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Theorem sinneg 11469
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 7986 . . 3 |- ( A e. CC -> -u A e. CC )
2 sinval 11445 . . 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 11445 . . . . 5 |- ( A e. CC -> ( sin ` A ) = ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
54negeqd 7981 . . . 4 |- ( A e. CC -> -u ( sin ` A ) = -u ( ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
6 ax-icn 7739 . . . . . . . 8 |- _i e. CC
7 mulcl 7771 . . . . . . . 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 11407 . . . . . . 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 7987 . . . . . . . 8 |- -u _i e. CC
12 mulcl 7771 . . . . . . . 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 11407 . . . . . . 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 8097 . . . . 5 |- ( A e. CC -> ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) e. CC )
17 2mulicn 8966 . . . . . 6 |- ( 2 x. _i ) e. CC
18 2muliap0 8968 . . . . . 6 |- ( 2 x. _i ) # 0
19 divnegap 8490 . . . . . 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 1308 . . . . 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 2173 . . 3 |- ( A e. CC -> -u ( sin ` A ) = ( -u ( ( exp ` ( _i x. A ) ) - ( exp ` ( -u _i x. A ) ) ) / ( 2 x. _i ) ) )
23 mulneg12 8183 . . . . . . . . 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 2146 . . . . . . 7 |- ( A e. CC -> ( _i x. -u A ) = ( -u _i x. A ) )
2625fveq2d 5433 . . . . . 6 |- ( A e. CC -> ( exp ` ( _i x. -u A ) ) = ( exp ` ( -u _i x. A ) ) )
27 mul2neg 8184 . . . . . . . 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 5433 . . . . . 6 |- ( A e. CC -> ( exp ` ( -u _i x. -u A ) ) = ( exp ` ( _i x. A ) ) )
3026, 29oveq12d 5800 . . . . 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 8113 . . . . 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 2176 . . . 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 5797 . . 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 2176 . 2 |- ( A e. CC -> -u ( sin ` A ) = ( ( ( exp ` ( _i x. -u A ) ) - ( exp ` ( -u _i x. -u A ) ) ) / ( 2 x. _i ) ) )
353, 34eqtr4d 2176 1 |- ( A e. CC -> ( sin ` -u A ) = -u ( sin ` A ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   = wceq 1332   e. wcel 1481   class class class wbr 3937   ` cfv 5131  (class class class)co 5782   CCcc 7642   0cc0 7644   _ici 7646   x. cmul 7649   - cmin 7957   -ucneg 7958   # cap 8367   / cdiv 8456   2c2 8795   expce 11385   sincsin 11387
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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054  ax-nul 4062  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-iinf 4510  ax-cnex 7735  ax-resscn 7736  ax-1cn 7737  ax-1re 7738  ax-icn 7739  ax-addcl 7740  ax-addrcl 7741  ax-mulcl 7742  ax-mulrcl 7743  ax-addcom 7744  ax-mulcom 7745  ax-addass 7746  ax-mulass 7747  ax-distr 7748  ax-i2m1 7749  ax-0lt1 7750  ax-1rid 7751  ax-0id 7752  ax-rnegex 7753  ax-precex 7754  ax-cnre 7755  ax-pre-ltirr 7756  ax-pre-ltwlin 7757  ax-pre-lttrn 7758  ax-pre-apti 7759  ax-pre-ltadd 7760  ax-pre-mulgt0 7761  ax-pre-mulext 7762  ax-arch 7763  ax-caucvg 7764
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2914  df-csb 3008  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-nul 3369  df-if 3480  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-int 3780  df-iun 3823  df-br 3938  df-opab 3998  df-mpt 3999  df-tr 4035  df-id 4223  df-po 4226  df-iso 4227  df-iord 4296  df-on 4298  df-ilim 4299  df-suc 4301  df-iom 4513  df-xp 4553  df-rel 4554  df-cnv 4555  df-co 4556  df-dm 4557  df-rn 4558  df-res 4559  df-ima 4560  df-iota 5096  df-fun 5133  df-fn 5134  df-f 5135  df-f1 5136  df-fo 5137  df-f1o 5138  df-fv 5139  df-isom 5140  df-riota 5738  df-ov 5785  df-oprab 5786  df-mpo 5787  df-1st 6046  df-2nd 6047  df-recs 6210  df-irdg 6275  df-frec 6296  df-1o 6321  df-oadd 6325  df-er 6437  df-en 6643  df-dom 6644  df-fin 6645  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829  df-le 7830  df-sub 7959  df-neg 7960  df-reap 8361  df-ap 8368  df-div 8457  df-inn 8745  df-2 8803  df-3 8804  df-4 8805  df-n0 9002  df-z 9079  df-uz 9351  df-q 9439  df-rp 9471  df-ico 9707  df-fz 9822  df-fzo 9951  df-seqfrec 10250  df-exp 10324  df-fac 10504  df-ihash 10554  df-cj 10646  df-re 10647  df-im 10648  df-rsqrt 10802  df-abs 10803  df-clim 11080  df-sumdc 11155  df-ef 11391  df-sin 11393
This theorem is referenced by:  tannegap  11471  sin0  11472  efmival  11476  sinsub  11483  cossub  11484  sincossq  11491  sin2pim  12942
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