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Theorem sinneg 11667
Description: The sine of a negative is the negative of the sine. (Contributed by NM, 30-Apr-2005.)
Assertion
Ref Expression
sinneg |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
Proof of Theorem sinneg
StepHypRef Expression
1 negcl 8098 . . 3 |- ( A e. CC -> -u A e. CC )
2 sinval 11643 . . 3 |- ( -u A e. CC -> ( sin ` -u A ) = ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) )
31, 2syl 14 . 2 |- ( A e. CC -> ( sin ` -u A ) = ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) )
4 sinval 11643 . . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
54negeqd 8093 . . . 4 |- ( A e. CC -> -u ( sin ` A ) = -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
6 ax-icn 7848 . . . . . . . 8 |- _i e. CC
7 mulcl 7880 . . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
86, 7mpan 421 . . . . . . 7 |- ( A e. CC -> ( _i x. A ) e. CC )
9 efcl 11605 . . . . . . 7 |- ( ( _i x. A ) e. CC -> ( exp ` ( _i x. A ) ) e. CC )
108, 9syl 14 . . . . . 6 |- ( A e. CC -> ( exp ` ( _i x. A ) ) e. CC )
11 negicn 8099 . . . . . . . 8 |- -u _i e. CC
12 mulcl 7880 . . . . . . . 8 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
1311, 12mpan 421 . . . . . . 7 |- ( A e. CC -> ( -u _i x. A ) e. CC )
14 efcl 11605 . . . . . . 7 |- ( ( -u _i x. A ) e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
1513, 14syl 14 . . . . . 6 |- ( A e. CC -> ( exp ` ( -u _i x. A ) ) e. CC )
1610, 15subcld 8209 . . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
17 2mulicn 9079 . . . . . 6 |- ( 2 x. _i ) e. CC
18 2muliap0 9081 . . . . . 6 |- ( 2 x. _i ) # 0
19 divnegap 8602 . . . . . 6 |- ( ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC /\ ( 2 x. _i ) e. CC /\ ( 2 x. _i ) # 0 ) -> -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
2017, 18, 19mp3an23 1319 . . . . 5 |- ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC -> -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
2116, 20syl 14 . . . 4 |- ( A e. CC -> -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
225, 21eqtrd 2198 . . 3 |- ( A e. CC -> -u ( sin ` A ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
23 mulneg12 8295 . . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
246, 23mpan 421 . . . . . . . 8 |- ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
2524eqcomd 2171 . . . . . . 7 |- ( A e. CC -> ( _i x. -u A ) = ( -u _i x. A ) )
2625fveq2d 5490 . . . . . 6 |- ( A e. CC -> ( exp ` ( _i x. -u A ) ) = ( exp ` ( -u _i x. A ) ) )
27 mul2neg 8296 . . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. -u A ) = ( _i x. A ) )
286, 27mpan 421 . . . . . . 7 |- ( A e. CC -> ( -u _i x. -u A ) = ( _i x. A ) )
2928fveq2d 5490 . . . . . 6 |- ( A e. CC -> ( exp ` ( -u _i x. -u A ) ) = ( exp ` ( _i x. A ) ) )
3026, 29oveq12d 5860 . . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) = ( ( exp ` ( -u _i x. A ) ) - ( exp ` ( _i x. A ) ) ) )
3110, 15negsubdi2d 8225 . . . . 5 |- ( A e. CC -> -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) = ( ( exp ` ( -u _i x. A ) ) - ( exp ` ( _i x. A ) ) ) )
3230, 31eqtr4d 2201 . . . 4 |- ( A e. CC -> ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) = -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) )
3332oveq1d 5857 . . 3 |- ( A e. CC -> ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
3422, 33eqtr4d 2201 . 2 |- ( A e. CC -> -u ( sin ` A ) = ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) )
353, 34eqtr4d 2201 1 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   class class class wbr 3982   ` cfv 5188  (class class class)co 5842   CCcc 7751   0cc0 7753   _ici 7755   x. cmul 7758   - cmin 8069   -ucneg 8070   # cap 8479   / cdiv 8568   2c2 8908   expce 11583   sincsin 11585
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-en 6707  df-dom 6708  df-fin 6709  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-ico 9830  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-ihash 10689  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-sin 11591
This theorem is referenced by:  tannegap  11669  sin0  11670  efmival  11674  sinsub  11681  cossub  11682  sincossq  11689  sin2pim  13384
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