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Theorem dvrval 18454
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 · = (.r𝑅)
dvrval.u 𝑈 = (Unit‘𝑅)
dvrval.i 𝐼 = (invr𝑅)
dvrval.d / = (/r𝑅)
Assertion
Ref Expression
dvrval ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))

Proof of Theorem dvrval
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6534 . 2 (𝑥 = 𝑋 → (𝑥 · (𝐼𝑦)) = (𝑋 · (𝐼𝑦)))
2 fveq2 6088 . . 3 (𝑦 = 𝑌 → (𝐼𝑦) = (𝐼𝑌))
32oveq2d 6543 . 2 (𝑦 = 𝑌 → (𝑋 · (𝐼𝑦)) = (𝑋 · (𝐼𝑌)))
4 dvrval.b . . 3 𝐵 = (Base‘𝑅)
5 dvrval.t . . 3 · = (.r𝑅)
6 dvrval.u . . 3 𝑈 = (Unit‘𝑅)
7 dvrval.i . . 3 𝐼 = (invr𝑅)
8 dvrval.d . . 3 / = (/r𝑅)
94, 5, 6, 7, 8dvrfval 18453 . 2 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
10 ovex 6555 . 2 (𝑋 · (𝐼𝑌)) ∈ V
111, 3, 9, 10ovmpt2 6672 1 ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  cfv 5790  (class class class)co 6527  Basecbs 15641  .rcmulr 15715  Unitcui 18408  invrcinvr 18440  /rcdvr 18451
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-dvr 18452
This theorem is referenced by:  dvrcl  18455  unitdvcl  18456  dvrid  18457  dvr1  18458  dvrass  18459  dvrcan1  18460  ringinvdv  18463  subrgdv  18566  abvdiv  18606  cnflddiv  19541  nmdvr  22217  sum2dchr  24716  dvrdir  28927  rdivmuldivd  28928  dvrcan5  28930
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