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Theorem dvrfval 20402
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 6906 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvrval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6906 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6 dvrval.u . . . . . 6 𝑈 = (Unit‘𝑅)
75, 6eqtr4di 2795 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8 fveq2 6906 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvrval.t . . . . . . 7 · = (.r𝑅)
108, 9eqtr4di 2795 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
11 eqidd 2738 . . . . . 6 (𝑟 = 𝑅𝑥 = 𝑥)
12 fveq2 6906 . . . . . . . 8 (𝑟 = 𝑅 → (invr𝑟) = (invr𝑅))
13 dvrval.i . . . . . . . 8 𝐼 = (invr𝑅)
1412, 13eqtr4di 2795 . . . . . . 7 (𝑟 = 𝑅 → (invr𝑟) = 𝐼)
1514fveq1d 6908 . . . . . 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 20401 . . . 4 /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
193fvexi 6920 . . . . 5 𝐵 ∈ V
206fvexi 6920 . . . . 5 𝑈 ∈ V
2119, 20mpoex 8104 . . . 4 (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) ∈ V
2217, 18, 21fvmpt 7016 . . 3 (𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
23 fvprc 6898 . . . 4 𝑅 ∈ V → (/r𝑅) = ∅)
24 fvprc 6898 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
253, 24eqtrid 2789 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
2625orcd 874 . . . . 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 2780 . . 3 𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
3022, 29pm2.61i 182 . 2 (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
311, 30eqtri 2765 1 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 848   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  cfv 6561  (class class class)co 7431  cmpo 7433  Basecbs 17247  .rcmulr 17298  Unitcui 20355  invrcinvr 20387  /rcdvr 20400
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-dvr 20401
This theorem is referenced by:  dvrval  20403  cnflddiv  21413  cnflddivOLD  21414  dvrcn  24192
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