MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  rrgval Structured version   Visualization version   GIF version

Theorem rrgval 20349
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 6735 . . . . . 6 (𝑟 = 𝑅 → (Base‘𝑟) = (Base‘𝑅))
3 rrgval.b . . . . . 6 𝐵 = (Base‘𝑅)
42, 3eqtr4di 2797 . . . . 5 (𝑟 = 𝑅 → (Base‘𝑟) = 𝐵)
5 fveq2 6735 . . . . . . . . . 10 (𝑟 = 𝑅 → (.r𝑟) = (.r𝑅))
6 rrgval.t . . . . . . . . . 10 · = (.r𝑅)
75, 6eqtr4di 2797 . . . . . . . . 9 (𝑟 = 𝑅 → (.r𝑟) = · )
87oveqd 7248 . . . . . . . 8 (𝑟 = 𝑅 → (𝑥(.r𝑟)𝑦) = (𝑥 · 𝑦))
9 fveq2 6735 . . . . . . . . 9 (𝑟 = 𝑅 → (0g𝑟) = (0g𝑅))
10 rrgval.z . . . . . . . . 9 0 = (0g𝑅)
119, 10eqtr4di 2797 . . . . . . . 8 (𝑟 = 𝑅 → (0g𝑟) = 0 )
128, 11eqeq12d 2754 . . . . . . 7 (𝑟 = 𝑅 → ((𝑥(.r𝑟)𝑦) = (0g𝑟) ↔ (𝑥 · 𝑦) = 0 ))
1311eqeq2d 2749 . . . . . . 7 (𝑟 = 𝑅 → (𝑦 = (0g𝑟) ↔ 𝑦 = 0 ))
1412, 13imbi12d 348 . . . . . 6 (𝑟 = 𝑅 → (((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ((𝑥 · 𝑦) = 0𝑦 = 0 )))
154, 14raleqbidv 3325 . . . . 5 (𝑟 = 𝑅 → (∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟)) ↔ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )))
164, 15rabeqbidv 3408 . . . 4 (𝑟 = 𝑅 → {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))} = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
17 df-rlreg 20345 . . . 4 RLReg = (𝑟 ∈ V ↦ {𝑥 ∈ (Base‘𝑟) ∣ ∀𝑦 ∈ (Base‘𝑟)((𝑥(.r𝑟)𝑦) = (0g𝑟) → 𝑦 = (0g𝑟))})
183fvexi 6749 . . . . 5 𝐵 ∈ V
1918rabex 5239 . . . 4 {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} ∈ V
2016, 17, 19fvmpt 6836 . . 3 (𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
21 fvprc 6727 . . . 4 𝑅 ∈ V → (RLReg‘𝑅) = ∅)
22 fvprc 6727 . . . . . . 7 𝑅 ∈ V → (Base‘𝑅) = ∅)
233, 22syl5eq 2791 . . . . . 6 𝑅 ∈ V → 𝐵 = ∅)
2423rabeqdv 3407 . . . . 5 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
25 rab0 4311 . . . . 5 {𝑥 ∈ ∅ ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅
2624, 25eqtrdi 2795 . . . 4 𝑅 ∈ V → {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )} = ∅)
2721, 26eqtr4d 2781 . . 3 𝑅 ∈ V → (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )})
2820, 27pm2.61i 185 . 2 (RLReg‘𝑅) = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
291, 28eqtri 2766 1 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1543  wcel 2111  wral 3062  {crab 3066  Vcvv 3420  c0 4251  cfv 6397  (class class class)co 7231  Basecbs 16784  .rcmulr 16827  0gc0g 16968  RLRegcrlreg 20341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2159  ax-12 2176  ax-ext 2709  ax-sep 5206  ax-nul 5213  ax-pr 5336
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2072  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2887  df-ne 2942  df-ral 3067  df-rex 3068  df-rab 3071  df-v 3422  df-dif 3883  df-un 3885  df-in 3887  df-ss 3897  df-nul 4252  df-if 4454  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4834  df-br 5068  df-opab 5130  df-mpt 5150  df-id 5469  df-xp 5571  df-rel 5572  df-cnv 5573  df-co 5574  df-dm 5575  df-iota 6355  df-fun 6399  df-fv 6405  df-ov 7234  df-rlreg 20345
This theorem is referenced by:  isrrg  20350  rrgeq0  20352  rrgss  20354
  Copyright terms: Public domain W3C validator