Theorem sinperlem 15128

Description: Lemma for sinper 15129 and cosper 15130. (Contributed by Paul Chapman, 23-Jan-2008.) (Revised by Mario Carneiro, 10-May-2014.)

Hypotheses

Ref	Expression
sinperlem.1	|- ( A e. CC -> ( F ` A ) = ( ( ( exp ` ( _i x. A ) ) O ( exp ` ( -u _i x. A ) ) ) / D ) )
sinperlem.2	|- ( ( A + ( K x. ( 2 x. pi ) ) ) e. CC -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) O ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) ) / D ) )

Assertion

Ref	Expression
sinperlem	|- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )

Proof of Theorem sinperlem

Step	Hyp	Ref	Expression
1		zcn 9348	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
2		2cn 9078	. . . . . . . . . 10 |- 2 e. CC
3		picn 15107	. . . . . . . . . 10 |- pi e. CC
4	2, 3	mulcli 8048	. . . . . . . . 9 |- ( 2 x. pi ) e. CC
5		mulcl 8023	. . . . . . . . 9 |- ( ( K e. CC /\ ( 2 x. pi ) e. CC ) -> ( K x. ( 2 x. pi ) ) e. CC )
6	1, 4, 5	sylancl 413	. . . . . . . 8 |- ( K e. ZZ -> ( K x. ( 2 x. pi ) ) e. CC )
7		ax-icn 7991	. . . . . . . . 9 |- _i e. CC
8		adddi 8028	. . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( _i x. A ) + ( _i x. ( K x. ( 2 x. pi ) ) ) ) )
9	7, 8	mp3an1 1335	. . . . . . . 8 |- ( ( A e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( _i x. A ) + ( _i x. ( K x. ( 2 x. pi ) ) ) ) )
10	6, 9	sylan2 286	. . . . . . 7 |- ( ( A e. CC /\ K e. ZZ ) -> ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( _i x. A ) + ( _i x. ( K x. ( 2 x. pi ) ) ) ) )
11		mul12 8172	. . . . . . . . . . . 12 |- ( ( _i e. CC /\ K e. CC /\ ( 2 x. pi ) e. CC ) -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( K x. ( _i x. ( 2 x. pi ) ) ) )
12	7, 4, 11	mp3an13 1339	. . . . . . . . . . 11 |- ( K e. CC -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( K x. ( _i x. ( 2 x. pi ) ) ) )
13	1, 12	syl 14	. . . . . . . . . 10 |- ( K e. ZZ -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( K x. ( _i x. ( 2 x. pi ) ) ) )
14	7, 4	mulcli 8048	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
15		mulcom 8025	. . . . . . . . . . 11 |- ( ( K e. CC /\ ( _i x. ( 2 x. pi ) ) e. CC ) -> ( K x. ( _i x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
16	1, 14, 15	sylancl 413	. . . . . . . . . 10 |- ( K e. ZZ -> ( K x. ( _i x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
17	13, 16	eqtrd 2229	. . . . . . . . 9 |- ( K e. ZZ -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
18	17	adantl 277	. . . . . . . 8 |- ( ( A e. CC /\ K e. ZZ ) -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
19	18	oveq2d 5941	. . . . . . 7 |- ( ( A e. CC /\ K e. ZZ ) -> ( ( _i x. A ) + ( _i x. ( K x. ( 2 x. pi ) ) ) ) = ( ( _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. K ) ) )
20	10, 19	eqtrd 2229	. . . . . 6 |- ( ( A e. CC /\ K e. ZZ ) -> ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. K ) ) )
21	20	fveq2d 5565	. . . . 5 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( ( _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. K ) ) ) )
22		mulcl 8023	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
23	7, 22	mpan 424	. . . . . 6 |- ( A e. CC -> ( _i x. A ) e. CC )
24		efper 15127	. . . . . 6 |- ( ( ( _i x. A ) e. CC /\ K e. ZZ ) -> ( exp ` ( ( _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. K ) ) ) = ( exp ` ( _i x. A ) ) )
25	23, 24	sylan 283	. . . . 5 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( ( _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. K ) ) ) = ( exp ` ( _i x. A ) ) )
26	21, 25	eqtrd 2229	. . . 4 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( _i x. A ) ) )
27		negicn 8244	. . . . . . . . 9 |- -u _i e. CC
28		adddi 8028	. . . . . . . . 9 |- ( ( -u _i e. CC /\ A e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( -u _i x. A ) + ( -u _i x. ( K x. ( 2 x. pi ) ) ) ) )
29	27, 28	mp3an1 1335	. . . . . . . 8 |- ( ( A e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( -u _i x. A ) + ( -u _i x. ( K x. ( 2 x. pi ) ) ) ) )
