Metamath Proof Explorer


Theorem ltrncnvat

Description: The converse of the lattice translation of an atom is an atom. (Contributed by NM, 9-May-2013)

Ref Expression
Hypotheses ltrnel.l
|- .<_ = ( le ` K )
ltrnel.a
|- A = ( Atoms ` K )
ltrnel.h
|- H = ( LHyp ` K )
ltrnel.t
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrncnvat
|- ( ( ( 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 ltrncnvatb
 |-  ( ( ( 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 )