Theorem sinmul 11747

Description: Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11740 and cossub 11744. (Contributed by David A. Wheeler, 26-May-2015.)

Assertion

Ref	Expression
sinmul	|- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) x. ( sin ` B ) ) = ( ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) / 2 ) )

Proof of Theorem sinmul

Step	Hyp	Ref	Expression
1		cossub 11744	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A - B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
2		cosadd 11740	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
3	1, 2	oveq12d 5892	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) = ( ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) - ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) )
4		coscl 11710	. . . . . 6 |- ( A e. CC -> ( cos ` A ) e. CC )
5		coscl 11710	. . . . . 6 |- ( B e. CC -> ( cos ` B ) e. CC )
6		mulcl 7937	. . . . . 6 |- ( ( ( cos ` A ) e. CC /\ ( cos ` B ) e. CC ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
7	4, 5, 6	syl2an 289	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` A ) x. ( cos ` B ) ) e. CC )
8		sincl 11709	. . . . . 6 |- ( A e. CC -> ( sin ` A ) e. CC )
9		sincl 11709	. . . . . 6 |- ( B e. CC -> ( sin ` B ) e. CC )
10		mulcl 7937	. . . . . 6 |- ( ( ( sin ` A ) e. CC /\ ( sin ` B ) e. CC ) -> ( ( sin ` A ) x. ( sin ` B ) ) e. CC )
11	8, 9, 10	syl2an 289	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) x. ( sin ` B ) ) e. CC )
12		pnncan 8196	. . . . . . 7 |- ( ( ( ( cos ` A ) x. ( cos ` B ) ) e. CC /\ ( ( sin ` A ) x. ( sin ` B ) ) e. CC /\ ( ( sin ` A ) x. ( sin ` B ) ) e. CC ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) - ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
13	12	3anidm23 1297	. . . . . 6 |- ( ( ( ( cos ` A ) x. ( cos ` B ) ) e. CC /\ ( ( sin ` A ) x. ( sin ` B ) ) e. CC ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) - ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
14		2times 9045	. . . . . . 7 |- ( ( ( sin ` A ) x. ( sin ` B ) ) e. CC -> ( 2 x. ( ( sin ` A ) x. ( sin ` B ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
15	14	adantl 277	. . . . . 6 |- ( ( ( ( cos ` A ) x. ( cos ` B ) ) e. CC /\ ( ( sin ` A ) x. ( sin ` B ) ) e. CC ) -> ( 2 x. ( ( sin ` A ) x. ( sin ` B ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
16	13, 15	eqtr4d 2213	. . . . 5 |- ( ( ( ( cos ` A ) x. ( cos ` B ) ) e. CC /\ ( ( sin ` A ) x. ( sin ` B ) ) e. CC ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) - ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) = ( 2 x. ( ( sin ` A ) x. ( sin ` B ) ) ) )
17	7, 11, 16	syl2anc 411	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) - ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) ) = ( 2 x. ( ( sin ` A ) x. ( sin ` B ) ) ) )
18		2cn 8988	. . . . 5 |- 2 e. CC
19		mulcom 7939	. . . . 5 |- ( ( 2 e. CC /\ ( ( sin ` A ) x. ( sin ` B ) ) e. CC ) -> ( 2 x. ( ( sin ` A ) x. ( sin ` B ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) )
20	18, 11, 19	sylancr 414	. . . 4 |- ( ( A e. CC /\ B e. CC ) -> ( 2 x. ( ( sin ` A ) x. ( sin ` B ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) )
21	3, 17, 20	3eqtrd 2214	. . 3 |- ( ( A e. CC /\ B e. CC ) -> ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) = ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) )
22	21	oveq1d 5889	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) / 2 ) = ( ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) / 2 ) )
23		2ap0 9010	. . . 4 |- 2 # 0
24		divcanap4 8654	. . . 4 |- ( ( ( ( sin ` A ) x. ( sin ` B ) ) e. CC /\ 2 e. CC /\ 2 # 0 ) -> ( ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) / 2 ) = ( ( sin ` A ) x. ( sin ` B ) ) )
25	18, 23, 24	mp3an23 1329	. . 3 |- ( ( ( sin ` A ) x. ( sin ` B ) ) e. CC -> ( ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) / 2 ) = ( ( sin ` A ) x. ( sin ` B ) ) )
26	11, 25	syl 14	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) / 2 ) = ( ( sin ` A ) x. ( sin ` B ) ) )
27	22, 26	eqtr2d 2211	1 |- ( ( A e. CC /\ B e. CC ) -> ( ( sin ` A ) x. ( sin ` B ) ) = ( ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) / 2 ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1353   e. wcel 2148   class class class wbr 4003   ` cfv 5216  (class class class)co 5874   CCcc 7808   0cc0 7810   + caddc 7813   x. cmul 7815   - cmin 8126   # cap 8536   / cdiv 8627   2c2 8968   sincsin 11647   cosccos 11648

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4118  ax-sep 4121  ax-nul 4129  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-iinf 4587  ax-cnex 7901  ax-resscn 7902  ax-1cn 7903  ax-1re 7904  ax-icn 7905  ax-addcl 7906  ax-addrcl 7907  ax-mulcl 7908  ax-mulrcl 7909  ax-addcom 7910  ax-mulcom 7911  ax-addass 7912  ax-mulass 7913  ax-distr 7914  ax-i2m1 7915  ax-0lt1 7916  ax-1rid 7917  ax-0id 7918  ax-rnegex 7919  ax-precex 7920  ax-cnre 7921  ax-pre-ltirr 7922  ax-pre-ltwlin 7923  ax-pre-lttrn 7924  ax-pre-apti 7925  ax-pre-ltadd 7926  ax-pre-mulgt0 7927  ax-pre-mulext 7928  ax-arch 7929  ax-caucvg 7930

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-int 3845  df-iun 3888  df-disj 3981  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-po 4296  df-iso 4297  df-iord 4366  df-on 4368  df-ilim 4369  df-suc 4371  df-iom 4590  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-isom 5225  df-riota 5830  df-ov 5877  df-oprab 5878  df-mpo 5879  df-1st 6140  df-2nd 6141  df-recs 6305  df-irdg 6370  df-frec 6391  df-1o 6416  df-oadd 6420  df-er 6534  df-en 6740  df-dom 6741  df-fin 6742  df-sup 6982  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996  df-sub 8128  df-neg 8129  df-reap 8530  df-ap 8537  df-div 8628  df-inn 8918  df-2 8976  df-3 8977  df-4 8978  df-n0 9175  df-z 9252  df-uz 9527  df-q 9618  df-rp 9652  df-ico 9892  df-fz 10007  df-fzo 10140  df-seqfrec 10443  df-exp 10517  df-fac 10701  df-bc 10723  df-ihash 10751  df-cj 10846  df-re 10847  df-im 10848  df-rsqrt 11002  df-abs 11003  df-clim 11282  df-sumdc 11357  df-ef 11651  df-sin 11653  df-cos 11654

This theorem is referenced by:  ptolemy  14138