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Theorem isrrg 19109
 Description: Membership in the set of left-regular elements. (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
isrrg (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
Distinct variable groups:   𝑦,𝐵   𝑦,𝑅   𝑦,𝑋
Allowed substitution hints:   · (𝑦)   𝐸(𝑦)   0 (𝑦)

Proof of Theorem isrrg
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 oveq1 6556 . . . . 5 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
21eqeq1d 2612 . . . 4 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
32imbi1d 330 . . 3 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0𝑦 = 0 )))
43ralbidv 2969 . 2 (𝑥 = 𝑋 → (∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
5 rrgval.e . . 3 𝐸 = (RLReg‘𝑅)
6 rrgval.b . . 3 𝐵 = (Base‘𝑅)
7 rrgval.t . . 3 · = (.r𝑅)
8 rrgval.z . . 3 0 = (0g𝑅)
95, 6, 7, 8rrgval 19108 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
104, 9elrab2 3333 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ‘cfv 5804  (class class class)co 6549  Basecbs 15695  .rcmulr 15769  0gc0g 15923  RLRegcrlreg 19100 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-rlreg 19104 This theorem is referenced by:  rrgeq0i  19110  unitrrg  19114  isdomn2  19120
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