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Theorem sinneg 11689
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 8119 . . 3 |- ( A e. CC -> -u A e. CC )
2 sinval 11665 . . 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 11665 . . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
54negeqd 8114 . . . 4 |- ( A e. CC -> -u ( sin ` A ) = -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
6 ax-icn 7869 . . . . . . . 8 |- _i e. CC
7 mulcl 7901 . . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
86, 7mpan 422 . . . . . . 7 |- ( A e. CC -> ( _i x. A ) e. CC )
9 efcl 11627 . . . . . . 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 8120 . . . . . . . 8 |- -u _i e. CC
12 mulcl 7901 . . . . . . . 8 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
1311, 12mpan 422 . . . . . . 7 |- ( A e. CC -> ( -u _i x. A ) e. CC )
14 efcl 11627 . . . . . . 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 8230 . . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
17 2mulicn 9100 . . . . . 6 |- ( 2 x. _i ) e. CC
18 2muliap0 9102 . . . . . 6 |- ( 2 x. _i ) # 0
19 divnegap 8623 . . . . . 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 1324 . . . . 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 2203 . . 3 |- ( A e. CC -> -u ( sin ` A ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
23 mulneg12 8316 . . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
246, 23mpan 422 . . . . . . . 8 |- ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
2524eqcomd 2176 . . . . . . 7 |- ( A e. CC -> ( _i x. -u A ) = ( -u _i x. A ) )
2625fveq2d 5500 . . . . . 6 |- ( A e. CC -> ( exp ` ( _i x. -u A ) ) = ( exp ` ( -u _i x. A ) ) )
27 mul2neg 8317 . . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. -u A ) = ( _i x. A ) )
286, 27mpan 422 . . . . . . 7 |- ( A e. CC -> ( -u _i x. -u A ) = ( _i x. A ) )
2928fveq2d 5500 . . . . . 6 |- ( A e. CC -> ( exp ` ( -u _i x. -u A ) ) = ( exp ` ( _i x. A ) ) )
3026, 29oveq12d 5871 . . . . 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 8246 . . . . 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 2206 . . . 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 5868 . . 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 2206 . 2 |- ( A e. CC -> -u ( sin ` A ) = ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) )
353, 34eqtr4d 2206 1 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1348   e. wcel 2141   class class class wbr 3989   ` cfv 5198  (class class class)co 5853   CCcc 7772   0cc0 7774   _ici 7776   x. cmul 7779   - cmin 8090   -ucneg 8091   # cap 8500   / cdiv 8589   2c2 8929   expce 11605   sincsin 11607
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572  ax-cnex 7865  ax-resscn 7866  ax-1cn 7867  ax-1re 7868  ax-icn 7869  ax-addcl 7870  ax-addrcl 7871  ax-mulcl 7872  ax-mulrcl 7873  ax-addcom 7874  ax-mulcom 7875  ax-addass 7876  ax-mulass 7877  ax-distr 7878  ax-i2m1 7879  ax-0lt1 7880  ax-1rid 7881  ax-0id 7882  ax-rnegex 7883  ax-precex 7884  ax-cnre 7885  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888  ax-pre-apti 7889  ax-pre-ltadd 7890  ax-pre-mulgt0 7891  ax-pre-mulext 7892  ax-arch 7893  ax-caucvg 7894
This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-reu 2455  df-rmo 2456  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-if 3527  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-ilim 4354  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-isom 5207  df-riota 5809  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-frec 6370  df-1o 6395  df-oadd 6399  df-er 6513  df-en 6719  df-dom 6720  df-fin 6721  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-sub 8092  df-neg 8093  df-reap 8494  df-ap 8501  df-div 8590  df-inn 8879  df-2 8937  df-3 8938  df-4 8939  df-n0 9136  df-z 9213  df-uz 9488  df-q 9579  df-rp 9611  df-ico 9851  df-fz 9966  df-fzo 10099  df-seqfrec 10402  df-exp 10476  df-fac 10660  df-ihash 10710  df-cj 10806  df-re 10807  df-im 10808  df-rsqrt 10962  df-abs 10963  df-clim 11242  df-sumdc 11317  df-ef 11611  df-sin 11613
This theorem is referenced by:  tannegap  11691  sin0  11692  efmival  11696  sinsub  11703  cossub  11704  sincossq  11711  sin2pim  13528
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