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 ) ) )