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Theorem rrgval 20773
Description: Value of the set or left-regular elements in a ring. (Contributed by Stefan O'Rear, 22-Mar-2015.)
Hypotheses
Ref Expression
rrgval.e 𝐸 = (RLRegβ€˜π‘…)
rrgval.b 𝐡 = (Baseβ€˜π‘…)
rrgval.t Β· = (.rβ€˜π‘…)
rrgval.z 0 = (0gβ€˜π‘…)
Assertion
Ref Expression
rrgval 𝐸 = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )}
Distinct variable groups:   π‘₯,𝐡,𝑦   π‘₯,𝑅,𝑦
Allowed substitution hints:   Β· (π‘₯,𝑦)   𝐸(π‘₯,𝑦)   0 (π‘₯,𝑦)

Proof of Theorem rrgval
Dummy variable π‘Ÿ is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . 2 𝐸 = (RLRegβ€˜π‘…)
2 fveq2 6843 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = (Baseβ€˜π‘…))
3 rrgval.b . . . . . 6 𝐡 = (Baseβ€˜π‘…)
42, 3eqtr4di 2791 . . . . 5 (π‘Ÿ = 𝑅 β†’ (Baseβ€˜π‘Ÿ) = 𝐡)
5 fveq2 6843 . . . . . . . . . 10 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = (.rβ€˜π‘…))
6 rrgval.t . . . . . . . . . 10 Β· = (.rβ€˜π‘…)
75, 6eqtr4di 2791 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (.rβ€˜π‘Ÿ) = Β· )
87oveqd 7375 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (π‘₯(.rβ€˜π‘Ÿ)𝑦) = (π‘₯ Β· 𝑦))
9 fveq2 6843 . . . . . . . . 9 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = (0gβ€˜π‘…))
10 rrgval.z . . . . . . . . 9 0 = (0gβ€˜π‘…)
119, 10eqtr4di 2791 . . . . . . . 8 (π‘Ÿ = 𝑅 β†’ (0gβ€˜π‘Ÿ) = 0 )
128, 11eqeq12d 2749 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ ((π‘₯(.rβ€˜π‘Ÿ)𝑦) = (0gβ€˜π‘Ÿ) ↔ (π‘₯ Β· 𝑦) = 0 ))
1311eqeq2d 2744 . . . . . . 7 (π‘Ÿ = 𝑅 β†’ (𝑦 = (0gβ€˜π‘Ÿ) ↔ 𝑦 = 0 ))
1412, 13imbi12d 345 . . . . . 6 (π‘Ÿ = 𝑅 β†’ (((π‘₯(.rβ€˜π‘Ÿ)𝑦) = (0gβ€˜π‘Ÿ) β†’ 𝑦 = (0gβ€˜π‘Ÿ)) ↔ ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )))
154, 14raleqbidv 3318 . . . . 5 (π‘Ÿ = 𝑅 β†’ (βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘₯(.rβ€˜π‘Ÿ)𝑦) = (0gβ€˜π‘Ÿ) β†’ 𝑦 = (0gβ€˜π‘Ÿ)) ↔ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )))
164, 15rabeqbidv 3423 . . . 4 (π‘Ÿ = 𝑅 β†’ {π‘₯ ∈ (Baseβ€˜π‘Ÿ) ∣ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘₯(.rβ€˜π‘Ÿ)𝑦) = (0gβ€˜π‘Ÿ) β†’ 𝑦 = (0gβ€˜π‘Ÿ))} = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )})
17 df-rlreg 20769 . . . 4 RLReg = (π‘Ÿ ∈ V ↦ {π‘₯ ∈ (Baseβ€˜π‘Ÿ) ∣ βˆ€π‘¦ ∈ (Baseβ€˜π‘Ÿ)((π‘₯(.rβ€˜π‘Ÿ)𝑦) = (0gβ€˜π‘Ÿ) β†’ 𝑦 = (0gβ€˜π‘Ÿ))})
183fvexi 6857 . . . . 5 𝐡 ∈ V
1918rabex 5290 . . . 4 {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )} ∈ V
2016, 17, 19fvmpt 6949 . . 3 (𝑅 ∈ V β†’ (RLRegβ€˜π‘…) = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )})
21 fvprc 6835 . . . 4 (Β¬ 𝑅 ∈ V β†’ (RLRegβ€˜π‘…) = βˆ…)
22 fvprc 6835 . . . . . . 7 (Β¬ 𝑅 ∈ V β†’ (Baseβ€˜π‘…) = βˆ…)
233, 22eqtrid 2785 . . . . . 6 (Β¬ 𝑅 ∈ V β†’ 𝐡 = βˆ…)
2423rabeqdv 3421 . . . . 5 (Β¬ 𝑅 ∈ V β†’ {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )} = {π‘₯ ∈ βˆ… ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )})
25 rab0 4343 . . . . 5 {π‘₯ ∈ βˆ… ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )} = βˆ…
2624, 25eqtrdi 2789 . . . 4 (Β¬ 𝑅 ∈ V β†’ {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )} = βˆ…)
2721, 26eqtr4d 2776 . . 3 (Β¬ 𝑅 ∈ V β†’ (RLRegβ€˜π‘…) = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )})
2820, 27pm2.61i 182 . 2 (RLRegβ€˜π‘…) = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )}
291, 28eqtri 2761 1 𝐸 = {π‘₯ ∈ 𝐡 ∣ βˆ€π‘¦ ∈ 𝐡 ((π‘₯ Β· 𝑦) = 0 β†’ 𝑦 = 0 )}
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  {crab 3406  Vcvv 3444  βˆ…c0 4283  β€˜cfv 6497  (class class class)co 7358  Basecbs 17088  .rcmulr 17139  0gc0g 17326  RLRegcrlreg 20765
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-rlreg 20769
This theorem is referenced by:  isrrg  20774  rrgeq0  20776  rrgss  20778
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