Theorem rspssp

Description: The ideal span of a set of elements in a ring is contained in any subring which contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015)

	Ref	Expression
Hypotheses	rspcl.k	|- K = ( RSpan ` R )
	rspssp.u	|- U = ( LIdeal ` R )
Assertion	rspssp	|- ( ( R e. Ring /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )

Proof

Step	Hyp	Ref	Expression
1		rspcl.k	|- K = ( RSpan ` R )
2		rspssp.u	|- U = ( LIdeal ` R )
3		rlmlmod	|- ( R e. Ring -> ( ringLMod ` R ) e. LMod )
4		lidlval	|- ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) )
5	2 4	eqtri	|- U = ( LSubSp ` ( ringLMod ` R ) )
6		rspval	|- ( RSpan ` R ) = ( LSpan ` ( ringLMod ` R ) )
7	1 6	eqtri	|- K = ( LSpan ` ( ringLMod ` R ) )
8	5 7	lspssp	|- ( ( ( ringLMod ` R ) e. LMod /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )
9	3 8	syl3an1	|- ( ( R e. Ring /\ I e. U /\ G C_ I ) -> ( K ` G ) C_ I )