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Theorem sinneg 12426
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 9052 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 sinval 12402 . . 3  |-  ( -u A  e.  CC  ->  ( sin `  -u A
)  =  ( ( ( exp `  (
_i  x.  -u A ) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
31, 2syl 15 . 2  |-  ( A  e.  CC  ->  ( sin `  -u A )  =  ( ( ( exp `  ( _i  x.  -u A
) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
4 sinval 12402 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
54negeqd 9046 . . . 4  |-  ( A  e.  CC  ->  -u ( sin `  A )  = 
-u ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) ) )
6 ax-icn 8796 . . . . . . . 8  |-  _i  e.  CC
7 mulcl 8821 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
86, 7mpan 651 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
9 efcl 12364 . . . . . . 7  |-  ( ( _i  x.  A )  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
108, 9syl 15 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  A ) )  e.  CC )
116negcli 9114 . . . . . . . 8  |-  -u _i  e.  CC
12 mulcl 8821 . . . . . . . 8  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1311, 12mpan 651 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
14 efcl 12364 . . . . . . 7  |-  ( (
-u _i  x.  A
)  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1513, 14syl 15 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  A ) )  e.  CC )
1610, 15subcld 9157 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
17 2cn 9816 . . . . . . 7  |-  2  e.  CC
1817, 6mulcli 8842 . . . . . 6  |-  ( 2  x.  _i )  e.  CC
19 2ne0 9829 . . . . . . 7  |-  2  =/=  0
20 ine0 9215 . . . . . . 7  |-  _i  =/=  0
2117, 6, 19, 20mulne0i 9411 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
22 divneg 9455 . . . . . 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 1269 . . . . 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 15 . . . 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 2315 . . 3  |-  ( A  e.  CC  ->  -u ( sin `  A )  =  ( -u ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) ) )
26 mulneg12 9218 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
276, 26mpan 651 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2827eqcomd 2288 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  ( -u _i  x.  A ) )
2928fveq2d 5529 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  -u A ) )  =  ( exp `  ( -u _i  x.  A ) ) )
30 mul2neg 9219 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  -u A )  =  ( _i  x.  A ) )
316, 30mpan 651 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  -u A
)  =  ( _i  x.  A ) )
3231fveq2d 5529 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  -u A ) )  =  ( exp `  (
_i  x.  A )
) )
3329, 32oveq12d 5876 . . . . 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 9173 . . . . 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 2318 . . . 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 5873 . . 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 2318 . 2  |-  ( A  e.  CC  ->  -u ( sin `  A )  =  ( ( ( exp `  ( _i  x.  -u A
) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
383, 37eqtr4d 2318 1  |-  ( A  e.  CC  ->  ( sin `  -u A )  = 
-u ( sin `  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   _ici 8739    x. cmul 8742    - cmin 9037   -ucneg 9038    / cdiv 9423   2c2 9795   expce 12343   sincsin 12345
This theorem is referenced by:  tanneg  12428  sin0  12429  efmival  12433  sinsub  12448  cossub  12449  sincossq  12456  sin2pim  19853  reasinsin  20192  atantan  20219  sinccvglem  24005
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-pm 6775  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-ico 10662  df-fz 10783  df-fzo 10871  df-fl 10925  df-seq 11047  df-exp 11105  df-fac 11289  df-hash 11338  df-shft 11562  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-limsup 11945  df-clim 11962  df-rlim 11963  df-sum 12159  df-ef 12349  df-sin 12351
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