Theorem diarnN

Description: Partial isomorphism A maps onto the set of all closed subspaces of partial vector space A. Part of Lemma M of Crawley p. 121 line 12, with closed subspaces rather than subspaces. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	dvadia.h	|- H = ( LHyp ` K )
	dvadia.u	|- U = ( ( DVecA ` K ) ` W )
	dvadia.i	|- I = ( ( DIsoA ` K ) ` W )
	dvadia.n	|- ._|_ = ( ( ocA ` K ) ` W )
	dvadia.s	|- S = ( LSubSp ` U )
Assertion	diarnN	|- ( ( K e. HL /\ W e. H ) -> ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )

Proof

Step	Hyp	Ref	Expression
1		dvadia.h	|- H = ( LHyp ` K )
2		dvadia.u	|- U = ( ( DVecA ` K ) ` W )
3		dvadia.i	|- I = ( ( DIsoA ` K ) ` W )
4		dvadia.n	|- ._|_ = ( ( ocA ` K ) ` W )
5		dvadia.s	|- S = ( LSubSp ` U )
6	1 2 3 5	diasslssN	|- ( ( K e. HL /\ W e. H ) -> ran I C_ S )
7		sseqin2	|- ( ran I C_ S <-> ( S i^i ran I ) = ran I )
8	6 7	sylib	|- ( ( K e. HL /\ W e. H ) -> ( S i^i ran I ) = ran I )
9	1 3 4	doca3N	|- ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( ._|_ ` ( ._|_ ` x ) ) = x )
10	9	ex	|- ( ( K e. HL /\ W e. H ) -> ( x e. ran I -> ( ._|_ ` ( ._|_ ` x ) ) = x ) )
11	10	adantr	|- ( ( ( K e. HL /\ W e. H ) /\ x e. S ) -> ( x e. ran I -> ( ._|_ ` ( ._|_ ` x ) ) = x ) )
12	1 2 3 4 5	dvadiaN	|- ( ( ( K e. HL /\ W e. H ) /\ ( x e. S /\ ( ._|_ ` ( ._|_ ` x ) ) = x ) ) -> x e. ran I )
13	12	expr	|- ( ( ( K e. HL /\ W e. H ) /\ x e. S ) -> ( ( ._|_ ` ( ._|_ ` x ) ) = x -> x e. ran I ) )
14	11 13	impbid	|- ( ( ( K e. HL /\ W e. H ) /\ x e. S ) -> ( x e. ran I <-> ( ._|_ ` ( ._|_ ` x ) ) = x ) )
15	14	rabbi2dva	|- ( ( K e. HL /\ W e. H ) -> ( S i^i ran I ) = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )
16	8 15	eqtr3d	|- ( ( K e. HL /\ W e. H ) -> ran I = { x e. S | ( ._|_ ` ( ._|_ ` x ) ) = x } )