Theorem islnr

Description: Property of a left-Noetherian ring. (Contributed by Stefan O'Rear, 24-Jan-2015)

	Ref	Expression
Assertion	islnr	⊢ ( 𝐴 ∈ LNoeR ↔ ( 𝐴 ∈ Ring ∧ ( ringLMod ‘ 𝐴 ) ∈ LNoeM ) )

Proof

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