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Theorem islnr 40672
Description: Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015.)
Assertion
Ref Expression
islnr (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))

Proof of Theorem islnr
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6736 . . 3 (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴))
21eleq1d 2823 . 2 (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM))
3 df-lnr 40671 . 2 LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
42, 3elrab2 3618 1 (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
Colors of variables: wff setvar class
Syntax hints:  wb 209  wa 399   = wceq 1543  wcel 2111  cfv 6398  Ringcrg 19590  ringLModcrglmod 20234  LNoeMclnm 40636  LNoeRclnr 40670
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-ext 2709
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-sb 2072  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3071  df-v 3423  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4253  df-if 4455  df-sn 4557  df-pr 4559  df-op 4563  df-uni 4835  df-br 5069  df-iota 6356  df-fv 6406  df-lnr 40671
This theorem is referenced by:  lnrring  40673  lnrlnm  40674  islnr2  40675
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