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Theorem sinperlem 22606
Description: Lemma for sinper 22607 and cosper 22608. (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
StepHypRef Expression
1 zcn 10865 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
2 2cn 10602 . . . . . . . . . 10  |-  2  e.  CC
3 picn 22586 . . . . . . . . . 10  |-  pi  e.  CC
42, 3mulcli 9597 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
5 mulcl 9572 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  ( 2  x.  pi )  e.  CC )  ->  ( K  x.  (
2  x.  pi ) )  e.  CC )
61, 4, 5sylancl 662 . . . . . . . 8  |-  ( K  e.  ZZ  ->  ( K  x.  ( 2  x.  pi ) )  e.  CC )
7 ax-icn 9547 . . . . . . . . 9  |-  _i  e.  CC
8 adddi 9577 . . . . . . . . 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 ) ) ) ) )
97, 8mp3an1 1311 . . . . . . . 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 ) ) ) ) )
106, 9sylan2 474 . . . . . . 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 9741 . . . . . . . . . . . 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 ) ) ) )
127, 4, 11mp3an13 1315 . . . . . . . . . . 11  |-  ( K  e.  CC  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( K  x.  (
_i  x.  ( 2  x.  pi ) ) ) )
131, 12syl 16 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( K  x.  (
_i  x.  ( 2  x.  pi ) ) ) )
147, 4mulcli 9597 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
15 mulcom 9574 . . . . . . . . . . 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 ) )
161, 14, 15sylancl 662 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( K  x.  ( _i  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K
) )
1713, 16eqtrd 2508 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K
) )
1817adantl 466 . . . . . . . 8  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) )
1918oveq2d 6298 . . . . . . 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 ) ) )
2010, 19eqtrd 2508 . . . . . 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
) ) )
2120fveq2d 5868 . . . . 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 9572 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
237, 22mpan 670 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
24 efper 22605 . . . . . 6  |-  ( ( ( _i  x.  A
)  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
( _i  x.  A
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  ( _i  x.  A
) ) )
2523, 24sylan 471 . . . . 5  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
( _i  x.  A
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  ( _i  x.  A
) ) )
2621, 25eqtrd 2508 . . . 4  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
_i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) )  =  ( exp `  ( _i  x.  A
) ) )
27 negicn 9817 . . . . . . . . 9  |-  -u _i  e.  CC
28 adddi 9577 . . . . . . . . 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 ) ) ) ) )
2927, 28mp3an1 1311 . . . . . . . 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 ) ) ) ) )
306, 29sylan2 474 . . . . . . 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 ) ) ) ) )
3117negeqd 9810 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  -u (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  = 
-u ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) )
32 mulneg1 9989 . . . . . . . . . . 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 ) ) ) )
337, 6, 32sylancr 663 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( -u _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  -u ( _i  x.  ( K  x.  (
2  x.  pi ) ) ) )
34 mulneg2 9990 . . . . . . . . . . 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 ) )
3514, 1, 34sylancr 663 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  (
( _i  x.  (
2  x.  pi ) )  x.  -u K
)  =  -u (
( _i  x.  (
2  x.  pi ) )  x.  K ) )
3631, 33, 353eqtr4d 2518 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  ( -u _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  -u K ) )
3736adantl 466 . . . . . . . 8  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( -u _i  x.  ( K  x.  (
2  x.  pi ) ) )  =  ( ( _i  x.  (
2  x.  pi ) )  x.  -u K
) )
3837oveq2d 6298 . . . . . . 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
) ) )
3930, 38eqtrd 2508 . . . . . 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
) ) )
4039fveq2d 5868 . . . . 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 9572 . . . . . . 7  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
4227, 41mpan 670 . . . . . 6  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
43 znegcl 10894 . . . . . 6  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
44 efper 22605 . . . . . 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 ) ) )
4542, 43, 44syl2an 477 . . . . 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 ) ) )
4640, 45eqtrd 2508 . . . 4  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  ( -u _i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) )  =  ( exp `  ( -u _i  x.  A ) ) )
4726, 46oveq12d 6300 . . 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 ) ) ) )
4847oveq1d 6297 . 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 9570 . . . 4  |-  ( ( A  e.  CC  /\  ( K  x.  (
2  x.  pi ) )  e.  CC )  ->  ( A  +  ( K  x.  (
2  x.  pi ) ) )  e.  CC )
506, 49sylan2 474 . . 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
) )
5250, 51syl 16 . 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
) )
5453adantr 465 . 2  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  A
)  =  ( ( ( exp `  (
_i  x.  A )
) O ( exp `  ( -u _i  x.  A ) ) )  /  D ) )
5548, 52, 543eqtr4d 2518 1  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( F `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   CCcc 9486   _ici 9490    + caddc 9491    x. cmul 9493   -ucneg 9802    / cdiv 10202   2c2 10581   ZZcz 10860   expce 13655   picpi 13660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-pm 7420  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-ioc 11530  df-ico 11531  df-icc 11532  df-fz 11669  df-fzo 11789  df-fl 11893  df-seq 12072  df-exp 12131  df-fac 12318  df-bc 12345  df-hash 12370  df-shft 12859  df-cj 12891  df-re 12892  df-im 12893  df-sqrt 13027  df-abs 13028  df-limsup 13253  df-clim 13270  df-rlim 13271  df-sum 13468  df-ef 13661  df-sin 13663  df-cos 13664  df-pi 13666  df-struct 14488  df-ndx 14489  df-slot 14490  df-base 14491  df-sets 14492  df-ress 14493  df-plusg 14564  df-mulr 14565  df-starv 14566  df-sca 14567  df-vsca 14568  df-ip 14569  df-tset 14570  df-ple 14571  df-ds 14573  df-unif 14574  df-hom 14575  df-cco 14576  df-rest 14674  df-topn 14675  df-0g 14693  df-gsum 14694  df-topgen 14695  df-pt 14696  df-prds 14699  df-xrs 14753  df-qtop 14758  df-imas 14759  df-xps 14761  df-mre 14837  df-mrc 14838  df-acs 14840  df-mnd 15728  df-submnd 15778  df-mulg 15861  df-cntz 16150  df-cmn 16596  df-psmet 18182  df-xmet 18183  df-met 18184  df-bl 18185  df-mopn 18186  df-fbas 18187  df-fg 18188  df-cnfld 18192  df-top 19166  df-bases 19168  df-topon 19169  df-topsp 19170  df-cld 19286  df-ntr 19287  df-cls 19288  df-nei 19365  df-lp 19403  df-perf 19404  df-cn 19494  df-cnp 19495  df-haus 19582  df-tx 19798  df-hmeo 19991  df-fil 20082  df-fm 20174  df-flim 20175  df-flf 20176  df-xms 20558  df-ms 20559  df-tms 20560  df-cncf 21117  df-limc 22005  df-dv 22006
This theorem is referenced by:  sinper  22607  cosper  22608
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