Theorem rspcl

Description: The span of a set of ring elements is an ideal. (Contributed by Stefan O'Rear, 3-Jan-2015) (Revised by Mario Carneiro, 2-Oct-2015)

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

Proof

Step	Hyp	Ref	Expression
1		rspcl.k	|- K = ( RSpan ` R )
2		rspcl.b	|- B = ( Base ` R )
3		rspcl.u	|- U = ( LIdeal ` R )
4		rlmlmod	|- ( R e. Ring -> ( ringLMod ` R ) e. LMod )
5		rlmbas	|- ( Base ` R ) = ( Base ` ( ringLMod ` R ) )
6	2 5	eqtri	|- B = ( Base ` ( ringLMod ` R ) )
7		lidlval	|- ( LIdeal ` R ) = ( LSubSp ` ( ringLMod ` R ) )
8	3 7	eqtri	|- U = ( LSubSp ` ( ringLMod ` R ) )
9		rspval	|- ( RSpan ` R ) = ( LSpan ` ( ringLMod ` R ) )
10	1 9	eqtri	|- K = ( LSpan ` ( ringLMod ` R ) )
11	6 8 10	lspcl	|- ( ( ( ringLMod ` R ) e. LMod /\ G C_ B ) -> ( K ` G ) e. U )
12	4 11	sylan	|- ( ( R e. Ring /\ G C_ B ) -> ( K ` G ) e. U )