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Theorem divmuldivap 7733
Description: Multiplication of two ratios. (Contributed by Jim Kingdon, 25-Feb-2020.)
Assertion
Ref Expression
divmuldivap  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )

Proof of Theorem divmuldivap
StepHypRef Expression
1 3anass 898 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )
2 3anass 898 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
3 divclap 7701 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
4 divclap 7701 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 7036 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 277 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( A  /  C
)  x.  ( B  /  D ) )  e.  CC )
7 mulcl 7036 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 486 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
983adantr1 1072 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1069 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulap0 7679 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
12113adantr1 1072 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
) #  0 )
13123adantl1 1069 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
14 divcanap3 7719 . . . . 5  |-  ( ( ( ( A  /  C )  x.  ( B  /  D ) )  e.  CC  /\  ( C  x.  D )  e.  CC  /\  ( C  x.  D ) #  0 )  ->  ( (
( C  x.  D
)  x.  ( ( A  /  C )  x.  ( B  /  D ) ) )  /  ( C  x.  D ) )  =  ( ( A  /  C )  x.  ( B  /  D ) ) )
156, 10, 13, 14syl3anc 1144 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( ( C  x.  D )  x.  (
( A  /  C
)  x.  ( B  /  D ) ) )  /  ( C  x.  D ) )  =  ( ( A  /  C )  x.  ( B  /  D
) ) )
16 simp2 914 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  C  e.  CC )
1716, 3jca 294 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 914 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  D  e.  CC )
1918, 4jca 294 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 7176 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  ( A  /  C
)  e.  CC )  /\  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )  -> 
( ( C  x.  ( A  /  C
) )  x.  ( D  x.  ( B  /  D ) ) )  =  ( ( C  x.  D )  x.  ( ( A  /  C )  x.  ( B  /  D ) ) ) )
2117, 19, 20syl2an 277 . . . . . 6  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( C  x.  ( A  /  C ) )  x.  ( D  x.  ( B  /  D
) ) )  =  ( ( C  x.  D )  x.  (
( A  /  C
)  x.  ( B  /  D ) ) ) )
22 divcanap2 7703 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcanap2 7703 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 5556 . . . . . 6  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( C  x.  ( A  /  C ) )  x.  ( D  x.  ( B  /  D
) ) )  =  ( A  x.  B
) )
2521, 24eqtr3d 2088 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( C  x.  D
)  x.  ( ( A  /  C )  x.  ( B  /  D ) ) )  =  ( A  x.  B ) )
2625oveq1d 5552 . . . 4  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( ( C  x.  D )  x.  (
( A  /  C
)  x.  ( B  /  D ) ) )  /  ( C  x.  D ) )  =  ( ( A  x.  B )  / 
( C  x.  D
) ) )
2715, 26eqtr3d 2088 . . 3  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  (
( A  /  C
)  x.  ( B  /  D ) )  =  ( ( A  x.  B )  / 
( C  x.  D
) ) )
281, 2, 27syl2anbr 280 . 2  |-  ( ( ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) )  /\  ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )
2928an4s 530 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC )  /\  ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) ) )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  =  ( ( A  x.  B
)  /  ( C  x.  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 101    /\ w3a 894    = wceq 1257    e. wcel 1407   class class class wbr 3789  (class class class)co 5537   CCcc 6915   0cc0 6917    x. cmul 6922   # cap 7616    / cdiv 7695
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 552  ax-in2 553  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-coll 3897  ax-sep 3900  ax-nul 3908  ax-pow 3952  ax-pr 3969  ax-un 4195  ax-setind 4287  ax-iinf 4336  ax-cnex 7003  ax-resscn 7004  ax-1cn 7005  ax-1re 7006  ax-icn 7007  ax-addcl 7008  ax-addrcl 7009  ax-mulcl 7010  ax-mulrcl 7011  ax-addcom 7012  ax-mulcom 7013  ax-addass 7014  ax-mulass 7015  ax-distr 7016  ax-i2m1 7017  ax-1rid 7019  ax-0id 7020  ax-rnegex 7021  ax-precex 7022  ax-cnre 7023  ax-pre-ltirr 7024  ax-pre-ltwlin 7025  ax-pre-lttrn 7026  ax-pre-apti 7027  ax-pre-ltadd 7028  ax-pre-mulgt0 7029  ax-pre-mulext 7030
This theorem depends on definitions:  df-bi 114  df-dc 752  df-3or 895  df-3an 896  df-tru 1260  df-fal 1263  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ne 2219  df-nel 2313  df-ral 2326  df-rex 2327  df-reu 2328  df-rmo 2329  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-dif 2945  df-un 2947  df-in 2949  df-ss 2956  df-nul 3250  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-int 3641  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-tr 3880  df-eprel 4051  df-id 4055  df-po 4058  df-iso 4059  df-iord 4128  df-on 4130  df-suc 4133  df-iom 4339  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-f1 4932  df-fo 4933  df-f1o 4934  df-fv 4935  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-irdg 5985  df-1o 6029  df-2o 6030  df-oadd 6033  df-omul 6034  df-er 6134  df-ec 6136  df-qs 6140  df-ni 6430  df-pli 6431  df-mi 6432  df-lti 6433  df-plpq 6470  df-mpq 6471  df-enq 6473  df-nqqs 6474  df-plqqs 6475  df-mqqs 6476  df-1nqqs 6477  df-rq 6478  df-ltnqqs 6479  df-enq0 6550  df-nq0 6551  df-0nq0 6552  df-plq0 6553  df-mq0 6554  df-inp 6592  df-i1p 6593  df-iplp 6594  df-iltp 6596  df-enr 6839  df-nr 6840  df-ltr 6843  df-0r 6844  df-1r 6845  df-0 6924  df-1 6925  df-r 6927  df-lt 6930  df-pnf 7091  df-mnf 7092  df-xr 7093  df-ltxr 7094  df-le 7095  df-sub 7217  df-neg 7218  df-reap 7610  df-ap 7617  df-div 7696
This theorem is referenced by:  divdivdivap  7734  divcanap5  7735  divmul13ap  7736  divmul24ap  7737  divmuldivapi  7793  divmuldivapd  7851  qmulcl  8639  mulexpzap  9425  expaddzap  9429  sqdivap  9449
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