ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  divmuldivap Unicode version

Theorem divmuldivap 8465
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 966 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )
2 3anass 966 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
3 divclap 8431 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
4 divclap 8431 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 7740 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 287 . . . . 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 7740 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 500 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
983adantr1 1140 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1137 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulap0 8408 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
12113adantr1 1140 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
) #  0 )
13123adantl1 1137 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
14 divcanap3 8451 . . . . 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 1216 . . . 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 982 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  C  e.  CC )
1716, 3jca 304 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 982 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  D  e.  CC )
1918, 4jca 304 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 7887 . . . . . . 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 287 . . . . . 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 8433 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcanap2 8433 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  x.  ( B  /  D ) )  =  B )
2422, 23oveqan12d 5786 . . . . . 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 2172 . . . . 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 5782 . . . 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 2172 . . 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 290 . 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 577 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 103    /\ w3a 962    = wceq 1331    e. wcel 1480   class class class wbr 3924  (class class class)co 5767   CCcc 7611   0cc0 7613    x. cmul 7618   # cap 8336    / cdiv 8425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-1cn 7706  ax-1re 7707  ax-icn 7708  ax-addcl 7709  ax-addrcl 7710  ax-mulcl 7711  ax-mulrcl 7712  ax-addcom 7713  ax-mulcom 7714  ax-addass 7715  ax-mulass 7716  ax-distr 7717  ax-i2m1 7718  ax-0lt1 7719  ax-1rid 7720  ax-0id 7721  ax-rnegex 7722  ax-precex 7723  ax-cnre 7724  ax-pre-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727  ax-pre-apti 7728  ax-pre-ltadd 7729  ax-pre-mulgt0 7730  ax-pre-mulext 7731
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-reu 2421  df-rmo 2422  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-riota 5723  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-sub 7928  df-neg 7929  df-reap 8330  df-ap 8337  df-div 8426
This theorem is referenced by:  divdivdivap  8466  divcanap5  8467  divmul13ap  8468  divmul24ap  8469  divmuldivapi  8525  divmuldivapd  8585  qmulcl  9422  mulexpzap  10326  expaddzap  10330  sqdivap  10350
  Copyright terms: Public domain W3C validator