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Theorem isrrg 20699
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 7439 . . . . 5 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
21eqeq1d 2738 . . . 4 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
32imbi1d 341 . . 3 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0𝑦 = 0 )))
43ralbidv 3177 . 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 20698 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
104, 9elrab2 3694 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  wral 3060  cfv 6560  (class class class)co 7432  Basecbs 17248  .rcmulr 17299  0gc0g 17485  RLRegcrlreg 20692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-rlreg 20695
This theorem is referenced by:  rrgeq0i  20700  unitrrg  20704  isdomn2  20712  isdomn2OLD  20713  rrgsubm  33288  zringidom  33580
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