Metamath Proof Explorer

Theorem dvadiaN

Description: Any closed subspace is a member of the range of partial isomorphism A, showing the isomorphism maps onto the set of closed subspaces of partial vector space A. (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 dvadiaN 
|- ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> X e. ran I )

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 simprr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> ( ._|_ ` ( ._|_ ` X ) ) = X )
7 eqid 
 |-  ( Base ` U ) = ( Base ` U )
8 7 5 lssss 
 |-  ( X e. S -> X C_ ( Base ` U ) )
9 8 ad2antrl 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> X C_ ( Base ` U ) )
10 eqid 
 |-  ( ( LTrn ` K ) ` W ) = ( ( LTrn ` K ) ` W )
11 1 10 2 7 dvavbase 
 |-  ( ( K e. HL /\ W e. H ) -> ( Base ` U ) = ( ( LTrn ` K ) ` W ) )
12 11 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> ( Base ` U ) = ( ( LTrn ` K ) ` W ) )
13 9 12 sseqtrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> X C_ ( ( LTrn ` K ) ` W ) )
14 1 10 3 4 docaclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X C_ ( ( LTrn ` K ) ` W ) ) -> ( ._|_ ` X ) e. ran I )
15 13 14 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> ( ._|_ ` X ) e. ran I )
16 1 10 3 diaelrnN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) e. ran I ) -> ( ._|_ ` X ) C_ ( ( LTrn ` K ) ` W ) )
17 15 16 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> ( ._|_ ` X ) C_ ( ( LTrn ` K ) ` W ) )
18 1 10 3 4 docaclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ._|_ ` X ) C_ ( ( LTrn ` K ) ` W ) ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
19 17 18 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> ( ._|_ ` ( ._|_ ` X ) ) e. ran I )
20 6 19 eqeltrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( X e. S /\ ( ._|_ ` ( ._|_ ` X ) ) = X ) ) -> X e. ran I )