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Metamath Proof Explorer

Theorem dibval

Description: The partial isomorphism B for a lattice K . (Contributed by NM, 8-Dec-2013)

Ref Expression
Hypotheses dibval.b 
|- B = ( Base ` K )
dibval.h 
|- H = ( LHyp ` K )
dibval.t 
|- T = ( ( LTrn ` K ) ` W )
dibval.o 
|- .0. = ( f e. T |-> ( _I |` B ) )
dibval.j 
|- J = ( ( DIsoA ` K ) ` W )
dibval.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibval 
|- ( ( ( K e. V /\ W e. H ) /\ X e. dom J ) -> ( I ` X ) = ( ( J ` X ) X. { .0. } ) )

Proof

Step Hyp Ref Expression
1 dibval.b 
 |-  B = ( Base ` K )
2 dibval.h 
 |-  H = ( LHyp ` K )
3 dibval.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 dibval.o 
 |-  .0. = ( f e. T |-> ( _I |` B ) )
5 dibval.j 
 |-  J = ( ( DIsoA ` K ) ` W )
6 dibval.i 
 |-  I = ( ( DIsoB ` K ) ` W )
7 1 2 3 4 5 6 dibfval 
 |-  ( ( K e. V /\ W e. H ) -> I = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) )
8 7 adantr 
 |-  ( ( ( K e. V /\ W e. H ) /\ X e. dom J ) -> I = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) )
9 8 fveq1d 
 |-  ( ( ( K e. V /\ W e. H ) /\ X e. dom J ) -> ( I ` X ) = ( ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) ` X ) )
10 fveq2 
 |-  ( x = X -> ( J ` x ) = ( J ` X ) )
11 10 xpeq1d 
 |-  ( x = X -> ( ( J ` x ) X. { .0. } ) = ( ( J ` X ) X. { .0. } ) )
12 eqid 
 |-  ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) = ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) )
13 fvex 
 |-  ( J ` X ) e. _V
14 snex 
 |-  { .0. } e. _V
15 13 14 xpex 
 |-  ( ( J ` X ) X. { .0. } ) e. _V
16 11 12 15 fvmpt 
 |-  ( X e. dom J -> ( ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) ` X ) = ( ( J ` X ) X. { .0. } ) )
17 16 adantl 
 |-  ( ( ( K e. V /\ W e. H ) /\ X e. dom J ) -> ( ( x e. dom J |-> ( ( J ` x ) X. { .0. } ) ) ` X ) = ( ( J ` X ) X. { .0. } ) )
18 9 17 eqtrd 
 |-  ( ( ( K e. V /\ W e. H ) /\ X e. dom J ) -> ( I ` X ) = ( ( J ` X ) X. { .0. } ) )