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Theorem dvrval 20209
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 𝐡 = (Baseβ€˜π‘…)
dvrval.t Β· = (.rβ€˜π‘…)
dvrval.u π‘ˆ = (Unitβ€˜π‘…)
dvrval.i 𝐼 = (invrβ€˜π‘…)
dvrval.d / = (/rβ€˜π‘…)
Assertion
Ref Expression
dvrval ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ) β†’ (𝑋 / π‘Œ) = (𝑋 Β· (πΌβ€˜π‘Œ)))

Proof of Theorem dvrval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 7412 . 2 (π‘₯ = 𝑋 β†’ (π‘₯ Β· (πΌβ€˜π‘¦)) = (𝑋 Β· (πΌβ€˜π‘¦)))
2 fveq2 6888 . . 3 (𝑦 = π‘Œ β†’ (πΌβ€˜π‘¦) = (πΌβ€˜π‘Œ))
32oveq2d 7421 . 2 (𝑦 = π‘Œ β†’ (𝑋 Β· (πΌβ€˜π‘¦)) = (𝑋 Β· (πΌβ€˜π‘Œ)))
4 dvrval.b . . 3 𝐡 = (Baseβ€˜π‘…)
5 dvrval.t . . 3 Β· = (.rβ€˜π‘…)
6 dvrval.u . . 3 π‘ˆ = (Unitβ€˜π‘…)
7 dvrval.i . . 3 𝐼 = (invrβ€˜π‘…)
8 dvrval.d . . 3 / = (/rβ€˜π‘…)
94, 5, 6, 7, 8dvrfval 20208 . 2 / = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦)))
10 ovex 7438 . 2 (𝑋 Β· (πΌβ€˜π‘Œ)) ∈ V
111, 3, 9, 10ovmpo 7564 1 ((𝑋 ∈ 𝐡 ∧ π‘Œ ∈ π‘ˆ) β†’ (𝑋 / π‘Œ) = (𝑋 Β· (πΌβ€˜π‘Œ)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  β€˜cfv 6540  (class class class)co 7405  Basecbs 17140  .rcmulr 17194  Unitcui 20161  invrcinvr 20193  /rcdvr 20206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-dvr 20207
This theorem is referenced by:  dvrcl  20210  unitdvcl  20211  dvrid  20212  dvr1  20213  dvrass  20214  dvrcan1  20215  dvrdir  20218  rdivmuldivd  20219  ringinvdv  20220  subrgdv  20372  abvdiv  20437  cnflddiv  20967  nmdvr  24178  sum2dchr  26766  dvrcan5  32373
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