MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sinmul Structured version   Unicode version

Theorem sinmul 12773
Description: Product of sines can be rewritten as half the difference of certain cosines. This follows from cosadd 12766 and cossub 12770. (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
StepHypRef Expression
1 cossub 12770 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  -  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  +  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
2 cosadd 12766 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( cos `  ( A  +  B )
)  =  ( ( ( cos `  A
)  x.  ( cos `  B ) )  -  ( ( sin `  A
)  x.  ( sin `  B ) ) ) )
31, 2oveq12d 6099 . . . 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 12728 . . . . . 6  |-  ( A  e.  CC  ->  ( cos `  A )  e.  CC )
5 coscl 12728 . . . . . 6  |-  ( B  e.  CC  ->  ( cos `  B )  e.  CC )
6 mulcl 9074 . . . . . 6  |-  ( ( ( cos `  A
)  e.  CC  /\  ( cos `  B )  e.  CC )  -> 
( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC )
74, 5, 6syl2an 464 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  A
)  x.  ( cos `  B ) )  e.  CC )
8 sincl 12727 . . . . . 6  |-  ( A  e.  CC  ->  ( sin `  A )  e.  CC )
9 sincl 12727 . . . . . 6  |-  ( B  e.  CC  ->  ( sin `  B )  e.  CC )
10 mulcl 9074 . . . . . 6  |-  ( ( ( sin `  A
)  e.  CC  /\  ( sin `  B )  e.  CC )  -> 
( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC )
118, 9, 10syl2an 464 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC )
12 pnncan 9342 . . . . . . 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 ) ) ) )
13123anidm23 1243 . . . . . 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 10099 . . . . . . 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 ) ) ) )
1514adantl 453 . . . . . 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 ) ) ) )
1613, 15eqtr4d 2471 . . . . 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 ) ) ) )
177, 11, 16syl2anc 643 . . . 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 10070 . . . . 5  |-  2  e.  CC
19 mulcom 9076 . . . . 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 ) )
2018, 11, 19sylancr 645 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( 2  x.  (
( sin `  A
)  x.  ( sin `  B ) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B
) )  x.  2 ) )
213, 17, 203eqtrd 2472 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( cos `  ( A  -  B )
)  -  ( cos `  ( A  +  B
) ) )  =  ( ( ( sin `  A )  x.  ( sin `  B ) )  x.  2 ) )
2221oveq1d 6096 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( cos `  ( A  -  B
) )  -  ( cos `  ( A  +  B ) ) )  /  2 )  =  ( ( ( ( sin `  A )  x.  ( sin `  B
) )  x.  2 )  /  2 ) )
23 2ne0 10083 . . . 4  |-  2  =/=  0
24 divcan4 9703 . . . 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 ) ) )
2518, 23, 24mp3an23 1271 . . 3  |-  ( ( ( sin `  A
)  x.  ( sin `  B ) )  e.  CC  ->  ( (
( ( sin `  A
)  x.  ( sin `  B ) )  x.  2 )  /  2
)  =  ( ( sin `  A )  x.  ( sin `  B
) ) )
2611, 25syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( ( ( sin `  A )  x.  ( sin `  B
) )  x.  2 )  /  2 )  =  ( ( sin `  A )  x.  ( sin `  B ) ) )
2722, 26eqtr2d 2469 1  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( sin `  A
)  x.  ( sin `  B ) )  =  ( ( ( cos `  ( A  -  B
) )  -  ( cos `  ( A  +  B ) ) )  /  2 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990    + caddc 8993    x. cmul 8995    - cmin 9291    / cdiv 9677   2c2 10049   sincsin 12666   cosccos 12667
This theorem is referenced by:  ptolemy  20404
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 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-pre-sup 9068  ax-addf 9069  ax-mulf 9070
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 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-pm 7021  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-ico 10922  df-fz 11044  df-fzo 11136  df-fl 11202  df-seq 11324  df-exp 11383  df-fac 11567  df-bc 11594  df-hash 11619  df-shft 11882  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-limsup 12265  df-clim 12282  df-rlim 12283  df-sum 12480  df-ef 12670  df-sin 12672  df-cos 12673
  Copyright terms: Public domain W3C validator