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

Theorem dvrval 19429
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 7157 . 2 (𝑥 = 𝑋 → (𝑥 · (𝐼𝑦)) = (𝑋 · (𝐼𝑦)))
2 fveq2 6665 . . 3 (𝑦 = 𝑌 → (𝐼𝑦) = (𝐼𝑌))
32oveq2d 7166 . 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 19428 . 2 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
10 ovex 7183 . 2 (𝑋 · (𝐼𝑌)) ∈ V
111, 3, 9, 10ovmpo 7304 1 ((𝑋𝐵𝑌𝑈) → (𝑋 / 𝑌) = (𝑋 · (𝐼𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  cfv 6350  (class class class)co 7150  Basecbs 16477  .rcmulr 16560  Unitcui 19383  invrcinvr 19415  /rcdvr 19426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7683  df-2nd 7684  df-dvr 19427
This theorem is referenced by:  dvrcl  19430  unitdvcl  19431  dvrid  19432  dvr1  19433  dvrass  19434  dvrcan1  19435  ringinvdv  19438  subrgdv  19546  abvdiv  19602  cnflddiv  20569  nmdvr  23273  sum2dchr  25844  dvrdir  30856  rdivmuldivd  30857  dvrcan5  30859
  Copyright terms: Public domain W3C validator