30	6, 29	sylan2 286	. . . . . . 7 |- ( ( A e. CC /\ K e. ZZ ) -> ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( -u _i x. A ) + ( -u _i x. ( K x. ( 2 x. pi ) ) ) ) )
31	17	negeqd 8238	. . . . . . . . . 10 |- ( K e. ZZ -> -u ( _i x. ( K x. ( 2 x. pi ) ) ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
32		mulneg1 8438	. . . . . . . . . . 11 |- ( ( _i e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = -u ( _i x. ( K x. ( 2 x. pi ) ) ) )
33	7, 6, 32	sylancr 414	. . . . . . . . . 10 |- ( K e. ZZ -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = -u ( _i x. ( K x. ( 2 x. pi ) ) ) )
34		mulneg2 8439	. . . . . . . . . . 11 |- ( ( ( _i x. ( 2 x. pi ) ) e. CC /\ K e. CC ) -> ( ( _i x. ( 2 x. pi ) ) x. -u K ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
35	14, 1, 34	sylancr 414	. . . . . . . . . 10 |- ( K e. ZZ -> ( ( _i x. ( 2 x. pi ) ) x. -u K ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
36	31, 33, 35	3eqtr4d 2239	. . . . . . . . 9 |- ( K e. ZZ -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. -u K ) )
37	36	adantl 277	. . . . . . . 8 |- ( ( A e. CC /\ K e. ZZ ) -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. -u K ) )
38	37	oveq2d 5941	. . . . . . 7 |- ( ( A e. CC /\ K e. ZZ ) -> ( ( -u _i x. A ) + ( -u _i x. ( K x. ( 2 x. pi ) ) ) ) = ( ( -u _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. -u K ) ) )
39	30, 38	eqtrd 2229	. . . . . 6 |- ( ( A e. CC /\ K e. ZZ ) -> ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( -u _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. -u K ) ) )
40	39	fveq2d 5565	. . . . 5 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( ( -u _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. -u K ) ) ) )
41		mulcl 8023	. . . . . . 7 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
42	27, 41	mpan 424	. . . . . 6 |- ( A e. CC -> ( -u _i x. A ) e. CC )
43		znegcl 9374	. . . . . 6 |- ( K e. ZZ -> -u K e. ZZ )
44		efper 15127	. . . . . 6 |- ( ( ( -u _i x. A ) e. CC /\ -u K e. ZZ ) -> ( exp ` ( ( -u _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. -u K ) ) ) = ( exp ` ( -u _i x. A ) ) )
45	42, 43, 44	syl2an 289	. . . . 5 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( ( -u _i x. A ) + ( ( _i x. ( 2 x. pi ) ) x. -u K ) ) ) = ( exp ` ( -u _i x. A ) ) )
46	40, 45	eqtrd 2229	. . . 4 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( -u _i x. A ) ) )
47	26, 46	oveq12d 5943	. . 3 |- ( ( A e. CC /\ K e. ZZ ) -> ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) O ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) ) = ( ( exp ` ( _i x. A ) ) O ( exp ` ( -u _i x. A ) ) ) )
48	47	oveq1d 5940	. 2 |- ( ( A e. CC /\ K e. ZZ ) -> ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) O ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) ) / D ) = ( ( ( exp ` ( _i x. A ) ) O ( exp ` ( -u _i x. A ) ) ) / D ) )
49		addcl 8021	. . . 4 |- ( ( A e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( A + ( K x. ( 2 x. pi ) ) ) e. CC )
50	6, 49	sylan2 286	. . 3 |- ( ( A e. CC /\ K e. ZZ ) -> ( A + ( K x. ( 2 x. pi ) ) ) e. CC )
51		sinperlem.2	. . 3 |- ( ( A + ( K x. ( 2 x. pi ) ) ) e. CC -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) O ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) ) / D ) )
52	50, 51	syl 14	. 2 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( ( ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) O ( exp ` ( -u _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) ) / D ) )
53		sinperlem.1	. . 3 |- ( A e. CC -> ( F ` A ) = ( ( ( exp ` ( _i x. A ) ) O ( exp ` ( -u _i x. A ) ) ) / D ) )
54	53	adantr 276	. 2 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` A ) = ( ( ( exp ` ( _i x. A ) ) O ( exp ` ( -u _i x. A ) ) ) / D ) )
55	48, 52, 54	3eqtr4d 2239	1 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2167   ` cfv 5259  (class class class)co 5925   CCcc 7894   _ici 7898   + caddc 7899   x. cmul 7901   -ucneg 8215   / cdiv 8716   2c2 9058   ZZcz 9343   expce 11824   picpi 11829

