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Theorem isrrg 20715
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 7438 . . . . 5 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
21eqeq1d 2737 . . . 4 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
32imbi1d 341 . . 3 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0𝑦 = 0 )))
43ralbidv 3176 . 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 20714 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
104, 9elrab2 3698 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  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:  rrgeq0i  20716  unitrrg  20720  isdomn2  20728  isdomn2OLD  20729  rrgsubm  33268  zringidom  33559
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