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Theorem dvrfval 18951
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 6375 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 dvrval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3syl6eqr 2817 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6375 . . . . . 6 (𝑟 = 𝑅 → (Unit‘𝑟) = (Unit‘𝑅))
6 dvrval.u . . . . . 6 𝑈 = (Unit‘𝑅)
75, 6syl6eqr 2817 . . . . 5 (𝑟 = 𝑅 → (Unit‘𝑟) = 𝑈)
8 fveq2 6375 . . . . . . 7 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
9 dvrval.t . . . . . . 7 · = (.r𝑅)
108, 9syl6eqr 2817 . . . . . 6 (𝑟 = 𝑅 → (.r𝑟) = · )
11 eqidd 2766 . . . . . 6 (𝑟 = 𝑅𝑥 = 𝑥)
12 fveq2 6375 . . . . . . . 8 (𝑟 = 𝑅 → (invr𝑟) = (invr𝑅))
13 dvrval.i . . . . . . . 8 𝐼 = (invr𝑅)
1412, 13syl6eqr 2817 . . . . . . 7 (𝑟 = 𝑅 → (invr𝑟) = 𝐼)
1514fveq1d 6377 . . . . . 6 (𝑟 = 𝑅 → ((invr𝑟)‘𝑦) = (𝐼𝑦))
1610, 11, 15oveq123d 6863 . . . . 5 (𝑟 = 𝑅 → (𝑥(.r𝑟)((invr𝑟)‘𝑦)) = (𝑥 · (𝐼𝑦)))
174, 7, 16mpt2eq123dv 6915 . . . 4 (𝑟 = 𝑅 → (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
18 df-dvr 18950 . . . 4 /r = (𝑟 ∈ V ↦ (𝑥 ∈ (Base‘𝑟), 𝑦 ∈ (Unit‘𝑟) ↦ (𝑥(.r𝑟)((invr𝑟)‘𝑦))))
193fvexi 6389 . . . . 5 𝐵 ∈ V
206fvexi 6389 . . . . 5 𝑈 ∈ V
2119, 20mpt2ex 7448 . . . 4 (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) ∈ V
2217, 18, 21fvmpt 6471 . . 3 (𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
23 fvprc 6368 . . . 4 𝑅 ∈ V → (/r𝑅) = ∅)
24 fvprc 6368 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
253, 24syl5eq 2811 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
26 eqid 2765 . . . . . 6 𝑈 = 𝑈
27 mpt2eq12 6913 . . . . . 6 ((𝐵 = ∅ ∧ 𝑈 = 𝑈) → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
2825, 26, 27sylancl 580 . . . . 5 𝑅 ∈ V → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = (𝑥 ∈ ∅, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
29 mpt20 6923 . . . . 5 (𝑥 ∈ ∅, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅
3028, 29syl6eq 2815 . . . 4 𝑅 ∈ V → (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))) = ∅)
3123, 30eqtr4d 2802 . . 3 𝑅 ∈ V → (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦))))
3222, 31pm2.61i 176 . 2 (/r𝑅) = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
331, 32eqtri 2787 1 / = (𝑥𝐵, 𝑦𝑈 ↦ (𝑥 · (𝐼𝑦)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1652  wcel 2155  Vcvv 3350  c0 4079  cfv 6068  (class class class)co 6842  cmpt2 6844  Basecbs 16132  .rcmulr 16217  Unitcui 18906  invrcinvr 18938  /rcdvr 18949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-dvr 18950
This theorem is referenced by:  dvrval  18952  cnflddiv  20049  dvrcn  22266
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