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Theorem divmuldivap 8694
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 984 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )
2 3anass 984 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
3 divclap 8660 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
4 divclap 8660 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 7963 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 289 . . . . 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 7963 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 509 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
983adantr1 1158 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1155 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulap0 8636 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
12113adantr1 1158 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
) #  0 )
13123adantl1 1155 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
14 divcanap3 8680 . . . . 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 1249 . . . 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 1000 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  C  e.  CC )
1716, 3jca 306 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 1000 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  D  e.  CC )
1918, 4jca 306 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 8114 . . . . . . 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 289 . . . . . 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 8662 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcanap2 8662 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 5911 . . . . . 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 2224 . . . . 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 5907 . . . 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 2224 . . 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 292 . 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 588 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 104    /\ w3a 980    = wceq 1364    e. wcel 2160   class class class wbr 4018  (class class class)co 5892   CCcc 7834   0cc0 7836    x. cmul 7841   # cap 8563    / cdiv 8654
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-cnex 7927  ax-resscn 7928  ax-1cn 7929  ax-1re 7930  ax-icn 7931  ax-addcl 7932  ax-addrcl 7933  ax-mulcl 7934  ax-mulrcl 7935  ax-addcom 7936  ax-mulcom 7937  ax-addass 7938  ax-mulass 7939  ax-distr 7940  ax-i2m1 7941  ax-0lt1 7942  ax-1rid 7943  ax-0id 7944  ax-rnegex 7945  ax-precex 7946  ax-cnre 7947  ax-pre-ltirr 7948  ax-pre-ltwlin 7949  ax-pre-lttrn 7950  ax-pre-apti 7951  ax-pre-ltadd 7952  ax-pre-mulgt0 7953  ax-pre-mulext 7954
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-reu 2475  df-rmo 2476  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-br 4019  df-opab 4080  df-id 4308  df-po 4311  df-iso 4312  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-iota 5193  df-fun 5234  df-fv 5240  df-riota 5848  df-ov 5895  df-oprab 5896  df-mpo 5897  df-pnf 8019  df-mnf 8020  df-xr 8021  df-ltxr 8022  df-le 8023  df-sub 8155  df-neg 8156  df-reap 8557  df-ap 8564  df-div 8655
This theorem is referenced by:  divdivdivap  8695  divcanap5  8696  divmul13ap  8697  divmul24ap  8698  divmuldivapi  8754  divmuldivapd  8814  qmulcl  9662  mulexpzap  10586  expaddzap  10590  sqdivap  10610
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