Theorem sinperlem 12937

Description: Lemma for sinper 12938 and cosper 12939. (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 9083	. . . . . . . . 9 |- ( K e. ZZ -> K e. CC )
2		2cn 8815	. . . . . . . . . 10 |- 2 e. CC
3		picn 12916	. . . . . . . . . 10 |- pi e. CC
4	2, 3	mulcli 7795	. . . . . . . . 9 |- ( 2 x. pi ) e. CC
5		mulcl 7771	. . . . . . . . 9 |- ( ( K e. CC /\ ( 2 x. pi ) e. CC ) -> ( K x. ( 2 x. pi ) ) e. CC )
6	1, 4, 5	sylancl 410	. . . . . . . 8 |- ( K e. ZZ -> ( K x. ( 2 x. pi ) ) e. CC )
7		ax-icn 7739	. . . . . . . . 9 |- _i e. CC
8		adddi 7776	. . . . . . . . 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 1303	. . . . . . . 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 7915	. . . . . . . . . . . 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 1307	. . . . . . . . . . 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 7795	. . . . . . . . . . 11 |- ( _i x. ( 2 x. pi ) ) e. CC
15		mulcom 7773	. . . . . . . . . . 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 410	. . . . . . . . . 10 |- ( K e. ZZ -> ( K x. ( _i x. ( 2 x. pi ) ) ) = ( ( _i x. ( 2 x. pi ) ) x. K ) )
17	13, 16	eqtrd 2173	. . . . . . . . 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 5798	. . . . . . 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 2173	. . . . . 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 5433	. . . . 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 7771	. . . . . . 7 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
23	7, 22	mpan 421	. . . . . 6 |- ( A e. CC -> ( _i x. A ) e. CC )
24		efper 12936	. . . . . 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 2173	. . . 4 |- ( ( A e. CC /\ K e. ZZ ) -> ( exp ` ( _i x. ( A + ( K x. ( 2 x. pi ) ) ) ) ) = ( exp ` ( _i x. A ) ) )
27		negicn 7987	. . . . . . . . 9 |- -u _i e. CC
28		adddi 7776	. . . . . . . . 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 1303	. . . . . . . 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 7981	. . . . . . . . . 10 |- ( K e. ZZ -> -u ( _i x. ( K x. ( 2 x. pi ) ) ) = -u ( ( _i x. ( 2 x. pi ) ) x. K ) )
32		mulneg1 8181	. . . . . . . . . . 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 411	. . . . . . . . . 10 |- ( K e. ZZ -> ( -u _i x. ( K x. ( 2 x. pi ) ) ) = -u ( _i x. ( K x. ( 2 x. pi ) ) ) )
34		mulneg2 8182	. . . . . . . . . . 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 411	. . . . . . . . . 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 2183	. . . . . . . . 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 5798	. . . . . . 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 2173	. . . . . 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 5433	. . . . 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 7771	. . . . . . 7 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
42	27, 41	mpan 421	. . . . . 6 |- ( A e. CC -> ( -u _i x. A ) e. CC )
43		znegcl 9109	. . . . . 6 |- ( K e. ZZ -> -u K e. ZZ )
44		efper 12936	. . . . . 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 2173	. . . 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 5800	. . 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 5797	. 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 7769	. . . 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 2183	1 |- ( ( A e. CC /\ K e. ZZ ) -> ( F ` ( A + ( K x. ( 2 x. pi ) ) ) ) = ( F ` A ) )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1332   e. wcel 1481   ` cfv 5131  (class class class)co 5782   CCcc 7642   _ici 7646   + caddc 7647   x. cmul 7649   -ucneg 7958   / cdiv 8456   2c2 8795   ZZcz 9078   expce 11385   picpi 11390

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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764  ax-pre-suploc 7765  ax-addf 7766  ax-mulf 7767

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-disj 3915  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-of 5990  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-map 6552  df-pm 6553  df-en 6643  df-dom 6644  df-fin 6645  df-sup 6879  df-inf 6880  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-5 8806  df-6 8807  df-7 8808  df-8 8809  df-9 8810  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-xneg 9589  df-xadd 9590  df-ioo 9705  df-ioc 9706  df-ico 9707  df-icc 9708  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-bc 10526  df-ihash 10554  df-shft 10619  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391  df-sin 11393  df-cos 11394  df-pi 11396  df-rest 12161  df-topgen 12180  df-psmet 12195  df-xmet 12196  df-met 12197  df-bl 12198  df-mopn 12199  df-top 12204  df-topon 12217  df-bases 12249  df-ntr 12304  df-cn 12396  df-cnp 12397  df-tx 12461  df-cncf 12766  df-limced 12833  df-dvap 12834

This theorem is referenced by:  sinper  12938  cosper  12939