Theorem sinperlem 14469

Description: Lemma for sinper 14470 and cosper 14471. (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 9271	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
2		2cn 9003	. . . . . . . . . 10 |- 2 e. CC
3		picn 14448	. . . . . . . . . 10 |- pi e. CC
4	2, 3	mulcli 7975	. . . . . . . . 9 |- ( 2 x. pi ) e. CC
5		mulcl 7951	. . . . . . . . 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 7919	. . . . . . . . 9 |- _i e. CC
8		adddi 7956	. . . . . . . . 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 1334	. . . . . . . 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 8099	. . . . . . . . . . . 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 1338	. . . . . . . . . . 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 7975	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
15		mulcom 7953	. . . . . . . . . . 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 2220	. . . . . . . . 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 5904	. . . . . . 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 2220	. . . . . 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 5531	. . . . 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 7951	. . . . . . 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 14468	. . . . . 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 2220	. . . 4 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( _i x. A ) ) )
27		negicn 8171	. . . . . . . . 9 |- -u _i e. CC
28		adddi 7956	. . . . . . . . 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 1334	. . . . . . . 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 8165	. . . . . . . . . 10 |- ( K e. ZZ -> -u ( _i x. ( K x. ( 2 x. pi ) ) ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
32		mulneg1 8365	. . . . . . . . . . 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 8366	. . . . . . . . . . 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 2230	. . . . . . . . 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 5904	. . . . . . 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 2220	. . . . . 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 5531	. . . . 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 7951	. . . . . . 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 9297	. . . . . 6 |- ( K e. ZZ -> -u K e. ZZ )
44		efper 14468	. . . . . 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 2220	. . . 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 5906	. . 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 5903	. 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 7949	. . . 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 2230	1 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   CCcc 7822   _ici 7826   + caddc 7827   x. cmul 7829   -ucneg 8142   / cdiv 8642   2c2 8983   ZZcz 9266   expce 11663   picpi 11668

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7915  ax-resscn 7916  ax-1cn 7917  ax-1re 7918  ax-icn 7919  ax-addcl 7920  ax-addrcl 7921  ax-mulcl 7922  ax-mulrcl 7923  ax-addcom 7924  ax-mulcom 7925  ax-addass 7926  ax-mulass 7927  ax-distr 7928  ax-i2m1 7929  ax-0lt1 7930  ax-1rid 7931  ax-0id 7932  ax-rnegex 7933  ax-precex 7934  ax-cnre 7935  ax-pre-ltirr 7936  ax-pre-ltwlin 7937  ax-pre-lttrn 7938  ax-pre-apti 7939  ax-pre-ltadd 7940  ax-pre-mulgt0 7941  ax-pre-mulext 7942  ax-arch 7943  ax-caucvg 7944  ax-pre-suploc 7945  ax-addf 7946  ax-mulf 7947

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-disj 3993  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-of 6096  df-1st 6154  df-2nd 6155  df-recs 6319  df-irdg 6384  df-frec 6405  df-1o 6430  df-oadd 6434  df-er 6548  df-map 6663  df-pm 6664  df-en 6754  df-dom 6755  df-fin 6756  df-sup 6996  df-inf 6997  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011  df-sub 8143  df-neg 8144  df-reap 8545  df-ap 8552  df-div 8643  df-inn 8933  df-2 8991  df-3 8992  df-4 8993  df-5 8994  df-6 8995  df-7 8996  df-8 8997  df-9 8998  df-n0 9190  df-z 9267  df-uz 9542  df-q 9633  df-rp 9667  df-xneg 9785  df-xadd 9786  df-ioo 9905  df-ioc 9906  df-ico 9907  df-icc 9908  df-fz 10022  df-fzo 10156  df-seqfrec 10459  df-exp 10533  df-fac 10719  df-bc 10741  df-ihash 10769  df-shft 10837  df-cj 10864  df-re 10865  df-im 10866  df-rsqrt 11020  df-abs 11021  df-clim 11300  df-sumdc 11375  df-ef 11669  df-sin 11671  df-cos 11672  df-pi 11674  df-rest 12707  df-topgen 12726  df-psmet 13673  df-xmet 13674  df-met 13675  df-bl 13676  df-mopn 13677  df-top 13738  df-topon 13751  df-bases 13783  df-ntr 13836  df-cn 13928  df-cnp 13929  df-tx 13993  df-cncf 14298  df-limced 14365  df-dvap 14366

This theorem is referenced by:  sinper  14470  cosper  14471