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

Metamath Proof Explorer

Theorem diaelval

Description: Member of the partial isomorphism A for a lattice K . (Contributed by NM, 3-Dec-2013)

Ref Expression
Hypotheses diaval.b 
|- B = ( Base ` K )
diaval.l 
|- .<_ = ( le ` K )
diaval.h 
|- H = ( LHyp ` K )
diaval.t 
|- T = ( ( LTrn ` K ) ` W )
diaval.r 
|- R = ( ( trL ` K ) ` W )
diaval.i 
|- I = ( ( DIsoA ` K ) ` W )
Assertion diaelval 
|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> ( F e. T /\ ( R ` F ) .<_ X ) ) )

Proof

Step Hyp Ref Expression
1 diaval.b 
 |-  B = ( Base ` K )
2 diaval.l 
 |-  .<_ = ( le ` K )
3 diaval.h 
 |-  H = ( LHyp ` K )
4 diaval.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 diaval.r 
 |-  R = ( ( trL ` K ) ` W )
6 diaval.i 
 |-  I = ( ( DIsoA ` K ) ` W )
7 1 2 3 4 5 6 diaval 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = { f e. T | ( R ` f ) .<_ X } )
8 7 eleq2d 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> F e. { f e. T | ( R ` f ) .<_ X } ) )
9 fveq2 
 |-  ( f = F -> ( R ` f ) = ( R ` F ) )
10 9 breq1d 
 |-  ( f = F -> ( ( R ` f ) .<_ X <-> ( R ` F ) .<_ X ) )
11 10 elrab 
 |-  ( F e. { f e. T | ( R ` f ) .<_ X } <-> ( F e. T /\ ( R ` F ) .<_ X ) )
12 8 11 bitrdi 
 |-  ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( F e. ( I ` X ) <-> ( F e. T /\ ( R ` F ) .<_ X ) ) )