Theorem lidlss

Description: An ideal is a subset of the base set. (Contributed by Stefan O'Rear, 28-Mar-2015)

	Ref	Expression
Hypotheses	lidlss.b	|- B = ( Base ` W )
	lidlss.i	|- I = ( LIdeal ` W )
Assertion	lidlss	|- ( U e. I -> U C_ B )

Proof

Step	Hyp	Ref	Expression
1		lidlss.b	|- B = ( Base ` W )
2		lidlss.i	|- I = ( LIdeal ` W )
3		rlmbas	|- ( Base ` W ) = ( Base ` ( ringLMod ` W ) )
4	1 3	eqtri	|- B = ( Base ` ( ringLMod ` W ) )
5		lidlval	|- ( LIdeal ` W ) = ( LSubSp ` ( ringLMod ` W ) )
6	2 5	eqtri	|- I = ( LSubSp ` ( ringLMod ` W ) )
7	4 6	lssss	|- ( U e. I -> U C_ B )