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Theorem dvrval 17118
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.)
Hypotheses
Ref Expression
dvrval.b  |-  B  =  ( Base `  R
)
dvrval.t  |-  .x.  =  ( .r `  R )
dvrval.u  |-  U  =  (Unit `  R )
dvrval.i  |-  I  =  ( invr `  R
)
dvrval.d  |-  ./  =  (/r
`  R )
Assertion
Ref Expression
dvrval  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )

Proof of Theorem dvrval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6289 . 2  |-  ( x  =  X  ->  (
x  .x.  ( I `  y ) )  =  ( X  .x.  (
I `  y )
) )
2 fveq2 5864 . . 3  |-  ( y  =  Y  ->  (
I `  y )  =  ( I `  Y ) )
32oveq2d 6298 . 2  |-  ( y  =  Y  ->  ( X  .x.  ( I `  y ) )  =  ( X  .x.  (
I `  Y )
) )
4 dvrval.b . . 3  |-  B  =  ( Base `  R
)
5 dvrval.t . . 3  |-  .x.  =  ( .r `  R )
6 dvrval.u . . 3  |-  U  =  (Unit `  R )
7 dvrval.i . . 3  |-  I  =  ( invr `  R
)
8 dvrval.d . . 3  |-  ./  =  (/r
`  R )
94, 5, 6, 7, 8dvrfval 17117 . 2  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
10 ovex 6307 . 2  |-  ( X 
.x.  ( I `  Y ) )  e. 
_V
111, 3, 9, 10ovmpt2 6420 1  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   ` cfv 5586  (class class class)co 6282   Basecbs 14486   .rcmulr 14552  Unitcui 17072   invrcinvr 17104  /rcdvr 17115
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-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-ral 2819  df-rex 2820  df-reu 2821  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-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  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-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-dvr 17116
This theorem is referenced by:  dvrcl  17119  unitdvcl  17120  dvrid  17121  dvr1  17122  dvrass  17123  dvrcan1  17124  rnginvdv  17127  subrgdv  17229  abvdiv  17269  cnflddiv  18219  nmdvr  20914  sum2dchr  23277  dvrdir  27443  rdivmuldivd  27444  dvrcan5  27446
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