Theorem islnr 43143

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.

Step	Hyp	Ref	Expression
1		fveq2 6822	. . 3 ⊢ (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴))
2	1	eleq1d 2816	. 2 ⊢ (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM))
3		df-lnr 43142	. 2 ⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
4	2, 3	elrab2 3650	1 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ‘cfv 6481  Ringcrg 20149  ringLModcrglmod 21104  LNoeMclnm 43107  LNoeRclnr 43141

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-iota 6437  df-fv 6489  df-lnr 43142

This theorem is referenced by:  lnrring  43144  lnrlnm  43145  islnr2  43146