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Theorem sinperlem 14632
Description: Lemma for sinper 14633 and cosper 14634. (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 9277 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
2 2cn 9009 . . . . . . . . . 10  |-  2  e.  CC
3 picn 14611 . . . . . . . . . 10  |-  pi  e.  CC
42, 3mulcli 7981 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
5 mulcl 7957 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  ( 2  x.  pi )  e.  CC )  ->  ( K  x.  (
2  x.  pi ) )  e.  CC )
61, 4, 5sylancl 413 . . . . . . . 8  |-  ( K  e.  ZZ  ->  ( K  x.  ( 2  x.  pi ) )  e.  CC )
7 ax-icn 7925 . . . . . . . . 9  |-  _i  e.  CC
8 adddi 7962 . . . . . . . . 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 1335 . . . . . . . 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 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 8105 . . . . . . . . . . . 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 1339 . . . . . . . . . . 11  |-  ( K  e.  CC  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( K  x.  (
_i  x.  ( 2  x.  pi ) ) ) )
131, 12syl 14 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( K  x.  (
_i  x.  ( 2  x.  pi ) ) ) )
147, 4mulcli 7981 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
15 mulcom 7959 . . . . . . . . . . 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 413 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( K  x.  ( _i  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K
) )
1713, 16eqtrd 2222 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K
) )
1817adantl 277 . . . . . . . 8  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) )
1918oveq2d 5907 . . . . . . 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 2222 . . . . . 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 5534 . . . . 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 7957 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
237, 22mpan 424 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
24 efper 14631 . . . . . 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 283 . . . . 5  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
( _i  x.  A
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  ( _i  x.  A
) ) )
2621, 25eqtrd 2222 . . . 4  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
_i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) )  =  ( exp `  ( _i  x.  A
) ) )
27 negicn 8177 . . . . . . . . 9  |-  -u _i  e.  CC
28 adddi 7962 . . . . . . . . 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 1335 . . . . . . . 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 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 ) ) ) ) )
3117negeqd 8171 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  -u (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  = 
-u ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) )
32 mulneg1 8371 . . . . . . . . . . 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 414 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( -u _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  -u ( _i  x.  ( K  x.  (
2  x.  pi ) ) ) )
34 mulneg2 8372 . . . . . . . . . . 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 414 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  (
( _i  x.  (
2  x.  pi ) )  x.  -u K
)  =  -u (
( _i  x.  (
2  x.  pi ) )  x.  K ) )
3631, 33, 353eqtr4d 2232 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  ( -u _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  -u K ) )
3736adantl 277 . . . . . . . 8  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( -u _i  x.  ( K  x.  (
2  x.  pi ) ) )  =  ( ( _i  x.  (
2  x.  pi ) )  x.  -u K
) )
3837oveq2d 5907 . . . . . . 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 2222 . . . . . 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 5534 . . . . 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 7957 . . . . . . 7  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
4227, 41mpan 424 . . . . . 6  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
43 znegcl 9303 . . . . . 6  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
44 efper 14631 . . . . . 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 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 ) ) )
4640, 45eqtrd 2222 . . . 4  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  ( -u _i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) )  =  ( exp `  ( -u _i  x.  A ) ) )
4726, 46oveq12d 5909 . . 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 5906 . 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 7955 . . . 4  |-  ( ( A  e.  CC  /\  ( K  x.  (
2  x.  pi ) )  e.  CC )  ->  ( A  +  ( K  x.  (
2  x.  pi ) ) )  e.  CC )
506, 49sylan2 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
) )
5250, 51syl 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
) )
5453adantr 276 . 2  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  A
)  =  ( ( ( exp `  (
_i  x.  A )
) O ( exp `  ( -u _i  x.  A ) ) )  /  D ) )
5548, 52, 543eqtr4d 2232 1  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) )  =  ( F `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    = wceq 1364    e. wcel 2160   ` cfv 5231  (class class class)co 5891   CCcc 7828   _ici 7832    + caddc 7833    x. cmul 7835   -ucneg 8148    / cdiv 8648   2c2 8989   ZZcz 9272   expce 11669   picpi 11674
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602  ax-cnex 7921  ax-resscn 7922  ax-1cn 7923  ax-1re 7924  ax-icn 7925  ax-addcl 7926  ax-addrcl 7927  ax-mulcl 7928  ax-mulrcl 7929  ax-addcom 7930  ax-mulcom 7931  ax-addass 7932  ax-mulass 7933  ax-distr 7934  ax-i2m1 7935  ax-0lt1 7936  ax-1rid 7937  ax-0id 7938  ax-rnegex 7939  ax-precex 7940  ax-cnre 7941  ax-pre-ltirr 7942  ax-pre-ltwlin 7943  ax-pre-lttrn 7944  ax-pre-apti 7945  ax-pre-ltadd 7946  ax-pre-mulgt0 7947  ax-pre-mulext 7948  ax-arch 7949  ax-caucvg 7950  ax-pre-suploc 7951  ax-addf 7952  ax-mulf 7953
This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-if 3550  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-iun 3903  df-disj 3996  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4308  df-po 4311  df-iso 4312  df-iord 4381  df-on 4383  df-ilim 4384  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-isom 5240  df-riota 5847  df-ov 5894  df-oprab 5895  df-mpo 5896  df-of 6101  df-1st 6159  df-2nd 6160  df-recs 6324  df-irdg 6389  df-frec 6410  df-1o 6435  df-oadd 6439  df-er 6553  df-map 6668  df-pm 6669  df-en 6759  df-dom 6760  df-fin 6761  df-sup 7002  df-inf 7003  df-pnf 8013  df-mnf 8014  df-xr 8015  df-ltxr 8016  df-le 8017  df-sub 8149  df-neg 8150  df-reap 8551  df-ap 8558  df-div 8649  df-inn 8939  df-2 8997  df-3 8998  df-4 8999  df-5 9000  df-6 9001  df-7 9002  df-8 9003  df-9 9004  df-n0 9196  df-z 9273  df-uz 9548  df-q 9639  df-rp 9673  df-xneg 9791  df-xadd 9792  df-ioo 9911  df-ioc 9912  df-ico 9913  df-icc 9914  df-fz 10028  df-fzo 10162  df-seqfrec 10465  df-exp 10539  df-fac 10725  df-bc 10747  df-ihash 10775  df-shft 10843  df-cj 10870  df-re 10871  df-im 10872  df-rsqrt 11026  df-abs 11027  df-clim 11306  df-sumdc 11381  df-ef 11675  df-sin 11677  df-cos 11678  df-pi 11680  df-rest 12718  df-topgen 12737  df-psmet 13823  df-xmet 13824  df-met 13825  df-bl 13826  df-mopn 13827  df-top 13901  df-topon 13914  df-bases 13946  df-ntr 13999  df-cn 14091  df-cnp 14092  df-tx 14156  df-cncf 14461  df-limced 14528  df-dvap 14529
This theorem is referenced by:  sinper  14633  cosper  14634
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