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

Theorem divmulasscomap 8629
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 8628 . 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 7918 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  =  ( B  x.  A ) )
323adant3 1017 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
43adantr 276 . . 3  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( A  x.  B )  =  ( B  x.  A ) )
54oveq1d 5883 . 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 1001 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  B  e.  CC )
7 simpl1 1000 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  A  e.  CC )
8 simp3 999 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  C  e.  CC )
98anim1i 340 . . . . . 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 982 . . . . . 6  |-  ( ( C  e.  CC  /\  D  e.  CC  /\  D #  0 )  <->  ( C  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) ) )
119, 10sylibr 134 . . . . 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 8611 . . . . 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 7958 . . 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 276 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  C  e.  CC )
16 simpr 110 . . . . . 6  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( D  e.  CC  /\  D #  0 ) )
17 divassap 8623 . . . . . 6  |-  ( ( A  e.  CC  /\  C  e.  CC  /\  ( D  e.  CC  /\  D #  0 ) )  -> 
( ( A  x.  C )  /  D
)  =  ( A  x.  ( C  /  D ) ) )
187, 15, 16, 17syl3anc 1238 . . . . 5  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( ( A  x.  C )  /  D )  =  ( A  x.  ( C  /  D ) ) )
1918eqcomd 2183 . . . 4  |-  ( ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  /\  ( D  e.  CC  /\  D #  0 ) )  ->  ( A  x.  ( C  /  D
) )  =  ( ( A  x.  C
)  /  D ) )
2019oveq2d 5884 . . 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 2210 . 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 2214 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 104    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4000  (class class class)co 5868   CCcc 7787   0cc0 7789    x. cmul 7794   # cap 8515    / cdiv 8605
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1cn 7882  ax-1re 7883  ax-icn 7884  ax-addcl 7885  ax-addrcl 7886  ax-mulcl 7887  ax-mulrcl 7888  ax-addcom 7889  ax-mulcom 7890  ax-addass 7891  ax-mulass 7892  ax-distr 7893  ax-i2m1 7894  ax-0lt1 7895  ax-1rid 7896  ax-0id 7897  ax-rnegex 7898  ax-precex 7899  ax-cnre 7900  ax-pre-ltirr 7901  ax-pre-ltwlin 7902  ax-pre-lttrn 7903  ax-pre-apti 7904  ax-pre-ltadd 7905  ax-pre-mulgt0 7906  ax-pre-mulext 7907
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-br 4001  df-opab 4062  df-id 4289  df-po 4292  df-iso 4293  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-iota 5173  df-fun 5213  df-fv 5219  df-riota 5824  df-ov 5871  df-oprab 5872  df-mpo 5873  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-sub 8107  df-neg 8108  df-reap 8509  df-ap 8516  df-div 8606
This theorem is referenced by:  cncongr2  12074
  Copyright terms: Public domain W3C validator