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Theorem divmuleqap 8384
Description: Cross-multiply in an equality of ratios. (Contributed by Jim Kingdon, 26-Feb-2020.)
Assertion
Ref Expression
divmuleqap  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  =  ( B  /  D
)  <->  ( A  x.  D )  =  ( B  x.  C ) ) )

Proof of Theorem divmuleqap
StepHypRef Expression
1 divclap 8345 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
213expb 1163 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  C )  e.  CC )
32ad2ant2r 498 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( A  /  C )  e.  CC )
4 divclap 8345 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
543expb 1163 . . . 4  |-  ( ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( B  /  D )  e.  CC )
65ad2ant2l 497 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  /  D )  e.  CC )
7 mulcl 7665 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 498 . . . . 5  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
9 mulap0 8322 . . . . 5  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
108, 9jca 302 . . . 4  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( C  x.  D )  e.  CC  /\  ( C  x.  D ) #  0 ) )
1110adantl 273 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( C  x.  D )  e.  CC  /\  ( C  x.  D ) #  0 ) )
12 mulcanap2 8334 . . 3  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC  /\  ( ( C  x.  D )  e.  CC  /\  ( C  x.  D
) #  0 ) )  ->  ( ( ( A  /  C )  x.  ( C  x.  D ) )  =  ( ( B  /  D )  x.  ( C  x.  D )
)  <->  ( A  /  C )  =  ( B  /  D ) ) )
133, 6, 11, 12syl3anc 1197 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  /  C )  x.  ( C  x.  D ) )  =  ( ( B  /  D )  x.  ( C  x.  D )
)  <->  ( A  /  C )  =  ( B  /  D ) ) )
14 simprll 509 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C  e.  CC )
15 simprrl 511 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D  e.  CC )
163, 14, 15mulassd 7707 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  /  C )  x.  C )  x.  D )  =  ( ( A  /  C
)  x.  ( C  x.  D ) ) )
17 divcanap1 8348 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  (
( A  /  C
)  x.  C )  =  A )
18173expb 1163 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  C )  x.  C )  =  A )
1918ad2ant2r 498 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  C )  =  A )
2019oveq1d 5741 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  /  C )  x.  C )  x.  D )  =  ( A  x.  D ) )
2116, 20eqtr3d 2147 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( C  x.  D
) )  =  ( A  x.  D ) )
2214, 15mulcomd 7705 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
2322oveq2d 5742 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( B  /  D )  x.  ( C  x.  D
) )  =  ( ( B  /  D
)  x.  ( D  x.  C ) ) )
246, 15, 14mulassd 7707 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( B  /  D )  x.  D )  x.  C )  =  ( ( B  /  D
)  x.  ( D  x.  C ) ) )
25 divcanap1 8348 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  (
( B  /  D
)  x.  D )  =  B )
26253expb 1163 . . . . . 6  |-  ( ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( B  /  D )  x.  D )  =  B )
2726ad2ant2l 497 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( B  /  D )  x.  D )  =  B )
2827oveq1d 5741 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( B  /  D )  x.  D )  x.  C )  =  ( B  x.  C ) )
2923, 24, 283eqtr2d 2151 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( B  /  D )  x.  ( C  x.  D
) )  =  ( B  x.  C ) )
3021, 29eqeq12d 2127 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( ( A  /  C )  x.  ( C  x.  D ) )  =  ( ( B  /  D )  x.  ( C  x.  D )
)  <->  ( A  x.  D )  =  ( B  x.  C ) ) )
3113, 30bitr3d 189 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  =  ( B  /  D
)  <->  ( A  x.  D )  =  ( B  x.  C ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1312    e. wcel 1461   class class class wbr 3893  (class class class)co 5726   CCcc 7539   0cc0 7541    x. cmul 7546   # cap 8255    / cdiv 8339
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-pow 4056  ax-pr 4089  ax-un 4313  ax-setind 4410  ax-cnex 7630  ax-resscn 7631  ax-1cn 7632  ax-1re 7633  ax-icn 7634  ax-addcl 7635  ax-addrcl 7636  ax-mulcl 7637  ax-mulrcl 7638  ax-addcom 7639  ax-mulcom 7640  ax-addass 7641  ax-mulass 7642  ax-distr 7643  ax-i2m1 7644  ax-0lt1 7645  ax-1rid 7646  ax-0id 7647  ax-rnegex 7648  ax-precex 7649  ax-cnre 7650  ax-pre-ltirr 7651  ax-pre-ltwlin 7652  ax-pre-lttrn 7653  ax-pre-apti 7654  ax-pre-ltadd 7655  ax-pre-mulgt0 7656  ax-pre-mulext 7657
This theorem depends on definitions:  df-bi 116  df-3an 945  df-tru 1315  df-fal 1318  df-nf 1418  df-sb 1717  df-eu 1976  df-mo 1977  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-ne 2281  df-nel 2376  df-ral 2393  df-rex 2394  df-reu 2395  df-rmo 2396  df-rab 2397  df-v 2657  df-sbc 2877  df-dif 3037  df-un 3039  df-in 3041  df-ss 3048  df-pw 3476  df-sn 3497  df-pr 3498  df-op 3500  df-uni 3701  df-br 3894  df-opab 3948  df-id 4173  df-po 4176  df-iso 4177  df-xp 4503  df-rel 4504  df-cnv 4505  df-co 4506  df-dm 4507  df-iota 5044  df-fun 5081  df-fv 5087  df-riota 5682  df-ov 5729  df-oprab 5730  df-mpo 5731  df-pnf 7720  df-mnf 7721  df-xr 7722  df-ltxr 7723  df-le 7724  df-sub 7852  df-neg 7853  df-reap 8249  df-ap 8256  df-div 8340
This theorem is referenced by:  qtri3or  9907
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