Theorem rspssid

Description: The span of a set of ring elements contains those elements. (Contributed by Stefan O'Rear, 3-Jan-2015)

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

Proof

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