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. } ) )