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Theorem dvrfval 19124
 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
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 6538 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvrval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2849 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6538 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6 dvrval.u . . . . . 6 𝑈 = (Unit‘𝑅)
75, 6syl6eqr 2849 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8 fveq2 6538 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvrval.t . . . . . . 7 · = (.r𝑅)
108, 9syl6eqr 2849 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
11 eqidd 2796 . . . . . 6 (𝑟 = 𝑅𝑥 = 𝑥)
12 fveq2 6538 . . . . . . . 8 (𝑟 = 𝑅 → (invr𝑟) = (invr𝑅))
13 dvrval.i . . . . . . . 8 𝐼 = (invr𝑅)
1412, 13syl6eqr 2849 . . . . . . 7 (𝑟 = 𝑅 → (invr𝑟) = 𝐼)
1514fveq1d 6540 . . . . . 6 (𝑟 = 𝑅 → ((invr𝑟)‘𝑦) = (𝐼𝑦))
1610, 11, 15oveq123d 7037 . . . . 5 (𝑟 = 𝑅 → (𝑥(.r𝑟)((invr𝑟)‘𝑦)) = (𝑥 · (𝐼𝑦)))
174, 7, 16mpoeq123dv 7087 . . . 4 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
18 df-dvr 19123 . . . 4 /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
193fvexi 6552 . . . . 5 𝐵 ∈ V
206fvexi 6552 . . . . 5 𝑈 ∈ V
2119, 20mpoex 7633 . . . 4 (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) ∈ V
2217, 18, 21fvmpt 6635 . . 3 (𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
23 fvprc 6531 . . . 4 𝑅 ∈ V → (/r𝑅) = ∅)
24 fvprc 6531 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
253, 24syl5eq 2843 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
26 eqid 2795 . . . . . 6 𝑈 = 𝑈
27 mpoeq12 7085 . . . . . 6 ((𝐵 = ∅ ∧ 𝑈 = 𝑈) → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
2825, 26, 27sylancl 586 . . . . 5 𝑅 ∈ V → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
29 mpo0 7095 . . . . 5 (𝑥 ∈ ∅, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅
3028, 29syl6eq 2847 . . . 4 𝑅 ∈ V → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅)
3123, 30eqtr4d 2834 . . 3 𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
3222, 31pm2.61i 183 . 2 (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
331, 32eqtri 2819 1 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   = wceq 1522   ∈ wcel 2081  Vcvv 3437  ∅c0 4211  ‘cfv 6225  (class class class)co 7016   ∈ cmpo 7018  Basecbs 16312  .rcmulr 16395  Unitcui 19079  invrcinvr 19111  /rcdvr 19122 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-dvr 19123 This theorem is referenced by:  dvrval  19125  cnflddiv  20257  dvrcn  22475
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