Theorem resinval 12401

Description: The sine of a real number in terms of the exponential function. (Contributed by NM, 30-Apr-2005.)

Assertion

Ref	Expression
resinval	|- ( A e. RR -> ( sin ` A ) = ( Im ` ( exp ` ( _i x. A ) ) ) )

Proof of Theorem resinval

Step	Hyp	Ref	Expression
1		ax-icn 8222	. . . . . . . 8 |- _i e. CC
2		recn 8260	. . . . . . . 8 |- ( A e. RR -> A e. CC )
3		cjmul 11570	. . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( * ` ( _i x. A ) ) = ( ( * ` _i ) x. ( * ` A ) ) )
4	1, 2, 3	sylancr 414	. . . . . . 7 |- ( A e. RR -> ( * ` ( _i x. A ) ) = ( ( * ` _i ) x. ( * ` A ) ) )
5		cji 11587	. . . . . . . . 9 |- ( * ` _i ) = -u _i
6	5	oveq1i 6060	. . . . . . . 8 |- ( ( * ` _i ) x. ( * ` A ) ) = ( -u _i x. ( * ` A ) )
7		cjre 11567	. . . . . . . . 9 |- ( A e. RR -> ( * ` A ) = A )
8	7	oveq2d 6066	. . . . . . . 8 |- ( A e. RR -> ( -u _i x. ( * ` A ) ) = ( -u _i x. A ) )
9	6, 8	eqtrid 2277	. . . . . . 7 |- ( A e. RR -> ( ( * ` _i ) x. ( * ` A ) ) = ( -u _i x. A ) )
10	4, 9	eqtrd 2265	. . . . . 6 |- ( A e. RR -> ( * ` ( _i x. A ) ) = ( -u _i x. A ) )
11	10	fveq2d 5674	. . . . 5 |- ( A e. RR -> ( exp ` ( * ` ( _i x. A ) ) ) = ( exp ` ( -u _i x. A ) ) )
12		mulcl 8254	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
13	1, 2, 12	sylancr 414	. . . . . 6 |- ( A e. RR -> ( _i x. A ) e. CC )
14		efcj 12359	. . . . . 6 |- ( ( _i x. A ) e. CC -> ( exp ` ( * ` ( _i x. A ) ) ) = ( * ` ( exp ` ( _i x. A ) ) ) )
15	13, 14	syl 14	. . . . 5 |- ( A e. RR -> ( exp ` ( * ` ( _i x. A ) ) ) = ( * ` ( exp ` ( _i x. A ) ) ) )
16	11, 15	eqtr3d 2267	. . . 4 |- ( A e. RR -> ( exp ` ( -u _i x. A ) ) = ( * ` ( exp ` ( _i x. A ) ) ) )
17	16	oveq2d 6066	. . 3 |- ( A e. RR -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = ( ( exp ` ( _i x. A ) ) - ( * ` ( exp ` ( _i x. A ) ) ) ) )
18	17	oveq1d 6065	. 2 |- ( A e. RR -> ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = ( ( ( exp ` ( _i x. A ) ) - ( * ` ( exp ` ( _i x. A ) ) ) ) / ( 2 x. _i ) ) )
19		sinval 12388	. . 3 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
20	2, 19	syl 14	. 2 |- ( A e. RR -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
21		efcl 12350	. . 3 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
22		imval2 11579	. . 3 |- ( ( exp ` ( _i x. A ) ) e. CC -> ( Im ` ( exp ` ( _i x. A ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( * ` ( exp ` ( _i x. A ) ) ) ) / ( 2 x. _i ) ) )
23	13, 21, 22	3syl 17	. 2 |- ( A e. RR -> ( Im ` ( exp ` ( _i x. A ) ) ) = ( ( ( exp ` ( _i x. A ) ) - ( * ` ( exp ` ( _i x. A ) ) ) ) / ( 2 x. _i ) ) )
24	18, 20, 23	3eqtr4d 2275	1 |- ( A e. RR -> ( sin ` A ) = ( Im ` ( exp ` ( _i x. A ) ) ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2203   ` cfv 5352  (class class class)co 6050   CCcc 8125   RRcr 8126   _ici 8129   x. cmul 8132   - cmin 8444   -ucneg 8445   / cdiv 8946   2c2 9288   *ccj 11524   Imcim 11526   expce 12328   sincsin 12330

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4225  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710  ax-cnex 8218  ax-resscn 8219  ax-1cn 8220  ax-1re 8221  ax-icn 8222  ax-addcl 8223  ax-addrcl 8224  ax-mulcl 8225  ax-mulrcl 8226  ax-addcom 8227  ax-mulcom 8228  ax-addass 8229  ax-mulass 8230  ax-distr 8231  ax-i2m1 8232  ax-0lt1 8233  ax-1rid 8234  ax-0id 8235  ax-rnegex 8236  ax-precex 8237  ax-cnre 8238  ax-pre-ltirr 8239  ax-pre-ltwlin 8240  ax-pre-lttrn 8241  ax-pre-apti 8242  ax-pre-ltadd 8243  ax-pre-mulgt0 8244  ax-pre-mulext 8245  ax-arch 8246  ax-caucvg 8247

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2815  df-sbc 3043  df-csb 3139  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-if 3621  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-iun 3993  df-br 4110  df-opab 4172  df-mpt 4173  df-tr 4209  df-id 4414  df-po 4417  df-iso 4418  df-iord 4487  df-on 4489  df-ilim 4490  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-isom 5361  df-riota 6003  df-ov 6053  df-oprab 6054  df-mpo 6055  df-1st 6334  df-2nd 6335  df-recs 6536  df-irdg 6601  df-frec 6622  df-1o 6647  df-oadd 6651  df-er 6767  df-en 6976  df-dom 6977  df-fin 6978  df-pnf 8310  df-mnf 8311  df-xr 8312  df-ltxr 8313  df-le 8314  df-sub 8446  df-neg 8447  df-reap 8849  df-ap 8856  df-div 8947  df-inn 9238  df-2 9296  df-3 9297  df-4 9298  df-n0 9497  df-z 9578  df-uz 9854  df-q 9952  df-rp 9987  df-ico 10227  df-fz 10343  df-fzo 10477  df-seqfrec 10810  df-exp 10901  df-fac 11088  df-ihash 11139  df-cj 11527  df-re 11528  df-im 11529  df-rsqrt 11683  df-abs 11684  df-clim 11964  df-sumdc 12039  df-ef 12334  df-sin 12336

This theorem is referenced by:  resin4p  12404  resincl  12406