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Theorem isrrg 19269
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 6642 . . . . 5 (𝑥 = 𝑋 → (𝑥 · 𝑦) = (𝑋 · 𝑦))
21eqeq1d 2622 . . . 4 (𝑥 = 𝑋 → ((𝑥 · 𝑦) = 0 ↔ (𝑋 · 𝑦) = 0 ))
32imbi1d 331 . . 3 (𝑥 = 𝑋 → (((𝑥 · 𝑦) = 0𝑦 = 0 ) ↔ ((𝑋 · 𝑦) = 0𝑦 = 0 )))
43ralbidv 2983 . 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 19268 . 2 𝐸 = {𝑥𝐵 ∣ ∀𝑦𝐵 ((𝑥 · 𝑦) = 0𝑦 = 0 )}
104, 9elrab2 3360 1 (𝑋𝐸 ↔ (𝑋𝐵 ∧ ∀𝑦𝐵 ((𝑋 · 𝑦) = 0𝑦 = 0 )))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  cfv 5876  (class class class)co 6635  Basecbs 15838  .rcmulr 15923  0gc0g 16081  RLRegcrlreg 19260
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-rlreg 19264
This theorem is referenced by:  rrgeq0i  19270  unitrrg  19274  isdomn2  19280
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