Theorem ldilval

Description: Value of a lattice dilation under its co-atom. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	ldilval.b	|- B = ( Base ` K )
	ldilval.l	|- .<_ = ( le ` K )
	ldilval.h	|- H = ( LHyp ` K )
	ldilval.d	|- D = ( ( LDil ` K ) ` W )
Assertion	ldilval	|- ( ( ( K e. V /\ W e. H ) /\ F e. D /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )

Proof

Step	Hyp	Ref	Expression
1		ldilval.b	|- B = ( Base ` K )
2		ldilval.l	|- .<_ = ( le ` K )
3		ldilval.h	|- H = ( LHyp ` K )
4		ldilval.d	|- D = ( ( LDil ` K ) ` W )
5		eqid	|- ( LAut ` K ) = ( LAut ` K )
6	1 2 3 5 4	isldil	|- ( ( K e. V /\ W e. H ) -> ( F e. D <-> ( F e. ( LAut ` K ) /\ A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) ) )
7		simpr	|- ( ( F e. ( LAut ` K ) /\ A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) -> A. x e. B ( x .<_ W -> ( F ` x ) = x ) )
8	6 7	syl6bi	|- ( ( K e. V /\ W e. H ) -> ( F e. D -> A. x e. B ( x .<_ W -> ( F ` x ) = x ) ) )
9		breq1	|- ( x = X -> ( x .<_ W <-> X .<_ W ) )
10		fveq2	|- ( x = X -> ( F ` x ) = ( F ` X ) )
11		id	|- ( x = X -> x = X )
12	10 11	eqeq12d	|- ( x = X -> ( ( F ` x ) = x <-> ( F ` X ) = X ) )
13	9 12	imbi12d	|- ( x = X -> ( ( x .<_ W -> ( F ` x ) = x ) <-> ( X .<_ W -> ( F ` X ) = X ) ) )
14	13	rspccv	|- ( A. x e. B ( x .<_ W -> ( F ` x ) = x ) -> ( X e. B -> ( X .<_ W -> ( F ` X ) = X ) ) )
15	14	impd	|- ( A. x e. B ( x .<_ W -> ( F ` x ) = x ) -> ( ( X e. B /\ X .<_ W ) -> ( F ` X ) = X ) )
16	8 15	syl6	|- ( ( K e. V /\ W e. H ) -> ( F e. D -> ( ( X e. B /\ X .<_ W ) -> ( F ` X ) = X ) ) )
17	16	3imp	|- ( ( ( K e. V /\ W e. H ) /\ F e. D /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )