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Theorem sinperlem 12902
Description: Lemma for sinper 12903 and cosper 12904. (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 9066 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  K  e.  CC )
2 2cn 8798 . . . . . . . . . 10  |-  2  e.  CC
3 picn 12881 . . . . . . . . . 10  |-  pi  e.  CC
42, 3mulcli 7778 . . . . . . . . 9  |-  ( 2  x.  pi )  e.  CC
5 mulcl 7754 . . . . . . . . 9  |-  ( ( K  e.  CC  /\  ( 2  x.  pi )  e.  CC )  ->  ( K  x.  (
2  x.  pi ) )  e.  CC )
61, 4, 5sylancl 409 . . . . . . . 8  |-  ( K  e.  ZZ  ->  ( K  x.  ( 2  x.  pi ) )  e.  CC )
7 ax-icn 7722 . . . . . . . . 9  |-  _i  e.  CC
8 adddi 7759 . . . . . . . . 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 1302 . . . . . . . 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 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 7898 . . . . . . . . . . . 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 1306 . . . . . . . . . . 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 7778 . . . . . . . . . . 11  |-  ( _i  x.  ( 2  x.  pi ) )  e.  CC
15 mulcom 7756 . . . . . . . . . . 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 409 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( K  x.  ( _i  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K
) )
1713, 16eqtrd 2172 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K
) )
1817adantl 275 . . . . . . . 8  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) )
1918oveq2d 5790 . . . . . . 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 2172 . . . . . 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 5425 . . . . 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 7754 . . . . . . 7  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
237, 22mpan 420 . . . . . 6  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
24 efper 12901 . . . . . 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 281 . . . . 5  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
( _i  x.  A
)  +  ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) ) )  =  ( exp `  ( _i  x.  A
) ) )
2621, 25eqtrd 2172 . . . 4  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  (
_i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) )  =  ( exp `  ( _i  x.  A
) ) )
27 negicn 7970 . . . . . . . . 9  |-  -u _i  e.  CC
28 adddi 7759 . . . . . . . . 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 1302 . . . . . . . 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 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 ) ) ) ) )
3117negeqd 7964 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  -u (
_i  x.  ( K  x.  ( 2  x.  pi ) ) )  = 
-u ( ( _i  x.  ( 2  x.  pi ) )  x.  K ) )
32 mulneg1 8164 . . . . . . . . . . 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 410 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  ( -u _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  -u ( _i  x.  ( K  x.  (
2  x.  pi ) ) ) )
34 mulneg2 8165 . . . . . . . . . . 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 410 . . . . . . . . . 10  |-  ( K  e.  ZZ  ->  (
( _i  x.  (
2  x.  pi ) )  x.  -u K
)  =  -u (
( _i  x.  (
2  x.  pi ) )  x.  K ) )
3631, 33, 353eqtr4d 2182 . . . . . . . . 9  |-  ( K  e.  ZZ  ->  ( -u _i  x.  ( K  x.  ( 2  x.  pi ) ) )  =  ( ( _i  x.  ( 2  x.  pi ) )  x.  -u K ) )
3736adantl 275 . . . . . . . 8  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( -u _i  x.  ( K  x.  (
2  x.  pi ) ) )  =  ( ( _i  x.  (
2  x.  pi ) )  x.  -u K
) )
