Theorem ltrnat

Description: The lattice translation of an atom is also an atom. TODO: See if this can shorten some ltrnel uses. (Contributed by NM, 25-May-2012)

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

Proof

Step	Hyp	Ref	Expression
1		ltrnel.l	|- .<_ = ( le ` K )
2		ltrnel.a	|- A = ( Atoms ` K )
3		ltrnel.h	|- H = ( LHyp ` K )
4		ltrnel.t	|- T = ( ( LTrn ` K ) ` W )
5		simp3	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> P e. A )
6		eqid	|- ( Base ` K ) = ( Base ` K )
7	6 2	atbase	|- ( P e. A -> P e. ( Base ` K ) )
8	6 2 3 4	ltrnatb	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. ( Base ` K ) ) -> ( P e. A <-> ( F ` P ) e. A ) )
9	7 8	syl3an3	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( P e. A <-> ( F ` P ) e. A ) )
10	5 9	mpbid	|- ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ P e. A ) -> ( F ` P ) e. A )