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 ) ) )