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Theorem dvrfval 20113
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 6842 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvrval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2794 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6842 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6 dvrval.u . . . . . 6 𝑈 = (Unit‘𝑅)
75, 6eqtr4di 2794 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8 fveq2 6842 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvrval.t . . . . . . 7 · = (.r𝑅)
108, 9eqtr4di 2794 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
11 eqidd 2737 . . . . . 6 (𝑟 = 𝑅𝑥 = 𝑥)
12 fveq2 6842 . . . . . . . 8 (𝑟 = 𝑅 → (invr𝑟) = (invr𝑅))
13 dvrval.i . . . . . . . 8 𝐼 = (invr𝑅)
1412, 13eqtr4di 2794 . . . . . . 7 (𝑟 = 𝑅 → (invr𝑟) = 𝐼)
1514fveq1d 6844 . . . . . 6 (𝑟 = 𝑅 → ((invr𝑟)‘𝑦) = (𝐼𝑦))
1610, 11, 15oveq123d 7378 . . . . 5 (𝑟 = 𝑅 → (𝑥(.r𝑟)((invr𝑟)‘𝑦)) = (𝑥 · (𝐼𝑦)))
174, 7, 16mpoeq123dv 7432 . . . 4 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
18 df-dvr 20112 . . . 4 /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
193fvexi 6856 . . . . 5 𝐵 ∈ V
206fvexi 6856 . . . . 5 𝑈 ∈ V
2119, 20mpoex 8012 . . . 4 (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) ∈ V
2217, 18, 21fvmpt 6948 . . 3 (𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
23 fvprc 6834 . . . 4 𝑅 ∈ V → (/r𝑅) = ∅)
24 fvprc 6834 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
253, 24eqtrid 2788 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
2625orcd 871 . . . . 5 𝑅 ∈ V → (𝐵 = ∅ ∨ 𝑈 = ∅))
27 0mpo0 7440 . . . . 5 ((𝐵 = ∅ ∨ 𝑈 = ∅) → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅)
2826, 27syl 17 . . . 4 𝑅 ∈ V → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅)
2923, 28eqtr4d 2779 . . 3 𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
3022, 29pm2.61i 182 . 2 (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
311, 30eqtri 2764 1 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wo 845   = wceq 1541  wcel 2106  Vcvv 3445  c0 4282  cfv 6496  (class class class)co 7357  cmpo 7359  Basecbs 17083  .rcmulr 17134  Unitcui 20068  invrcinvr 20100  /rcdvr 20111
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-oprab 7361  df-mpo 7362  df-1st 7921  df-2nd 7922  df-dvr 20112
This theorem is referenced by:  dvrval  20114  cnflddiv  20827  dvrcn  23535
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