Theorem sinperlem 13523

Description: Lemma for sinper 13524 and cosper 13525. (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 9217	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
2		2cn 8949	. . . . . . . . . 10 |- 2 e. CC
3		picn 13502	. . . . . . . . . 10 |- pi e. CC
4	2, 3	mulcli 7925	. . . . . . . . 9 |- ( 2 x. pi ) e. CC
5		mulcl 7901	. . . . . . . . 9 |- ( ( K e. CC /\ ( 2 x. pi ) e. CC ) -> ( K x. ( 2 x. pi ) ) e. CC )
6	1, 4, 5	sylancl 411	. . . . . . . 8 |- ( K e. ZZ -> ( K x. ( 2 x. pi ) ) e. CC )
7		ax-icn 7869	. . . . . . . . 9 |- _i e. CC
8		adddi 7906	. . . . . . . . 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 1319	. . . . . . . 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 284	. . . . . . 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 8048	. . . . . . . . . . . 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 1323	. . . . . . . . . . 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 7925	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
15		mulcom 7903	. . . . . . . . . . 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 411	. . . . . . . . . 10 |- ( K e. ZZ -> ( K x. ( _i x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
17	13, 16	eqtrd 2203	. . . . . . . . 9 |- ( K e. ZZ -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
18	17	adantl 275	. . . . . . . 8 |- ( ( A e. CC /\ K e. ZZ ) -> ( _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
19	18	oveq2d 5869	. . . . . . 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 2203	. . . . . 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 5500	. . . . 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 7901	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
23	7, 22	mpan 422	. . . . . 6 |- ( A e. CC -> ( _i x. A ) e. CC )
24		efper 13522	. . . . . 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 281	. . . . 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 2203	. . . 4 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( _i x. A ) ) )
27		negicn 8120	. . . . . . . . 9 |- -u _i e. CC
28		adddi 7906	. . . . . . . . 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 1319	. . . . . . . 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 284	. . . . . . 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 8114	. . . . . . . . . 10 |- ( K e. ZZ -> -u ( _i x. ( K x. ( 2 x. pi ) ) ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
32		mulneg1 8314	. . . . . . . . . . 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 412	. . . . . . . . . 10 |- ( K e. ZZ -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = -u ( _i x. ( K x. ( 2 x. pi ) ) ) )
34		mulneg2 8315	. . . . . . . . . . 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 412	. . . . . . . . . 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 2213	. . . . . . . . 9 |- ( K e. ZZ -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. -u K ) )
37	36	adantl 275	. . . . . . . 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 5869	. . . . . . 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 2203	. . . . . 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 5500	. . . . 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 7901	. . . . . . 7 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
42	27, 41	mpan 422	. . . . . 6 |- ( A e. CC -> ( -u _i x. A ) e. CC )
43		znegcl 9243	. . . . . 6 |- ( K e. ZZ -> -u K e. ZZ )
44		efper 13522	. . . . . 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 287	. . . . 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 2203	. . . 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 5871	. . 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 5868	. 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 7899	. . . 4 |- ( ( A e. CC /\ ( K x. ( 2 x. pi ) ) e. CC ) -> ( A + ( K x. ( 2 x. pi ) ) ) e. CC )
50	6, 49	sylan2 284	. . 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 274	. 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 2213	1 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1348   e. wcel 2141   ` cfv 5198  (class class class)co 5853   CCcc 7772   _ici 7776   + caddc 7777   x. cmul 7779   -ucneg 8091   / cdiv 8589   2c2 8929   ZZcz 9212   expce 11605   picpi 11610

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894  ax-pre-suploc 7895  ax-addf 7896  ax-mulf 7897

This theorem depends on definitions:  df-bi 116  df-stab 826  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-disj 3967  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-of 6061  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-map 6628  df-pm 6629  df-en 6719  df-dom 6720  df-fin 6721  df-sup 6961  df-inf 6962  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-5 8940  df-6 8941  df-7 8942  df-8 8943  df-9 8944  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-xneg 9729  df-xadd 9730  df-ioo 9849  df-ioc 9850  df-ico 9851  df-icc 9852  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-bc 10682  df-ihash 10710  df-shft 10779  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-sin 11613  df-cos 11614  df-pi 11616  df-rest 12581  df-topgen 12600  df-psmet 12781  df-xmet 12782  df-met 12783  df-bl 12784  df-mopn 12785  df-top 12790  df-topon 12803  df-bases 12835  df-ntr 12890  df-cn 12982  df-cnp 12983  df-tx 13047  df-cncf 13352  df-limced 13419  df-dvap 13420

This theorem is referenced by:  sinper  13524  cosper  13525