 8867	. . . . . . 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 1365	. . . . . 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 8181	. . . . . . . 8 |- ( 1 x. _i ) = _i
42	41	oveq1i 6027	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( _i / ( 2 x. _i ) )
43		iap0 9366	. . . . . . . . . . 11 |- _i # 0
44	1, 43	dividapi 8924	. . . . . . . . . 10 |- ( _i / _i ) = 1
45	44	oveq2i 6028	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 / 2 ) x. 1 )
46		ax-1cn 8124	. . . . . . . . . 10 |- 1 e. CC
47	46, 13, 1, 1, 14, 43	divmuldivapi 8951	. . . . . . . . 9 |- ( ( 1 / 2 ) x. ( _i / _i ) ) = ( ( 1 x. _i ) / ( 2 x. _i ) )
48	45, 47	eqtr3i 2254	. . . . . . . 8 |- ( ( 1 / 2 ) x. 1 ) = ( ( 1 x. _i ) / ( 2 x. _i ) )
49		halfcn 9357	. . . . . . . . 9 |- ( 1 / 2 ) e. CC
50	49	mulridi 8180	. . . . . . . 8 |- ( ( 1 / 2 ) x. 1 ) = ( 1 / 2 )
51	48, 50	eqtr3i 2254	. . . . . . 7 |- ( ( 1 x. _i ) / ( 2 x. _i ) ) = ( 1 / 2 )
52	42, 51	eqtr3i 2254	. . . . . 6 |- ( _i / ( 2 x. _i ) ) = ( 1 / 2 )
53	52	oveq2i 6028	. . . . 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 2282	. . . 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 2274	. . 3 |- ( A e. CC -> ( _i x. ( sin ` A ) ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / 2 ) )
56	29, 55	oveq12d 6035	. 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 2274	1 |- ( A e. CC -> ( exp ` ( _i x. A ) ) = ( ( cos ` A ) + ( _i x. ( sin ` A ) ) ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1397   e. wcel 2202   class class class wbr 4088   ` cfv 5326  (class class class)co 6017   CCcc 8029   0cc0 8031   1c1 8032   _ici 8033   + caddc 8034   x. cmul 8036   - cmin 8349   -ucneg 8350   # cap 8760   / cdiv 8851   2c2 9193   expce 12202   sincsin 12204   cosccos 12205

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8122  ax-resscn 8123  ax-1cn 8124  ax-1re 8125  ax-icn 8126  ax-addcl 8127  ax-addrcl 8128  ax-mulcl 8129  ax-mulrcl 8130  ax-addcom 8131  ax-mulcom 8132  ax-addass 8133  ax-mulass 8134  ax-distr 8135  ax-i2m1 8136  ax-0lt1 8137  ax-1rid 8138  ax-0id 8139  ax-rnegex 8140  ax-precex 8141  ax-cnre 8142  ax-pre-ltirr 8143  ax-pre-ltwlin 8144  ax-pre-lttrn 8145  ax-pre-apti 8146  ax-pre-ltadd 8147  ax-pre-mulgt0 8148  ax-pre-mulext 8149  ax-arch 8150  ax-caucvg 8151

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-isom 5335  df-riota 5970  df-ov 6020  df-oprab 6021  df-mpo 6022  df-1st 6302  df-2nd 6303  df-recs 6470  df-irdg 6535  df-frec 6556  df-1o 6581  df-oadd 6585  df-er 6701  df-en 6909  df-dom 6910  df-fin 6911  df-pnf 8215  df-mnf 8216  df-xr 8217  df-ltxr 8218  df-le 8219  df-sub 8351  df-neg 8352  df-reap 8754  df-ap 8761  df-div 8852  df-inn 9143  df-2 9201  df-3 9202  df-4 9203  df-n0 9402  df-z 9479  df-uz 9755  df-q 9853  df-rp 9888  df-ico 10128  df-fz 10243  df-fzo 10377  df-seqfrec 10709  df-exp 10800  df-fac 10987  df-ihash 11037  df-cj 11402  df-re 11403  df-im 11404  df-rsqrt 11558  df-abs 11559  df-clim 11839  df-sumdc 11914  df-ef 12208  df-sin 12210  df-cos 12211

This theorem is referenced by:  efmival  12293  efeul  12294  efieq  12295  sinadd  12296  cosadd  12297  absefi  12329  demoivre  12333  efhalfpi  15522  efipi  15524  ef2pi  15528  efimpi  15542