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Theorem dvrfval 20066
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 𝐡 = (Baseβ€˜π‘…)
dvrval.t Β· = (.rβ€˜π‘…)
dvrval.u π‘ˆ = (Unitβ€˜π‘…)
dvrval.i 𝐼 = (invrβ€˜π‘…)
dvrval.d / = (/rβ€˜π‘…)
Assertion
Ref Expression
dvrfval / = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦)))
Distinct variable groups:   π‘₯,𝑦,𝐡   π‘₯,𝐼,𝑦   π‘₯,𝑅,𝑦   π‘₯, Β· ,𝑦   π‘₯,π‘ˆ,𝑦
Allowed substitution hints:   / (π‘₯,𝑦)

Proof of Theorem dvrfval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 dvrval.d . 2 / = (/rβ€˜π‘…)
2 fveq2 6839 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
3 dvrval.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
42, 3eqtr4di 2795 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
5 fveq2 6839 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
6 dvrval.u . . . . . 6 π‘ˆ = (Unitβ€˜π‘…)
75, 6eqtr4di 2795 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
8 fveq2 6839 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
9 dvrval.t . . . . . . 7 Β· = (.rβ€˜π‘…)
108, 9eqtr4di 2795 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
11 eqidd 2738 . . . . . 6 (π‘Ÿ = 𝑅 β†’ π‘₯ = π‘₯)
12 fveq2 6839 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (invrβ€˜π‘Ÿ) = (invrβ€˜π‘…))
13 dvrval.i . . . . . . . 8 𝐼 = (invrβ€˜π‘…)
1412, 13eqtr4di 2795 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (invrβ€˜π‘Ÿ) = 𝐼)
1514fveq1d 6841 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((invrβ€˜π‘Ÿ)β€˜π‘¦) = (πΌβ€˜π‘¦))
1610, 11, 15oveq123d 7372 . . . . 5 (π‘Ÿ = 𝑅 β†’ (π‘₯(.rβ€˜π‘Ÿ)((invrβ€˜π‘Ÿ)β€˜π‘¦)) = (π‘₯ Β· (πΌβ€˜π‘¦)))
174, 7, 16mpoeq123dv 7426 . . . 4 (π‘Ÿ = 𝑅 β†’ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Unitβ€˜π‘Ÿ) ↦ (π‘₯(.rβ€˜π‘Ÿ)((invrβ€˜π‘Ÿ)β€˜π‘¦))) = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))))
18 df-dvr 20065 . . . 4 /r = (π‘Ÿ ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Unitβ€˜π‘Ÿ) ↦ (π‘₯(.rβ€˜π‘Ÿ)((invrβ€˜π‘Ÿ)β€˜π‘¦))))
193fvexi 6853 . . . . 5 𝐡 ∈ V
206fvexi 6853 . . . . 5 π‘ˆ ∈ V
2119, 20mpoex 8004 . . . 4 (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))) ∈ V
2217, 18, 21fvmpt 6945 . . 3 (𝑅 ∈ V β†’ (/rβ€˜π‘…) = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))))
23 fvprc 6831 . . . 4 (Β¬ 𝑅 ∈ V β†’ (/rβ€˜π‘…) = βˆ…)
24 fvprc 6831 . . . . . . 7 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘…) = βˆ…)
253, 24eqtrid 2789 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ 𝐡 = βˆ…)
2625orcd 871 . . . . 5 (Β¬ 𝑅 ∈ V β†’ (𝐡 = βˆ… ∨ π‘ˆ = βˆ…))
27 0mpo0 7434 . . . . 5 ((𝐡 = βˆ… ∨ π‘ˆ = βˆ…) β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))) = βˆ…)
2826, 27syl 17 . . . 4 (Β¬ 𝑅 ∈ V β†’ (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))) = βˆ…)
2923, 28eqtr4d 2780 . . 3 (Β¬ 𝑅 ∈ V β†’ (/rβ€˜π‘…) = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))))
3022, 29pm2.61i 182 . 2 (/rβ€˜π‘…) = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦)))
311, 30eqtri 2765 1 / = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∨ wo 845   = wceq 1541   ∈ wcel 2106  Vcvv 3443  βˆ…c0 4280  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  Basecbs 17043  .rcmulr 17094  Unitcui 20021  invrcinvr 20053  /rcdvr 20064
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 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664
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 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7354  df-oprab 7355  df-mpo 7356  df-1st 7913  df-2nd 7914  df-dvr 20065
This theorem is referenced by:  dvrval  20067  cnflddiv  20780  dvrcn  23487
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