Theorem sinmul 11754

Description: Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 11747 and cossub 11751. (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 11751	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A - B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) + ( ( sin ` A ) x. ( sin ` B ) ) ) )
2		cosadd 11747	. . . . 5 |- ( ( A e. CC /\ B e. CC ) -> ( cos ` ( A + B ) ) = ( ( ( cos ` A ) x. ( cos ` B ) ) - ( ( sin ` A ) x. ( sin ` B ) ) ) )
3	1, 2	oveq12d 5895	. . . 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 11717	. . . . . 6 |- ( A e. CC -> ( cos ` A ) e. CC )
5		coscl 11717	. . . . . 6 |- ( B e. CC -> ( cos ` B ) e. CC )
6		mulcl 7940	. . . . . 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 11716	. . . . . 6 |- ( A e. CC -> ( sin ` A ) e. CC )
9		sincl 11716	. . . . . 6 |- ( B e. CC -> ( sin ` B ) e. CC )
10		mulcl 7940	. . . . . 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 8200	. . . . . . 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 9049	. . . . . . 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 8992	. . . . 5 |- 2 e. CC
19		mulcom 7942	. . . . 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 5892	. 2 |- ( ( A e. CC /\ B e. CC ) -> ( ( ( cos ` ( A - B ) ) - ( cos ` ( A + B ) ) ) / 2 ) = ( ( ( ( sin ` A ) x. ( sin ` B ) ) x. 2 ) / 2 ) )
23		2ap0 9014	. . . 4 |- 2 # 0
24		divcanap4 8658	. . . 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 4005   ` cfv 5218  (class class class)co 5877   CCcc 7811   0cc0 7813   + caddc 7816   x. cmul 7818   - cmin 8130   # cap 8540   / cdiv 8631   2c2 8972   sincsin 11654   cosccos 11655

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 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7904  ax-resscn 7905  ax-1cn 7906  ax-1re 7907  ax-icn 7908  ax-addcl 7909  ax-addrcl 7910  ax-mulcl 7911  ax-mulrcl 7912  ax-addcom 7913  ax-mulcom 7914  ax-addass 7915  ax-mulass 7916  ax-distr 7917  ax-i2m1 7918  ax-0lt1 7919  ax-1rid 7920  ax-0id 7921  ax-rnegex 7922  ax-precex 7923  ax-cnre 7924  ax-pre-ltirr 7925  ax-pre-ltwlin 7926  ax-pre-lttrn 7927  ax-pre-apti 7928  ax-pre-ltadd 7929  ax-pre-mulgt0 7930  ax-pre-mulext 7931  ax-arch 7932  ax-caucvg 7933

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 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-disj 3983  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5833  df-ov 5880  df-oprab 5881  df-mpo 5882  df-1st 6143  df-2nd 6144  df-recs 6308  df-irdg 6373  df-frec 6394  df-1o 6419  df-oadd 6423  df-er 6537  df-en 6743  df-dom 6744  df-fin 6745  df-sup 6985  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999  df-le 8000  df-sub 8132  df-neg 8133  df-reap 8534  df-ap 8541  df-div 8632  df-inn 8922  df-2 8980  df-3 8981  df-4 8982  df-n0 9179  df-z 9256  df-uz 9531  df-q 9622  df-rp 9656  df-ico 9896  df-fz 10011  df-fzo 10145  df-seqfrec 10448  df-exp 10522  df-fac 10708  df-bc 10730  df-ihash 10758  df-cj 10853  df-re 10854  df-im 10855  df-rsqrt 11009  df-abs 11010  df-clim 11289  df-sumdc 11364  df-ef 11658  df-sin 11660  df-cos 11661

This theorem is referenced by:  ptolemy  14330