Theorem sinperlem 15251

Description: Lemma for sinper 15252 and cosper 15253. (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 9376	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
2		2cn 9106	. . . . . . . . . 10 |- 2 e. CC
3		picn 15230	. . . . . . . . . 10 |- pi e. CC
4	2, 3	mulcli 8076	. . . . . . . . 9 |- ( 2 x. pi ) e. CC
5		mulcl 8051	. . . . . . . . 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 8019	. . . . . . . . 9 |- _i e. CC
8		adddi 8056	. . . . . . . . 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 1336	. . . . . . . 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 8200	. . . . . . . . . . . 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 1340	. . . . . . . . . . 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 8076	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
15		mulcom 8053	. . . . . . . . . . 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 2237	. . . . . . . . 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 5959	. . . . . . 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 2237	. . . . . 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 5579	. . . . 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 8051	. . . . . . 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 15250	. . . . . 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 2237	. . . 4 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( _i x. A ) ) )
27		negicn 8272	. . . . . . . . 9 |- -u _i e. CC
28		adddi 8056	. . . . . . . . 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 1336	. . . . . . . 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 8266	. . . . . . . . . 10 |- ( K e. ZZ -> -u ( _i x. ( K x. ( 2 x. pi ) ) ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
32		mulneg1 8466	. . . . . . . . . . 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 8467	. . . . . . . . . . 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 2247	. . . . . . . . 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 5959	. . . . . . 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 2237	. . . . . 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 5579	. . . . 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 8051	. . . . . . 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 9402	. . . . . 6 |- ( K e. ZZ -> -u K e. ZZ )
44		efper 15250	. . . . . 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 2237	. . . 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 5961	. . 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 5958	. 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 8049	. . . 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 2247	1 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1372   e. wcel 2175   ` cfv 5270  (class class class)co 5943   CCcc 7922   _ici 7926   + caddc 7927   x. cmul 7929   -ucneg 8243   / cdiv 8744   2c2 9086   ZZcz 9371   expce 11924   picpi 11929

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-nul 4169  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-iinf 4635  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-mulrcl 8023  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-0lt1 8030  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-precex 8034  ax-cnre 8035  ax-pre-ltirr 8036  ax-pre-ltwlin 8037  ax-pre-lttrn 8038  ax-pre-apti 8039  ax-pre-ltadd 8040  ax-pre-mulgt0 8041  ax-pre-mulext 8042  ax-arch 8043  ax-caucvg 8044  ax-pre-suploc 8045  ax-addf 8046  ax-mulf 8047

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-nel 2471  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-disj 4021  df-br 4044  df-opab 4105  df-mpt 4106  df-tr 4142  df-id 4339  df-po 4342  df-iso 4343  df-iord 4412  df-on 4414  df-ilim 4415  df-suc 4417  df-iom 4638  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-isom 5279  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-of 6157  df-1st 6225  df-2nd 6226  df-recs 6390  df-irdg 6455  df-frec 6476  df-1o 6501  df-oadd 6505  df-er 6619  df-map 6736  df-pm 6737  df-en 6827  df-dom 6828  df-fin 6829  df-sup 7085  df-inf 7086  df-pnf 8108  df-mnf 8109  df-xr 8110  df-ltxr 8111  df-le 8112  df-sub 8244  df-neg 8245  df-reap 8647  df-ap 8654  df-div 8745  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-9 9101  df-n0 9295  df-z 9372  df-uz 9648  df-q 9740  df-rp 9775  df-xneg 9893  df-xadd 9894  df-ioo 10013  df-ioc 10014  df-ico 10015  df-icc 10016  df-fz 10130  df-fzo 10264  df-seqfrec 10591  df-exp 10682  df-fac 10869  df-bc 10891  df-ihash 10919  df-shft 11097  df-cj 11124  df-re 11125  df-im 11126  df-rsqrt 11280  df-abs 11281  df-clim 11561  df-sumdc 11636  df-ef 11930  df-sin 11932  df-cos 11933  df-pi 11935  df-rest 13044  df-topgen 13063  df-psmet 14276  df-xmet 14277  df-met 14278  df-bl 14279  df-mopn 14280  df-top 14441  df-topon 14454  df-bases 14486  df-ntr 14539  df-cn 14631  df-cnp 14632  df-tx 14696  df-cncf 15014  df-limced 15099  df-dvap 15100

This theorem is referenced by:  sinper  15252  cosper  15253