Theorem dihsslss

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	dihsslss	|- ( ( K e. HL /\ W e. H ) -> ran I C_ 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 3	dihcnvid2	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) = x )
6		eqid	|- ( Base ` K ) = ( Base ` K )
7	6 1 3	dihcnvcl	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( `' I ` x ) e. ( Base ` K ) )
8	6 1 3 2 4	dihlss	|- ( ( ( K e. HL /\ W e. H ) /\ ( `' I ` x ) e. ( Base ` K ) ) -> ( I ` ( `' I ` x ) ) e. S )
9	7 8	syldan	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) e. S )
10	5 9	eqeltrrd	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> x e. S )
11	10	ex	|- ( ( K e. HL /\ W e. H ) -> ( x e. ran I -> x e. S ) )
12	11	ssrdv	|- ( ( K e. HL /\ W e. H ) -> ran I C_ S )