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Theorem sinneg 12739
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 9298 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 sinval 12715 . . 3  |-  ( -u A  e.  CC  ->  ( sin `  -u A
)  =  ( ( ( exp `  (
_i  x.  -u A ) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
31, 2syl 16 . 2  |-  ( A  e.  CC  ->  ( sin `  -u A )  =  ( ( ( exp `  ( _i  x.  -u A
) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
4 sinval 12715 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
54negeqd 9292 . . . 4  |-  ( A  e.  CC  ->  -u ( sin `  A )  = 
-u ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) ) )
6 ax-icn 9041 . . . . . . . 8  |-  _i  e.  CC
7 mulcl 9066 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
86, 7mpan 652 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
9 efcl 12677 . . . . . . 7  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
108, 9syl 16 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
116negcli 9360 . . . . . . . 8  |-  -u _i  e.  CC
12 mulcl 9066 . . . . . . . 8  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1311, 12mpan 652 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
14 efcl 12677 . . . . . . 7  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1513, 14syl 16 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1610, 15subcld 9403 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
17 2cn 10062 . . . . . . 7  |-  2  e.  CC
1817, 6mulcli 9087 . . . . . 6  |-  ( 2  x.  _i )  e.  CC
19 2ne0 10075 . . . . . . 7  |-  2  =/=  0
20 ine0 9461 . . . . . . 7  |-  _i  =/=  0
2117, 6, 19, 20mulne0i 9657 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
22 divneg 9701 . . . . . 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 ) ) )
2318, 21, 22mp3an23 1271 . . . . 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 ) ) )
2416, 23syl 16 . . . 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 ) ) )
255, 24eqtrd 2467 . . 3  |-  ( A  e.  CC  ->  -u ( sin `  A )  =  ( -u ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) ) )
26 mulneg12 9464 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
276, 26mpan 652 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2827eqcomd 2440 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  ( -u _i  x.  A ) )
2928fveq2d 5724 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  -u A ) )  =  ( exp `  ( -u _i  x.  A ) ) )
30 mul2neg 9465 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  -u A )  =  ( _i  x.  A ) )
316, 30mpan 652 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  -u A
)  =  ( _i  x.  A ) )
3231fveq2d 5724 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  -u A ) )  =  ( exp `  (
_i  x.  A )
) )
3329, 32oveq12d 6091 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  -u A ) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  =  ( ( exp `  ( -u _i  x.  A ) )  -  ( exp `  ( _i  x.  A ) ) ) )
3410, 15negsubdi2d 9419 . . . . 5  |-  ( A  e.  CC  ->  -u (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  =  ( ( exp `  ( -u _i  x.  A ) )  -  ( exp `  ( _i  x.  A ) ) ) )
3533, 34eqtr4d 2470 . . . 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 ) ) ) )
3635oveq1d 6088 . . 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 ) ) )
3725, 36eqtr4d 2470 . 2  |-  ( A  e.  CC  ->  -u ( sin `  A )  =  ( ( ( exp `  ( _i  x.  -u A
) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
383, 37eqtr4d 2470 1  |-  ( A  e.  CC  ->  ( sin `  -u A )  = 
-u ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725    =/= wne 2598   ` cfv 5446  (class class class)co 6073   CCcc 8980   0cc0 8982   _ici 8984    x. cmul 8987    - cmin 9283   -ucneg 9284    / cdiv 9669   2c2 10041   expce 12656   sincsin 12658
This theorem is referenced by:  tanneg  12741  sin0  12742  efmival  12746  sinsub  12761  cossub  12762  sincossq  12769  sin2pim  20385  reasinsin  20728  atantan  20755  sinccvglem  25101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-inf2 7588  ax-cnex 9038  ax-resscn 9039  ax-1cn 9040  ax-icn 9041  ax-addcl 9042  ax-addrcl 9043  ax-mulcl 9044  ax-mulrcl 9045  ax-mulcom 9046  ax-addass 9047  ax-mulass 9048  ax-distr 9049  ax-i2m1 9050  ax-1ne0 9051  ax-1rid 9052  ax-rnegex 9053  ax-rrecex 9054  ax-cnre 9055  ax-pre-lttri 9056  ax-pre-lttrn 9057  ax-pre-ltadd 9058  ax-pre-mulgt0 9059  ax-pre-sup 9060  ax-addf 9061  ax-mulf 9062
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-se 4534  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-isom 5455  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-oadd 6720  df-er 6897  df-pm 7013  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-oi 7471  df-card 7818  df-pnf 9114  df-mnf 9115  df-xr 9116  df-ltxr 9117  df-le 9118  df-sub 9285  df-neg 9286  df-div 9670  df-nn 9993  df-2 10050  df-3 10051  df-n0 10214  df-z 10275  df-uz 10481  df-rp 10605  df-ico 10914  df-fz 11036  df-fzo 11128  df-fl 11194  df-seq 11316  df-exp 11375  df-fac 11559  df-hash 11611  df-shft 11874  df-cj 11896  df-re 11897  df-im 11898  df-sqr 12032  df-abs 12033  df-limsup 12257  df-clim 12274  df-rlim 12275  df-sum 12472  df-ef 12662  df-sin 12664
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