Theorem ltrnval1

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

	Ref	Expression
Hypotheses	ltrnval1.b	|- B = ( Base ` K )
	ltrnval1.l	|- .<_ = ( le ` K )
	ltrnval1.h	|- H = ( LHyp ` K )
	ltrnval1.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	ltrnval1	|- ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )

Proof

Step	Hyp	Ref	Expression
1		ltrnval1.b	|- B = ( Base ` K )
2		ltrnval1.l	|- .<_ = ( le ` K )
3		ltrnval1.h	|- H = ( LHyp ` K )
4		ltrnval1.t	|- T = ( ( LTrn ` K ) ` W )
5		eqid	|- ( ( LDil ` K ) ` W ) = ( ( LDil ` K ) ` W )
6	3 5 4	ltrnldil	|- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. ( ( LDil ` K ) ` W ) )
7	6	3adant3	|- ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ X .<_ W ) ) -> F e. ( ( LDil ` K ) ` W ) )
8	1 2 3 5	ldilval	|- ( ( ( K e. V /\ W e. H ) /\ F e. ( ( LDil ` K ) ` W ) /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )
9	7 8	syld3an2	|- ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ X .<_ W ) ) -> ( F ` X ) = X )