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Theorem dvrval 19414
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 7140 . 2 (𝑥 = 𝑋 → (𝑥 · (𝐼𝑦)) = (𝑋 · (𝐼𝑦)))
2 fveq2 6646 . . 3 (𝑦 = 𝑌 → (𝐼𝑦) = (𝐼𝑌))
32oveq2d 7149 . 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 19413 . 2 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
10 ovex 7166 . 2 (𝑋 · (𝐼𝑌)) ∈ V
111, 3, 9, 10ovmpo 7287 1 ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1537  wcel 2114  cfv 6331  (class class class)co 7133  Basecbs 16462  .rcmulr 16545  Unitcui 19368  invrcinvr 19400  /rcdvr 19411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-reu 3132  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-dvr 19412
This theorem is referenced by:  dvrcl  19415  unitdvcl  19416  dvrid  19417  dvr1  19418  dvrass  19419  dvrcan1  19420  ringinvdv  19423  subrgdv  19528  abvdiv  19584  cnflddiv  20551  nmdvr  23255  sum2dchr  25837  dvrdir  30869  rdivmuldivd  30870  dvrcan5  30872
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