Users' Mathboxes Mathbox for BJ < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bj-ldiv Structured version   Unicode version

Theorem bj-ldiv 32591
Description: Left-division. (Contributed by BJ, 6-Jun-2019.)
Hypotheses
Ref Expression
bj-ldiv.a  |-  ( ph  ->  A  e.  CC )
bj-ldiv.b  |-  ( ph  ->  B  e.  CC )
bj-ldiv.c  |-  ( ph  ->  C  e.  CC )
bj-ldiv.bn0  |-  ( ph  ->  B  =/=  0 )
Assertion
Ref Expression
bj-ldiv  |-  ( ph  ->  ( ( A  x.  B )  =  C  <-> 
A  =  ( C  /  B ) ) )

Proof of Theorem bj-ldiv
StepHypRef Expression
1 oveq1 6096 . . 3  |-  ( ( A  x.  B )  =  C  ->  (
( A  x.  B
)  /  B )  =  ( C  /  B ) )
2 bj-ldiv.a . . . . 5  |-  ( ph  ->  A  e.  CC )
3 bj-ldiv.b . . . . 5  |-  ( ph  ->  B  e.  CC )
4 bj-ldiv.bn0 . . . . 5  |-  ( ph  ->  B  =/=  0 )
52, 3, 4divcan4d 10111 . . . 4  |-  ( ph  ->  ( ( A  x.  B )  /  B
)  =  A )
65eqeq1d 2449 . . 3  |-  ( ph  ->  ( ( ( A  x.  B )  /  B )  =  ( C  /  B )  <-> 
A  =  ( C  /  B ) ) )
71, 6syl5ib 219 . 2  |-  ( ph  ->  ( ( A  x.  B )  =  C  ->  A  =  ( C  /  B ) ) )
8 oveq1 6096 . . 3  |-  ( A  =  ( C  /  B )  ->  ( A  x.  B )  =  ( ( C  /  B )  x.  B ) )
9 bj-ldiv.c . . . . 5  |-  ( ph  ->  C  e.  CC )
109, 3, 4divcan1d 10106 . . . 4  |-  ( ph  ->  ( ( C  /  B )  x.  B
)  =  C )
1110eqeq2d 2452 . . 3  |-  ( ph  ->  ( ( A  x.  B )  =  ( ( C  /  B
)  x.  B )  <-> 
( A  x.  B
)  =  C ) )
128, 11syl5ib 219 . 2  |-  ( ph  ->  ( A  =  ( C  /  B )  ->  ( A  x.  B )  =  C ) )
137, 12impbid 191 1  |-  ( ph  ->  ( ( A  x.  B )  =  C  <-> 
A  =  ( C  /  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756    =/= wne 2604  (class class class)co 6089   CCcc 9278   0cc0 9280    x. cmul 9285    / cdiv 9991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pow 4468  ax-pr 4529  ax-un 6370  ax-resscn 9337  ax-1cn 9338  ax-icn 9339  ax-addcl 9340  ax-addrcl 9341  ax-mulcl 9342  ax-mulrcl 9343  ax-mulcom 9344  ax-addass 9345  ax-mulass 9346  ax-distr 9347  ax-i2m1 9348  ax-1ne0 9349  ax-1rid 9350  ax-rnegex 9351  ax-rrecex 9352  ax-cnre 9353  ax-pre-lttri 9354  ax-pre-lttrn 9355  ax-pre-ltadd 9356  ax-pre-mulgt0 9357
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3185  df-csb 3287  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-pw 3860  df-sn 3876  df-pr 3878  df-op 3882  df-uni 4090  df-br 4291  df-opab 4349  df-mpt 4350  df-id 4634  df-po 4639  df-so 4640  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847  df-dm 4848  df-rn 4849  df-res 4850  df-ima 4851  df-iota 5379  df-fun 5418  df-fn 5419  df-f 5420  df-f1 5421  df-fo 5422  df-f1o 5423  df-fv 5424  df-riota 6050  df-ov 6092  df-oprab 6093  df-mpt2 6094  df-er 7099  df-en 7309  df-dom 7310  df-sdom 7311  df-pnf 9418  df-mnf 9419  df-xr 9420  df-ltxr 9421  df-le 9422  df-sub 9595  df-neg 9596  df-div 9992
This theorem is referenced by:  bj-rdiv  32592  bj-mdiv  32593
  Copyright terms: Public domain W3C validator