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Theorem rrgval 20714
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 = (RLReg‘𝑅)
rrgval.b 𝐵 = (Base‘𝑅)
rrgval.t · = (.r𝑅)
rrgval.z 0 = (0g𝑅)
Assertion
Ref Expression
rrgval 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝑅,𝑦
Allowed substitution hints:   · (𝑥,𝑦)   𝐸(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem rrgval
Dummy variable 𝑟 is distinct from all other variables.
StepHypRef Expression
1 rrgval.e . 2 𝐸 = (RLReg‘𝑅)
2 fveq2 6907 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 rrgval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2793 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6907 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
6 rrgval.t . . . . . . . . . 10 · = (.r𝑅)
75, 6eqtr4di 2793 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
87oveqd 7448 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
9 fveq2 6907 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
10 rrgval.z . . . . . . . . 9 0 = (0g𝑅)
119, 10eqtr4di 2793 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
128, 11eqeq12d 2751 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(.r𝑟)𝑦) = (0g𝑟) ↔ (𝑥 · 𝑦) = 0 ))
1311eqeq2d 2746 . . . . . . 7 (𝑟 = 𝑅 → (𝑦 = (0g𝑟) ↔ 𝑦 = 0 ))
1412, 13imbi12d 344 . . . . . 6 (𝑟 = 𝑅 → (((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ((𝑥 · 𝑦) = 0𝑦 = 0 )))
154, 14raleqbidv 3344 . . . . 5 (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )))
164, 15rabeqbidv 3452 . . . 4 (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
17 df-rlreg 20711 . . . 4 RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
183fvexi 6921 . . . . 5 𝐵 ∈ V
1918rabex 5345 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} ∈ V
2016, 17, 19fvmpt 7016 . . 3 (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
21 fvprc 6899 . . . 4 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22 fvprc 6899 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
233, 22eqtrid 2787 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
2423rabeqdv 3449 . . . . 5 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
25 rab0 4392 . . . . 5 {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅
2624, 25eqtrdi 2791 . . . 4 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅)
2721, 26eqtr4d 2778 . . 3 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
2820, 27pm2.61i 182 . 2 (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
291, 28eqtri 2763 1 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  c0 4339  cfv 6563  (class class class)co 7431  Basecbs 17245  .rcmulr 17299  0gc0g 17486  RLRegcrlreg 20708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-rlreg 20711
This theorem is referenced by:  isrrg  20715  rrgeq0  20717  rrgss  20719
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