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Theorem dvrval 15467
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 5865 . 2  |-  ( x  =  X  ->  (
x  .x.  ( I `  y ) )  =  ( X  .x.  (
I `  y )
) )
2 fveq2 5525 . . 3  |-  ( y  =  Y  ->  (
I `  y )  =  ( I `  Y ) )
32oveq2d 5874 . 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 15466 . 2  |-  ./  =  ( x  e.  B ,  y  e.  U  |->  ( x  .x.  (
I `  y )
) )
10 ovex 5883 . 2  |-  ( X 
.x.  ( I `  Y ) )  e. 
_V
111, 3, 9, 10ovmpt2 5983 1  |-  ( ( X  e.  B  /\  Y  e.  U )  ->  ( X  ./  Y
)  =  ( X 
.x.  ( I `  Y ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1623    e. wcel 1684   ` cfv 5255  (class class class)co 5858   Basecbs 13148   .rcmulr 13209  Unitcui 15421   invrcinvr 15453  /rcdvr 15464
This theorem is referenced by:  dvrcl  15468  unitdvcl  15469  dvrid  15470  dvr1  15471  dvrass  15472  dvrcan1  15473  rnginvdv  15476  subrgdv  15562  abvdiv  15602  cnflddiv  16404  nmdvr  18181  sum2dchr  20513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-dvr 15465
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