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Theorem divmuldivap 7856
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 924 . . 3  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  <->  ( A  e.  CC  /\  ( C  e.  CC  /\  C #  0 ) ) )
2 3anass 924 . . 3  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( B  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
3 divclap 7822 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( A  /  C )  e.  CC )
4 divclap 7822 . . . . . 6  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( B  /  D )  e.  CC )
5 mulcl 7151 . . . . . 6  |-  ( ( ( A  /  C
)  e.  CC  /\  ( B  /  D
)  e.  CC )  ->  ( ( A  /  C )  x.  ( B  /  D
) )  e.  CC )
63, 4, 5syl2an 283 . . . . 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 7151 . . . . . . . 8  |-  ( ( C  e.  CC  /\  D  e.  CC )  ->  ( C  x.  D
)  e.  CC )
87ad2ant2r 493 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
983adantr1 1098 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
)  e.  CC )
1093adantl1 1095 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D )  e.  CC )
11 mulap0 7800 . . . . . . 7  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
12113adantr1 1098 . . . . . 6  |-  ( ( ( C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  -> 
( C  x.  D
) #  0 )
13123adantl1 1095 . . . . 5  |-  ( ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  /\  ( B  e.  CC  /\  D  e.  CC  /\  D #  0 ) )  ->  ( C  x.  D ) #  0 )
14 divcanap3 7842 . . . . 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 1170 . . . 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 940 . . . . . . . 8  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  C  e.  CC )
1716, 3jca 300 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  e.  CC  /\  ( A  /  C )  e.  CC ) )
18 simp2 940 . . . . . . . 8  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  D  e.  CC )
1918, 4jca 300 . . . . . . 7  |-  ( ( B  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( D  e.  CC  /\  ( B  /  D )  e.  CC ) )
20 mul4 7296 . . . . . . 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 283 . . . . . 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 7824 . . . . . . 7  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  C #  0 )  ->  ( C  x.  ( A  /  C ) )  =  A )
23 divcanap2 7824 . . . . . . 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 2116 . . . . 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 2116 . . 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 286 . 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 553 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 102    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3787  (class class class)co 5537   CCcc 7030   0cc0 7032    x. cmul 7037   # cap 7737    / cdiv 7816
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-br 3788  df-opab 3842  df-id 4050  df-po 4053  df-iso 4054  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-iota 4891  df-fun 4928  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817
This theorem is referenced by:  divdivdivap  7857  divcanap5  7858  divmul13ap  7859  divmul24ap  7860  divmuldivapi  7916  divmuldivapd  7974  qmulcl  8792  mulexpzap  9602  expaddzap  9606  sqdivap  9626
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