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Theorem bj-ldiv 33746
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 6289 . . 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 10322 . . . 4  |-  ( ph  ->  ( ( A  x.  B )  /  B
)  =  A )
65eqeq1d 2469 . . 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 6289 . . 3  |-  ( A  =  ( C  /  B )  ->  ( A  x.  B )  =  ( ( C  /  B )  x.  B ) )
9 bj-ldiv.c . . . . 5  |-  ( ph  ->  C  e.  CC )
109, 3, 4divcan1d 10317 . . . 4  |-  ( ph  ->  ( ( C  /  B )  x.  B
)  =  C )
1110eqeq2d 2481 . . 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 1379    e. wcel 1767    =/= wne 2662  (class class class)co 6282   CCcc 9486   0cc0 9488    x. cmul 9493    / cdiv 10202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203
This theorem is referenced by:  bj-rdiv  33747  bj-mdiv  33748
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