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Metamath Proof Explorer

Theorem dibval3N

Description: Value of the partial isomorphism B for a lattice K . (Contributed by NM, 24-Feb-2014) (New usage is discouraged.)

Ref Expression
Hypotheses dibval3.b 
|- B = ( Base ` K )
dibval3.l 
|- .<_ = ( le ` K )
dibval3.h 
|- H = ( LHyp ` K )
dibval3.t 
|- T = ( ( LTrn ` K ) ` W )
dibval3.r 
|- R = ( ( trL ` K ) ` W )
dibval3.o 
|- .0. = ( g e. T |-> ( _I |` B ) )
dibval3.i 
|- I = ( ( DIsoB ` K ) ` W )
Assertion dibval3N 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( { f e. T | ( R ` f ) .<_ X } X. { .0. } ) )

Proof

Step Hyp Ref Expression
1 dibval3.b 
 |-  B = ( Base ` K )
2 dibval3.l 
 |-  .<_ = ( le ` K )
3 dibval3.h 
 |-  H = ( LHyp ` K )
4 dibval3.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 dibval3.r 
 |-  R = ( ( trL ` K ) ` W )
6 dibval3.o 
 |-  .0. = ( g e. T |-> ( _I |` B ) )
7 dibval3.i 
 |-  I = ( ( DIsoB ` K ) ` W )
8 eqid 
 |-  ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
9 1 2 3 4 6 8 7 dibval2 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
10 1 2 3 4 5 8 diaval 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( DIsoA ` K ) ` W ) ` X ) = { f e. T | ( R ` f ) .<_ X } )
11 10 xpeq1d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) = ( { f e. T | ( R ` f ) .<_ X } X. { .0. } ) )
12 9 11 eqtrd 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( { f e. T | ( R ` f ) .<_ X } X. { .0. } ) )