Theorem islnr

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

	Ref	Expression
Assertion	islnr	|- ( A e. LNoeR <-> ( A e. Ring /\ ( ringLMod ` A ) e. LNoeM ) )

Proof

Step	Hyp	Ref	Expression
1		fveq2	|- ( a = A -> ( ringLMod ` a ) = ( ringLMod ` A ) )
2	1	eleq1d	|- ( a = A -> ( ( ringLMod ` a ) e. LNoeM <-> ( ringLMod ` A ) e. LNoeM ) )
3		df-lnr	|- LNoeR = { a e. Ring | ( ringLMod ` a ) e. LNoeM }
4	2 3	elrab2	|- ( A e. LNoeR <-> ( A e. Ring /\ ( ringLMod ` A ) e. LNoeM ) )