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Metamath Proof Explorer

Theorem dihcnvid2

Description: The isomorphism of a converse isomorphism. (Contributed by NM, 5-Aug-2014)

Ref Expression
Hypotheses dihcnvid2.h 
|- H = ( LHyp ` K )
dihcnvid2.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihcnvid2 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )

Proof

Step Hyp Ref Expression
1 dihcnvid2.h 
 |-  H = ( LHyp ` K )
2 dihcnvid2.i 
 |-  I = ( ( DIsoH ` K ) ` W )
3 eqid 
 |-  ( Base ` K ) = ( Base ` K )
4 eqid 
 |-  ( ( DVecH ` K ) ` W ) = ( ( DVecH ` K ) ` W )
5 eqid 
 |-  ( LSubSp ` ( ( DVecH ` K ) ` W ) ) = ( LSubSp ` ( ( DVecH ` K ) ` W ) )
6 3 1 2 4 5 dihf11 
 |-  ( ( K e. HL /\ W e. H ) -> I : ( Base ` K ) -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) )
7 f1f1orn 
 |-  ( I : ( Base ` K ) -1-1-> ( LSubSp ` ( ( DVecH ` K ) ` W ) ) -> I : ( Base ` K ) -1-1-onto-> ran I )
8 6 7 syl 
 |-  ( ( K e. HL /\ W e. H ) -> I : ( Base ` K ) -1-1-onto-> ran I )
9 f1ocnvfv2 
 |-  ( ( I : ( Base ` K ) -1-1-onto-> ran I /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )
10 8 9 sylan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. ran I ) -> ( I ` ( `' I ` X ) ) = X )