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Theorem divmulasscomap 7903
Description: An associative/commutative law for division and multiplication. (Contributed by Jim Kingdon, 24-Jan-2022.)
Assertion
Ref Expression
divmulasscomap  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( A  x.  ( B  /  D ) )  x.  C )  =  ( B  x.  ( ( A  x.  C )  /  D ) ) )

Proof of Theorem divmulasscomap
StepHypRef Expression
1 divmulassap 7902 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( A  x.  ( B  /  D ) )  x.  C )  =  ( ( A  x.  B
)  x.  ( C  /  D ) ) )
2 mulcom 7216 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
323adant3 959 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
43adantr 270 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( A  x.  B )  =  ( B  x.  A ) )
54oveq1d 5578 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( A  x.  B )  x.  ( C  /  D
) )  =  ( ( B  x.  A
)  x.  ( C  /  D ) ) )
6 simpl2 943 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  B  e.  CC )
7 simpl1 942 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  A  e.  CC )
8 simp3 941 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
98anim1i 333 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
10 3anass 924 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( C  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
119, 10sylibr 132 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  e.  CC  /\  D  e.  CC  /\  D #  0 ) )
12 divclap 7885 . . . . 5  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D #  0 )  ->  ( C  /  D )  e.  CC )
1311, 12syl 14 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( C  /  D )  e.  CC )
146, 7, 13mulassd 7256 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( B  x.  A )  x.  ( C  /  D
) )  =  ( B  x.  ( A  x.  ( C  /  D ) ) ) )
158adantr 270 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  C  e.  CC )
16 simpr 108 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( D  e.  CC  /\  D #  0 ) )
17 divassap 7897 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) )  -> 
( ( A  x.  C )  /  D
)  =  ( A  x.  ( C  /  D ) ) )
187, 15, 16, 17syl3anc 1170 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( A  x.  C )  /  D )  =  ( A  x.  ( C  /  D ) ) )
1918eqcomd 2088 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( A  x.  ( C  /  D
) )  =  ( ( A  x.  C
)  /  D ) )
2019oveq2d 5579 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( B  x.  ( A  x.  ( C  /  D ) ) )  =  ( B  x.  ( ( A  x.  C )  /  D ) ) )
2114, 20eqtrd 2115 . 2  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( B  x.  A )  x.  ( C  /  D
) )  =  ( B  x.  ( ( A  x.  C )  /  D ) ) )
221, 5, 213eqtrd 2119 1  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( A  x.  ( B  /  D ) )  x.  C )  =  ( B  x.  ( ( A  x.  C )  /  D ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285    e. wcel 1434   class class class wbr 3805  (class class class)co 5563   CCcc 7093   0cc0 7095    x. cmul 7100   # cap 7800    / cdiv 7879
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 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308  ax-cnex 7181  ax-resscn 7182  ax-1cn 7183  ax-1re 7184  ax-icn 7185  ax-addcl 7186  ax-addrcl 7187  ax-mulcl 7188  ax-mulrcl 7189  ax-addcom 7190  ax-mulcom 7191  ax-addass 7192  ax-mulass 7193  ax-distr 7194  ax-i2m1 7195  ax-0lt1 7196  ax-1rid 7197  ax-0id 7198  ax-rnegex 7199  ax-precex 7200  ax-cnre 7201  ax-pre-ltirr 7202  ax-pre-ltwlin 7203  ax-pre-lttrn 7204  ax-pre-apti 7205  ax-pre-ltadd 7206  ax-pre-mulgt0 7207  ax-pre-mulext 7208
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-nel 2345  df-ral 2358  df-rex 2359  df-reu 2360  df-rmo 2361  df-rab 2362  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-po 4079  df-iso 4080  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-iota 4917  df-fun 4954  df-fv 4960  df-riota 5519  df-ov 5566  df-oprab 5567  df-mpt2 5568  df-pnf 7269  df-mnf 7270  df-xr 7271  df-ltxr 7272  df-le 7273  df-sub 7400  df-neg 7401  df-reap 7794  df-ap 7801  df-div 7880
This theorem is referenced by:  cncongr2  10693
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