3837oveq2d 5790 . . . . . . 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 2172 . . . . . 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 5425 . . . . 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 7754 . . . . . . 7  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
4227, 41mpan 420 . . . . . 6  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
43 znegcl 9092 . . . . . 6  |-  ( K  e.  ZZ  ->  -u K  e.  ZZ )
44 efper 12901 . . . . . 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 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 ) ) )
4640, 45eqtrd 2172 . . . 4  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( exp `  ( -u _i  x.  ( A  +  ( K  x.  ( 2  x.  pi ) ) ) ) )  =  ( exp `  ( -u _i  x.  A ) ) )
4726, 46oveq12d 5792 . . 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 5789 . 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 7752 . . . 4  |-  ( ( A  e.  CC  /\  ( K  x.  (
2  x.  pi ) )  e.  CC )  ->  ( A  +  ( K  x.  (
2  x.  pi ) ) )  e.  CC )
506, 49sylan2 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
) )
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 274 . 2  |-  ( ( A  e.  CC  /\  K  e.  ZZ )  ->  ( F `  A
)  =  ( ( ( exp `  (
_i  x.  A )
) O ( exp `  ( -u _i  x.  A ) ) )  /  D ) )
5548, 52, 543eqtr4d 2182 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 103    = wceq 1331    e. wcel 1480   ` cfv 5123  (class class class)co 5774   CCcc 7625   _ici 7629    + caddc 7630    x. cmul 7632   -ucneg 7941    / cdiv 8439   2c2 8778   ZZcz 9061   expce 11355   picpi 11360
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502  ax-cnex 7718  ax-resscn 7719  ax-1cn 7720  ax-1re 7721  ax-icn 7722  ax-addcl 7723  ax-addrcl 7724  ax-mulcl 7725  ax-mulrcl 7726  ax-addcom 7727  ax-mulcom 7728  ax-addass 7729  ax-mulass 7730  ax-distr 7731  ax-i2m1 7732  ax-0lt1 7733  ax-1rid 7734  ax-0id 7735  ax-rnegex 7736  ax-precex 7737  ax-cnre 7738  ax-pre-ltirr 7739  ax-pre-ltwlin 7740  ax-pre-lttrn 7741  ax-pre-apti 7742  ax-pre-ltadd 7743  ax-pre-mulgt0 7744  ax-pre-mulext 7745  ax-arch 7746  ax-caucvg 7747  ax-pre-suploc 7748  ax-addf 7749  ax-mulf 7750
This theorem depends on definitions:  df-bi 116  df-stab 816  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-reu 2423  df-rmo 2424  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-disj 3907  df-br 3930  df-opab 3990  df-mpt 3991  df-tr 4027  df-id 4215  df-po 4218  df-iso 4219  df-iord 4288  df-on 4290  df-ilim 4291  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132  df-riota 5730  df-ov 5777  df-oprab 5778  df-mpo 5779  df-of 5982  df-1st 6038  df-2nd 6039  df-recs 6202  df-irdg 6267  df-frec 6288  df-1o 6313  df-oadd 6317  df-er 6429  df-map 6544  df-pm 6545  df-en 6635  df-dom 6636  df-fin 6637  df-sup 6871  df-inf 6872  df-pnf 7809  df-mnf 7810  df-xr 7811  df-ltxr 7812  df-le 7813  df-sub 7942  df-neg 7943  df-reap 8344  df-ap 8351  df-div 8440  df-inn 8728  df-2 8786  df-3 8787  df-4 8788  df-5 8789  df-6 8790  df-7 8791  df-8 8792  df-9 8793  df-n0 8985  df-z 9062  df-uz 9334  df-q 9419  df-rp 9449  df-xneg 9566  df-xadd 9567  df-ioo 9682  df-ioc 9683  df-ico 9684  df-icc 9685  df-fz 9798  df-fzo 9927  df-seqfrec 10226  df-exp 10300  df-fac 10479  df-bc 10501  df-ihash 10529  df-shft 10594  df-cj 10621  df-re 10622  df-im 10623  df-rsqrt 10777  df-abs 10778  df-clim 11055  df-sumdc 11130  df-ef 11361  df-sin 11363  df-cos 11364  df-pi 11366  df-rest 12132  df-topgen 12151  df-psmet 12166  df-xmet 12167  df-met 12168  df-bl 12169  df-mopn 12170  df-top 12175  df-topon 12188  df-bases 12220  df-ntr 12275  df-cn 12367  df-cnp 12368  df-tx 12432  df-cncf 12737  df-limced 12804  df-dvap 12805
This theorem is referenced by:  sinper  12903  cosper  12904
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