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Metamath Proof Explorer

Theorem dihcnvcl

Description: Closure of isomorphism H converse. (Contributed by NM, 8-Mar-2014)

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

Proof

Step Hyp Ref Expression
1 dihfn.b 
 |-  B = ( Base ` K )
2 dihfn.h 
 |-  H = ( LHyp ` K )
3 dihfn.i 
 |-  I = ( ( DIsoH ` K ) ` W )
4 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
5 eqid 
 |-  ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
6 1 2 3 4 5 dihf11 
 |-  ( ( K e. HL /\ W e. H ) -> I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) )
7 f1f1orn 
 |-  ( I : B -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) -> I : B -1-1-onto-> ran I )
8 6 7 syl 
 |-  ( ( K e. HL /\ W e. H ) -> I : B -1-1-onto-> ran I )
9 f1ocnvdm 
 |-  ( ( I : B -1-1-onto-> ran I /\ X e. ran I ) -> ( `' I ` X ) e. B )
10 8 9 sylan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( `' I ` X ) e. B )