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Metamath Proof Explorer

Theorem diasslssN

Description: The partial isomorphism A maps to subspaces of partial vector space A. (Contributed by NM, 17-Jan-2014) (New usage is discouraged.)

Ref Expression
Hypotheses diasslss.h 
|- H = ( LHyp ` K )
diasslss.u 
|- U = ( ( DVecA ` K ) ` W )
diasslss.i 
|- I = ( ( DIsoA ` K ) ` W )
diasslss.s 
|- S = ( LSubSp ` U )
Assertion diasslssN 
|- ( ( K e. HL /\ W e. H ) -> ran I C_ S )

Proof

Step Hyp Ref Expression
1 diasslss.h 
 |-  H = ( LHyp ` K )
2 diasslss.u 
 |-  U = ( ( DVecA ` K ) ` W )
3 diasslss.i 
 |-  I = ( ( DIsoA ` K ) ` W )
4 diasslss.s 
 |-  S = ( LSubSp ` U )
5 1 3 diaf11N 
 |-  ( ( K e. HL /\ W e. H ) -> I : dom I -1-1-onto-> ran I )
6 f1ocnvfv2 
 |-  ( ( I : dom I -1-1-onto-> ran I /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) = x )
7 5 6 sylan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) = x )
8 1 3 diacnvclN 
 |-  ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( `' I ` x ) e. dom I )
9 eqid 
 |-  ( Base ` K ) = ( Base ` K )
10 eqid 
 |-  ( le ` K ) = ( le ` K )
11 9 10 1 3 diaeldm 
 |-  ( ( K e. HL /\ W e. H ) -> ( ( `' I ` x ) e. dom I <-> ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) ) )
12 11 adantr 
 |-  ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( ( `' I ` x ) e. dom I <-> ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) ) )
13 8 12 mpbid 
 |-  ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) )
14 9 10 1 2 3 4 dialss 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( ( `' I ` x ) e. ( Base ` K ) /\ ( `' I ` x ) ( le ` K ) W ) ) -> ( I ` ( `' I ` x ) ) e. S )
15 13 14 syldan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> ( I ` ( `' I ` x ) ) e. S )
16 7 15 eqeltrrd 
 |-  ( ( ( K e. HL /\ W e. H ) /\ x e. ran I ) -> x e. S )
17 16 ex 
 |-  ( ( K e. HL /\ W e. H ) -> ( x e. ran I -> x e. S ) )
18 17 ssrdv 
 |-  ( ( K e. HL /\ W e. H ) -> ran I C_ S )