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Theorem islnr 41786
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 6888 . . 3 (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴))
21eleq1d 2819 . 2 (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM))
3 df-lnr 41785 . 2 LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
42, 3elrab2 3685 1 (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wcel 2107  cfv 6540  Ringcrg 20047  ringLModcrglmod 20770  LNoeMclnm 41750  LNoeRclnr 41784
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-lnr 41785
This theorem is referenced by:  lnrring  41787  lnrlnm  41788  islnr2  41789
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