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Theorem dvrfval 20304
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 6885 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
3 dvrval.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
42, 3eqtr4di 2784 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
5 fveq2 6885 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = (Unitβ€˜π‘…))
6 dvrval.u . . . . . 6 π‘ˆ = (Unitβ€˜π‘…)
75, 6eqtr4di 2784 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Unitβ€˜π‘Ÿ) = π‘ˆ)
8 fveq2 6885 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
9 dvrval.t . . . . . . 7 Β· = (.rβ€˜π‘…)
108, 9eqtr4di 2784 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
11 eqidd 2727 . . . . . 6 (π‘Ÿ = 𝑅 β†’ π‘₯ = π‘₯)
12 fveq2 6885 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (invrβ€˜π‘Ÿ) = (invrβ€˜π‘…))
13 dvrval.i . . . . . . . 8 𝐼 = (invrβ€˜π‘…)
1412, 13eqtr4di 2784 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (invrβ€˜π‘Ÿ) = 𝐼)
1514fveq1d 6887 . . . . . 6 (π‘Ÿ = 𝑅 β†’ ((invrβ€˜π‘Ÿ)β€˜π‘¦) = (πΌβ€˜π‘¦))
1610, 11, 15oveq123d 7426 . . . . 5 (π‘Ÿ = 𝑅 β†’ (π‘₯(.rβ€˜π‘Ÿ)((invrβ€˜π‘Ÿ)β€˜π‘¦)) = (π‘₯ Β· (πΌβ€˜π‘¦)))
174, 7, 16mpoeq123dv 7480 . . . 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 20303 . . . 4 /r = (π‘Ÿ ∈ V ↦ (π‘₯ ∈ (Baseβ€˜π‘Ÿ), 𝑦 ∈ (Unitβ€˜π‘Ÿ) ↦ (π‘₯(.rβ€˜π‘Ÿ)((invrβ€˜π‘Ÿ)β€˜π‘¦))))
193fvexi 6899 . . . . 5 𝐡 ∈ V
206fvexi 6899 . . . . 5 π‘ˆ ∈ V
2119, 20mpoex 8065 . . . 4 (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))) ∈ V
2217, 18, 21fvmpt 6992 . . 3 (𝑅 ∈ V β†’ (/rβ€˜π‘…) = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦))))
23 fvprc 6877 . . . 4 (Β¬ 𝑅 ∈ V β†’ (/rβ€˜π‘…) = βˆ…)
24 fvprc 6877 . . . . . . 7 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘…) = βˆ…)
253, 24eqtrid 2778 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ 𝐡 = βˆ…)
2625orcd 870 . . . . 5 (Β¬ 𝑅 ∈ V β†’ (𝐡 = βˆ… ∨ π‘ˆ = βˆ…))
27 0mpo0 7488 . . . . 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 2769 . . 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 2754 1 / = (π‘₯ ∈ 𝐡, 𝑦 ∈ π‘ˆ ↦ (π‘₯ Β· (πΌβ€˜π‘¦)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   ∨ wo 844   = wceq 1533   ∈ wcel 2098  Vcvv 3468  βˆ…c0 4317  β€˜cfv 6537  (class class class)co 7405   ∈ cmpo 7407  Basecbs 17153  .rcmulr 17207  Unitcui 20257  invrcinvr 20289  /rcdvr 20302
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-dvr 20303
This theorem is referenced by:  dvrval  20305  cnflddiv  21289  cnflddivOLD  21290  dvrcn  24043
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