This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem ltrncoat

Description: Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel , ltrnat uses. (Contributed by NM, 1-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 ltrncoat 
|- ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( F ` ( G ` 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 simp1 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( K e. HL /\ W e. H ) )
6 simp2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> F e. T )
7 1 2 3 4 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ P e. A ) -> ( G ` P ) e. A )
8 7 3adant2l 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( G ` P ) e. A )
9 1 2 3 4 ltrnat 
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( G ` P ) e. A ) -> ( F ` ( G ` P ) ) e. A )
10 5 6 8 9 syl3anc 
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ P e. A ) -> ( F ` ( G ` P ) ) e. A )