Theorem islnr 42403

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 6882	. . 3 ⊢ (𝑎 = 𝐴 → (ringLMod‘𝑎) = (ringLMod‘𝐴))
2	1	eleq1d 2810	. 2 ⊢ (𝑎 = 𝐴 → ((ringLMod‘𝑎) ∈ LNoeM ↔ (ringLMod‘𝐴) ∈ LNoeM))
3		df-lnr 42402	. 2 ⊢ LNoeR = {𝑎 ∈ Ring ∣ (ringLMod‘𝑎) ∈ LNoeM}
4	2, 3	elrab2 3679	1 ⊢ (𝐴 ∈ LNoeR ↔ (𝐴 ∈ Ring ∧ (ringLMod‘𝐴) ∈ LNoeM))

Syntax hints:   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ‘cfv 6534  Ringcrg 20134  ringLModcrglmod 21016  LNoeMclnm 42367  LNoeRclnr 42401

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-iota 6486  df-fv 6542  df-lnr 42402

This theorem is referenced by:  lnrring  42404  lnrlnm  42405  islnr2  42406