ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  sinneg Unicode version
Theorem sinneg 11872
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 8221 . . 3 |- ( A e. CC -> -u A e. CC )
2 sinval 11848 . . 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 11848 . . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
54negeqd 8216 . . . 4 |- ( A e. CC -> -u ( sin ` A ) = -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
6 ax-icn 7969 . . . . . . . 8 |- _i e. CC
7 mulcl 8001 . . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( _i x. A ) e. CC )
86, 7mpan 424 . . . . . . 7 |- ( A e. CC -> ( _i x. A ) e. CC )
9 efcl 11810 . . . . . . 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 8222 . . . . . . . 8 |- -u _i e. CC
12 mulcl 8001 . . . . . . . 8 |- ( ( -u _i e. CC /\ A e. CC ) -> ( -u _i x. A ) e. CC )
1311, 12mpan 424 . . . . . . 7 |- ( A e. CC -> ( -u _i x. A ) e. CC )
14 efcl 11810 . . . . . . 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 8332 . . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
17 2mulicn 9207 . . . . . 6 |- ( 2 x. _i ) e. CC
18 2muliap0 9209 . . . . . 6 |- ( 2 x. _i ) # 0
19 divnegap 8727 . . . . . 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 1340 . . . . 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 2226 . . 3 |- ( A e. CC -> -u ( sin ` A ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
23 mulneg12 8418 . . . . . . . . 9 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. A ) = ( _i x. -u A ) )
246, 23mpan 424 . . . . . . . 8 |- ( A e. CC -> ( -u _i x. A ) = ( _i x. -u A ) )
2524eqcomd 2199 . . . . . . 7 |- ( A e. CC -> ( _i x. -u A ) = ( -u _i x. A ) )
2625fveq2d 5559 . . . . . 6 |- ( A e. CC -> ( exp ` ( _i x. -u A ) ) = ( exp ` ( -u _i x. A ) ) )
27 mul2neg 8419 . . . . . . . 8 |- ( ( _i e. CC /\ A e. CC ) -> ( -u _i x. -u A ) = ( _i x. A ) )
286, 27mpan 424 . . . . . . 7 |- ( A e. CC -> ( -u _i x. -u A ) = ( _i x. A ) )
2928fveq2d 5559 . . . . . 6 |- ( A e. CC -> ( exp ` ( -u _i x. -u A ) ) = ( exp ` ( _i x. A ) ) )
3026, 29oveq12d 5937 . . . . 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 8348 . . . . 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 2229 . . . 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 5934 . . 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 2229 . 2 |- ( A e. CC -> -u ( sin ` A ) = ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) )
353, 34eqtr4d 2229 1 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1364   e. wcel 2164   class class class wbr 4030   ` cfv 5255  (class class class)co 5919   CCcc 7872   0cc0 7874   _ici 7876   x. cmul 7879   - cmin 8192   -ucneg 8193   # cap 8602   / cdiv 8693   2c2 9035   expce 11788   sincsin 11790
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 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ico 9963  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-ihash 10850  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ef 11794  df-sin 11796
This theorem is referenced by:  tannegap  11874  sin0  11875  efmival  11879  sinsub  11886  cossub  11887  sincossq  11894  sin2pim  14989
  Copyright terms: Public domain W3C validator