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Metamath Proof Explorer

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 )