MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dvrval Structured version   Visualization version   GIF version

Theorem dvrval 20205
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 7410 . 2 (𝑥 = 𝑋 → (𝑥 · (𝐼𝑦)) = (𝑋 · (𝐼𝑦)))
2 fveq2 6887 . . 3 (𝑦 = 𝑌 → (𝐼𝑦) = (𝐼𝑌))
32oveq2d 7419 . 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 20204 . 2 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
10 ovex 7436 . 2 (𝑋 · (𝐼𝑌)) ∈ V
111, 3, 9, 10ovmpo 7562 1 ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cfv 6539  (class class class)co 7403  Basecbs 17139  .rcmulr 17193  Unitcui 20157  invrcinvr 20189  /rcdvr 20202
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5283  ax-sep 5297  ax-nul 5304  ax-pow 5361  ax-pr 5425  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4527  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4907  df-iun 4997  df-br 5147  df-opab 5209  df-mpt 5230  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6491  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7406  df-oprab 7407  df-mpo 7408  df-1st 7969  df-2nd 7970  df-dvr 20203
This theorem is referenced by:  dvrcl  20206  unitdvcl  20207  dvrid  20208  dvr1  20209  dvrass  20210  dvrcan1  20211  dvrdir  20214  rdivmuldivd  20215  ringinvdv  20216  subrgdv  20367  abvdiv  20432  cnflddiv  20959  nmdvr  24168  sum2dchr  26756  dvrcan5  32359
  Copyright terms: Public domain W3C validator