This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem dihcnvid1

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

Ref Expression
Hypotheses dihcnvid1.b 
|- B = ( Base ` K )
dihcnvid1.h 
|- H = ( LHyp ` K )
dihcnvid1.i 
|- I = ( ( DIsoH ` K ) ` W )
Assertion dihcnvid1 
|- ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )

Proof

Step Hyp Ref Expression
1 dihcnvid1.b 
 |-  B = ( Base ` K )
2 dihcnvid1.h 
 |-  H = ( LHyp ` K )
3 dihcnvid1.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 f1ocnvfv1 
 |-  ( ( I : B -1-1-onto-> ran I /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )
10 8 9 sylan 
 |-  ( ( ( K e. HL /\ W e. H ) /\ X e. B ) -> ( `' I ` ( I ` X ) ) = X )