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Theorem dvrval 18893
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 6803 . 2 (𝑥 = 𝑋 → (𝑥 · (𝐼𝑦)) = (𝑋 · (𝐼𝑦)))
2 fveq2 6333 . . 3 (𝑦 = 𝑌 → (𝐼𝑦) = (𝐼𝑌))
32oveq2d 6812 . 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 18892 . 2 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
10 ovex 6827 . 2 (𝑋 · (𝐼𝑌)) ∈ V
111, 3, 9, 10ovmpt2 6947 1 ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  cfv 6030  (class class class)co 6796  Basecbs 16064  .rcmulr 16150  Unitcui 18847  invrcinvr 18879  /rcdvr 18890
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-1st 7319  df-2nd 7320  df-dvr 18891
This theorem is referenced by:  dvrcl  18894  unitdvcl  18895  dvrid  18896  dvr1  18897  dvrass  18898  dvrcan1  18899  ringinvdv  18902  subrgdv  19007  abvdiv  19047  cnflddiv  19991  nmdvr  22694  sum2dchr  25220  dvrdir  30130  rdivmuldivd  30131  dvrcan5  30133
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