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Theorem divmuleqap 8158
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 8119 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
213expb 1144 . . . 4  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( A  /  C )  e.  CC )
32ad2ant2r 493 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( A  /  C )  e.  CC )
4 divclap 8119 . . . . 5  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
543expb 1144 . . . 4  |-  ( ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( B  /  D )  e.  CC )
65ad2ant2l 492 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( B  /  D )  e.  CC )
7 mulcl 7448 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 493 . . . . 5  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
9 mulap0 8097 . . . . 5  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
108, 9jca 300 . . . 4  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( C  x.  D )  e.  CC  /\  ( C  x.  D ) #  0 ) )
1110adantl 271 . . 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 8109 . . 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 1174 . 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 504 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  C  e.  CC )
15 simprrl 506 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  D  e.  CC )
163, 14, 15mulassd 7490 . . . 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 8122 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  (
( A  /  C
)  x.  C )  =  A )
18173expb 1144 . . . . . 6  |-  ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  ->  ( ( A  /  C )  x.  C )  =  A )
1918ad2ant2r 493 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  C )  =  A )
2019oveq1d 5649 . . . 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 2122 . . 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 7488 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( C  x.  D )  =  ( D  x.  C ) )
2322oveq2d 5650 . . . 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 7490 . . . 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 8122 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  (
( B  /  D
)  x.  D )  =  B )
26253expb 1144 . . . . . 6  |-  ( ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( B  /  D )  x.  D )  =  B )
2726ad2ant2l 492 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( B  /  D )  x.  D )  =  B )
2827oveq1d 5649 . . . 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 2126 . . 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 2102 . 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 188 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 102    <-> wb 103    = wceq 1289    e. wcel 1438   class class class wbr 3837  (class class class)co 5634   CCcc 7327   0cc0 7329    x. cmul 7334   # cap 8034    / cdiv 8113
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343  ax-cnex 7415  ax-resscn 7416  ax-1cn 7417  ax-1re 7418  ax-icn 7419  ax-addcl 7420  ax-addrcl 7421  ax-mulcl 7422  ax-mulrcl 7423  ax-addcom 7424  ax-mulcom 7425  ax-addass 7426  ax-mulass 7427  ax-distr 7428  ax-i2m1 7429  ax-0lt1 7430  ax-1rid 7431  ax-0id 7432  ax-rnegex 7433  ax-precex 7434  ax-cnre 7435  ax-pre-ltirr 7436  ax-pre-ltwlin 7437  ax-pre-lttrn 7438  ax-pre-apti 7439  ax-pre-ltadd 7440  ax-pre-mulgt0 7441  ax-pre-mulext 7442
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-nel 2351  df-ral 2364  df-rex 2365  df-reu 2366  df-rmo 2367  df-rab 2368  df-v 2621  df-sbc 2839  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-iso 4115  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-iota 4967  df-fun 5004  df-fv 5010  df-riota 5590  df-ov 5637  df-oprab 5638  df-mpt2 5639  df-pnf 7503  df-mnf 7504  df-xr 7505  df-ltxr 7506  df-le 7507  df-sub 7634  df-neg 7635  df-reap 8028  df-ap 8035  df-div 8114
This theorem is referenced by:  qtri3or  9619
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