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Theorem dvrfval 20419
Description: Division operation in a ring. (Contributed by Mario Carneiro, 2-Jul-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Proof shortened by AV, 2-Mar-2024.)
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
dvrfval / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐼,𝑦   𝑥,𝑅,𝑦   𝑥, · ,𝑦   𝑥,𝑈,𝑦
Allowed substitution hints:   / (𝑥,𝑦)

Proof of Theorem dvrfval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 dvrval.d . 2 / = (/r𝑅)
2 fveq2 6907 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvrval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6907 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6 dvrval.u . . . . . 6 𝑈 = (Unit‘𝑅)
75, 6eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8 fveq2 6907 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvrval.t . . . . . . 7 · = (.r𝑅)
108, 9eqtr4di 2793 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
11 eqidd 2736 . . . . . 6 (𝑟 = 𝑅𝑥 = 𝑥)
12 fveq2 6907 . . . . . . . 8 (𝑟 = 𝑅 → (invr𝑟) = (invr𝑅))
13 dvrval.i . . . . . . . 8 𝐼 = (invr𝑅)
1412, 13eqtr4di 2793 . . . . . . 7 (𝑟 = 𝑅 → (invr𝑟) = 𝐼)
1514fveq1d 6909 . . . . . 6 (𝑟 = 𝑅 → ((invr𝑟)‘𝑦) = (𝐼𝑦))
1610, 11, 15oveq123d 7452 . . . . 5 (𝑟 = 𝑅 → (𝑥(.r𝑟)((invr𝑟)‘𝑦)) = (𝑥 · (𝐼𝑦)))
174, 7, 16mpoeq123dv 7508 . . . 4 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
18 df-dvr 20418 . . . 4 /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
193fvexi 6921 . . . . 5 𝐵 ∈ V
206fvexi 6921 . . . . 5 𝑈 ∈ V
2119, 20mpoex 8103 . . . 4 (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) ∈ V
2217, 18, 21fvmpt 7016 . . 3 (𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
23 fvprc 6899 . . . 4 𝑅 ∈ V → (/r𝑅) = ∅)
24 fvprc 6899 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
253, 24eqtrid 2787 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
2625orcd 873 . . . . 5 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅))
27 0mpo0 7516 . . . . 5 ((𝐵 = ∅ ∨ 𝑈 = ∅) → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅)
2826, 27syl 17 . . . 4 𝑅 ∈ V → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅)
2923, 28eqtr4d 2778 . . 3 𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
3022, 29pm2.61i 182 . 2 (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
311, 30eqtri 2763 1 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 847   = wceq 1537  wcel 2106  Vcvv 3478  c0 4339  cfv 6563  (class class class)co 7431  cmpo 7433  Basecbs 17245  .rcmulr 17299  Unitcui 20372  invrcinvr 20404  /rcdvr 20417
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-dvr 20418
This theorem is referenced by:  dvrval  20420  cnflddiv  21431  cnflddivOLD  21432  dvrcn  24208
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