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Metamath Proof Explorer

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 } )