Theorem idlss

Description: An ideal of R is a subset of R . (Contributed by Jeff Madsen, 10-Jun-2010)

	Ref	Expression
Hypotheses	idlss.1	|- G = ( 1st ` R )
	idlss.2	|- X = ran G
Assertion	idlss	|- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> I C_ X )

Proof

Step	Hyp	Ref	Expression
1		idlss.1	|- G = ( 1st ` R )
2		idlss.2	|- X = ran G
3		eqid	|- ( 2nd ` R ) = ( 2nd ` R )
4		eqid	|- ( GId ` G ) = ( GId ` G )
5	1 3 2 4	isidl	|- ( R e. RingOps -> ( I e. ( Idl ` R ) <-> ( I C_ X /\ ( GId ` G ) e. I /\ A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. X ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) ) ) )
6	5	biimpa	|- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> ( I C_ X /\ ( GId ` G ) e. I /\ A. x e. I ( A. y e. I ( x G y ) e. I /\ A. z e. X ( ( z ( 2nd ` R ) x ) e. I /\ ( x ( 2nd ` R ) z ) e. I ) ) ) )
7	6	simp1d	|- ( ( R e. RingOps /\ I e. ( Idl ` R ) ) -> I C_ X )