Theorem dibelval2nd

Description: Membership in value of the partial isomorphism B for a lattice K . (Contributed by NM, 13-Feb-2014)

	Ref	Expression
Hypotheses	dibelval2nd.b	|- B = ( Base ` K )
	dibelval2nd.l	|- .<_ = ( le ` K )
	dibelval2nd.h	|- H = ( LHyp ` K )
	dibelval2nd.t	|- T = ( ( LTrn ` K ) ` W )
	dibelval2nd.o	|- .0. = ( f e. T |-> ( _I |` B ) )
	dibelval2nd.i	|- I = ( ( DIsoB ` K ) ` W )
Assertion	dibelval2nd	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 2nd ` Y ) = .0. )

Proof

Step	Hyp	Ref	Expression
1		dibelval2nd.b	|- B = ( Base ` K )
2		dibelval2nd.l	|- .<_ = ( le ` K )
3		dibelval2nd.h	|- H = ( LHyp ` K )
4		dibelval2nd.t	|- T = ( ( LTrn ` K ) ` W )
5		dibelval2nd.o	|- .0. = ( f e. T |-> ( _I |` B ) )
6		dibelval2nd.i	|- I = ( ( DIsoB ` K ) ` W )
7		eqid	|- ( ( DIsoA ` K ) ` W ) = ( ( DIsoA ` K ) ` W )
8	1 2 3 4 5 7 6	dibval2	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( I ` X ) = ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
9	8	eleq2d	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) ) -> ( Y e. ( I ` X ) <-> Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) ) )
10	9	biimp3a	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) )
11		xp2nd	|- ( Y e. ( ( ( ( DIsoA ` K ) ` W ) ` X ) X. { .0. } ) -> ( 2nd ` Y ) e. { .0. } )
12		elsni	|- ( ( 2nd ` Y ) e. { .0. } -> ( 2nd ` Y ) = .0. )
13	10 11 12	3syl	|- ( ( ( K e. V /\ W e. H ) /\ ( X e. B /\ X .<_ W ) /\ Y e. ( I ` X ) ) -> ( 2nd ` Y ) = .0. )