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625  ax-cnex 7987  ax-resscn 7988  ax-1cn 7989  ax-1re 7990  ax-icn 7991  ax-addcl 7992  ax-addrcl 7993  ax-mulcl 7994  ax-mulrcl 7995  ax-addcom 7996  ax-mulcom 7997  ax-addass 7998  ax-mulass 7999  ax-distr 8000  ax-i2m1 8001  ax-0lt1 8002  ax-1rid 8003  ax-0id 8004  ax-rnegex 8005  ax-precex 8006  ax-cnre 8007  ax-pre-ltirr 8008  ax-pre-ltwlin 8009  ax-pre-lttrn 8010  ax-pre-apti 8011  ax-pre-ltadd 8012  ax-pre-mulgt0 8013  ax-pre-mulext 8014  ax-arch 8015  ax-caucvg 8016  ax-pre-suploc 8017  ax-addf 8018  ax-mulf 8019

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-disj 4012  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-po 4332  df-iso 4333  df-iord 4402  df-on 4404  df-ilim 4405  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-isom 5268  df-riota 5880  df-ov 5928  df-oprab 5929  df-mpo 5930  df-of 6139  df-1st 6207  df-2nd 6208  df-recs 6372  df-irdg 6437  df-frec 6458  df-1o 6483  df-oadd 6487  df-er 6601  df-map 6718  df-pm 6719  df-en 6809  df-dom 6810  df-fin 6811  df-sup 7059  df-inf 7060  df-pnf 8080  df-mnf 8081  df-xr 8082  df-ltxr 8083  df-le 8084  df-sub 8216  df-neg 8217  df-reap 8619  df-ap 8626  df-div 8717  df-inn 9008  df-2 9066  df-3 9067  df-4 9068  df-5 9069  df-6 9070  df-7 9071  df-8 9072  df-9 9073  df-n0 9267  df-z 9344  df-uz 9619  df-q 9711  df-rp 9746  df-xneg 9864  df-xadd 9865  df-ioo 9984  df-ioc 9985  df-ico 9986  df-icc 9987  df-fz 10101  df-fzo 10235  df-seqfrec 10557  df-exp 10648  df-fac 10835  df-bc 10857  df-ihash 10885  df-shft 10997  df-cj 11024  df-re 11025  df-im 11026  df-rsqrt 11180  df-abs 11181  df-clim 11461  df-sumdc 11536  df-ef 11830  df-sin 11832  df-cos 11833  df-pi 11835  df-rest 12943  df-topgen 12962  df-psmet 14175  df-xmet 14176  df-met 14177  df-bl 14178  df-mopn 14179  df-top 14318  df-topon 14331  df-bases 14363  df-ntr 14416  df-cn 14508  df-cnp 14509  df-tx 14573  df-cncf 14891  df-limced 14976  df-dvap 14977

This theorem is referenced by:  sinper  15129  cosper  15130