Theorem dihcnvid2

Description: The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014)

	Ref	Expression
Hypotheses	dihcnvid2.h	|- H = ( LHyp ` K )
	dihcnvid2.i	|- I = ( ( DIsoH ` K ) ` W )
Assertion	dihcnvid2	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )

Proof

Step	Hyp	Ref	Expression
1		dihcnvid2.h	|- H = ( LHyp ` K )
2		dihcnvid2.i	|- I = ( ( DIsoH ` K ) ` W )
3		eqid	|- ( Base ` K ) = ( Base ` K )
4		eqid	|- ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
5		eqid	|- ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
6	3 1 2 4 5	dihf11	|- ( ( K e. HL /\ W e. H ) -> I : ( Base ` K ) -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) )
7		f1f1orn	|- ( I : ( Base ` K ) -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) -> I : ( Base ` K ) -1-1-onto-> ran I )
8	6 7	syl	|- ( ( K e. HL /\ W e. H ) -> I : ( Base ` K ) -1-1-onto-> ran I )
9		f1ocnvfv2	|- ( ( I : ( Base ` K ) -1-1-onto-> ran I /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
10	8 9	sylan	|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )