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. } ) )