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Theorem sinneg 13759
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 9832 . . 3  |-  ( A  e.  CC  ->  -u A  e.  CC )
2 sinval 13735 . . 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 13735 . . . . 5  |-  ( A  e.  CC  ->  ( sin `  A )  =  ( ( ( exp `  ( _i  x.  A
) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  ( 2  x.  _i ) ) )
54negeqd 9826 . . . 4  |-  ( A  e.  CC  ->  -u ( sin `  A )  = 
-u ( ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) ) )
6 ax-icn 9563 . . . . . . . 8  |-  _i  e.  CC
7 mulcl 9588 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( _i  x.  A
)  e.  CC )
86, 7mpan 670 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  A )  e.  CC )
9 efcl 13697 . . . . . . 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 )
11 negicn 9833 . . . . . . . 8  |-  -u _i  e.  CC
12 mulcl 9588 . . . . . . . 8  |-  ( (
-u _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  e.  CC )
1311, 12mpan 670 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  e.  CC )
14 efcl 13697 . . . . . . 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 9942 . . . . 5  |-  ( A  e.  CC  ->  (
( exp `  (
_i  x.  A )
)  -  ( exp `  ( -u _i  x.  A ) ) )  e.  CC )
17 2mulicn 10774 . . . . . 6  |-  ( 2  x.  _i )  e.  CC
18 2muline0 10775 . . . . . 6  |-  ( 2  x.  _i )  =/=  0
19 divneg 10251 . . . . . 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 1316 . . . . 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 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 ) ) )
225, 21eqtrd 2508 . . 3  |-  ( A  e.  CC  ->  -u ( sin `  A )  =  ( -u ( ( exp `  ( _i  x.  A ) )  -  ( exp `  ( -u _i  x.  A ) ) )  /  (
2  x.  _i ) ) )
23 mulneg12 10007 . . . . . . . . 9  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A
) )
246, 23mpan 670 . . . . . . . 8  |-  ( A  e.  CC  ->  ( -u _i  x.  A )  =  ( _i  x.  -u A ) )
2524eqcomd 2475 . . . . . . 7  |-  ( A  e.  CC  ->  (
_i  x.  -u A )  =  ( -u _i  x.  A ) )
2625fveq2d 5876 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( _i  x.  -u A ) )  =  ( exp `  ( -u _i  x.  A ) ) )
27 mul2neg 10008 . . . . . . . 8  |-  ( ( _i  e.  CC  /\  A  e.  CC )  ->  ( -u _i  x.  -u A )  =  ( _i  x.  A ) )
286, 27mpan 670 . . . . . . 7  |-  ( A  e.  CC  ->  ( -u _i  x.  -u A
)  =  ( _i  x.  A ) )
2928fveq2d 5876 . . . . . 6  |-  ( A  e.  CC  ->  ( exp `  ( -u _i  x.  -u A ) )  =  ( exp `  (
_i  x.  A )
) )
3026, 29oveq12d 6313 . . . . 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 9958 . . . . 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 2511 . . . 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 6310 . . 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 2511 . 2  |-  ( A  e.  CC  ->  -u ( sin `  A )  =  ( ( ( exp `  ( _i  x.  -u A
) )  -  ( exp `  ( -u _i  x.  -u A ) ) )  /  ( 2  x.  _i ) ) )
353, 34eqtr4d 2511 1  |-  ( A  e.  CC  ->  ( sin `  -u A )  = 
-u ( sin `  A
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767    =/= wne 2662   ` cfv 5594  (class class class)co 6295   CCcc 9502   0cc0 9504   _ici 9506    x. cmul 9509    - cmin 9817   -ucneg 9818    / cdiv 10218   2c2 10597   expce 13676   sincsin 13678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582  ax-addf 9583  ax-mulf 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-se 4845  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-isom 5603  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-pm 7435  df-en 7529  df-dom 7530  df-sdom 7531  df-fin 7532  df-sup 7913  df-oi 7947  df-card 8332  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-ico 11547  df-fz 11685  df-fzo 11805  df-fl 11909  df-seq 12088  df-exp 12147  df-fac 12334  df-hash 12386  df-shft 12880  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-limsup 13274  df-clim 13291  df-rlim 13292  df-sum 13489  df-ef 13682  df-sin 13684
This theorem is referenced by:  tanneg  13761  sin0  13762  efmival  13766  sinsub  13781  cossub  13782  sincossq  13789  sin2pim  22744  reasinsin  23093  atantan  23120  sinccvglem  28863  dirkertrigeqlem2  31722  fourierdlem43  31773  fourierdlem44  31774  sqwvfoura  31852
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