Theorem dihcl

Description: Closure of isomorphism H. (Contributed by NM, 8-Mar-2014)

	Ref	Expression
Hypotheses	dihfn.b	|- B = ( Base ` K )
	dihfn.h	|- H = ( LHyp ` K )
	dihfn.i	|- I = ( ( DIsoH ` K ) ` W )
Assertion	dihcl	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. ran I )

Proof

Step	Hyp	Ref	Expression
1		dihfn.b	|- B = ( Base ` K )
2		dihfn.h	|- H = ( LHyp ` K )
3		dihfn.i	|- I = ( ( DIsoH ` K ) ` W )
4		eqid	|- ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
5		eqid	|- ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
6	1 2 3 4 5	dihf11	|- ( ( K e. HL /\ W e. H ) -> I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) )
7	6	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) )
8		f1fn	|- ( I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) -> I Fn B )
9	7 8	syl	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> I Fn B )
10		fnfvelrn	|- ( ( I Fn B /\ X e. B ) -> ( I ` X ) e. ran I )
11	9 10	sylancom	|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( I ` X ) e. ran I )