Theorem ltrn2ateq

Description: Property of the equality of a lattice translation with its value. (Contributed by NM, 27-May-2012)

	Ref	Expression
Hypotheses	ltrn2eq.l	|- .<_ = ( le ` K )
	ltrn2eq.a	|- A = ( Atoms ` K )
	ltrn2eq.h	|- H = ( LHyp ` K )
	ltrn2eq.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	ltrn2ateq	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) )

Proof

Step	Hyp	Ref	Expression
1		ltrn2eq.l	|- .<_ = ( le ` K )
2		ltrn2eq.a	|- A = ( Atoms ` K )
3		ltrn2eq.h	|- H = ( LHyp ` K )
4		ltrn2eq.t	|- T = ( ( LTrn ` K ) ` W )
5		eqid	|- ( Base ` K ) = ( Base ` K )
6	5 1 2 3 4	ltrnideq	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
7	6	3adant3r3	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` P ) = P ) )
8	5 1 2 3 4	ltrnideq	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( Q e. A /\ -. Q .<_ W ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` Q ) = Q ) )
9	8	3adant3r2	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( F = ( _I |` ( Base ` K ) ) <-> ( F ` Q ) = Q ) )
10	7 9	bitr3d	|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ ( P e. A /\ -. P .<_ W ) /\ ( Q e. A /\ -. Q .<_ W ) ) ) -> ( ( F ` P ) = P <-> ( F ` Q ) = Q ) )