Theorem dihrnlss

Description: The isomorphism H maps to subspaces. (Contributed by NM, 14-Mar-2014)

	Ref	Expression
Hypotheses	dihsslss.h	|- H = ( LHyp ` K )
	dihsslss.u	|- U = ( ( DVecH ` K ) ` W )
	dihsslss.i	|- I = ( ( DIsoH ` K ) ` W )
	dihsslss.s	|- S = ( LSubSp ` U )
Assertion	dihrnlss	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X e. S )

Proof

Step	Hyp	Ref	Expression
1		dihsslss.h	|- H = ( LHyp ` K )
2		dihsslss.u	|- U = ( ( DVecH ` K ) ` W )
3		dihsslss.i	|- I = ( ( DIsoH ` K ) ` W )
4		dihsslss.s	|- S = ( LSubSp ` U )
5	1 2 3 4	dihsslss	|- ( ( K e. HL /\ W e. H ) -> ran I C_ S )
6	5	sselda	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> X e. S )