Metamath Proof Explorer


Theorem ltrncoelN

Description: Composition of lattice translations of an atom. TODO: See if this can shorten some ltrnel uses. (Contributed by NM, 1-May-2013) (New usage is discouraged.)

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

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 /\ -. P .<_ W ) ) -> ( K e. HL /\ W e. H ) )
6 simp2l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> F e. T )
7 1 2 3 4 ltrnel
 |-  ( ( ( K e. HL /\ W e. H ) /\ G e. T /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
8 7 3adant2l
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) )
9 1 2 3 4 ltrnel
 |-  ( ( ( K e. HL /\ W e. H ) /\ F e. T /\ ( ( G ` P ) e. A /\ -. ( G ` P ) .<_ W ) ) -> ( ( F ` ( G ` P ) ) e. A /\ -. ( F ` ( G ` P ) ) .<_ W ) )
10 5 6 8 9 syl3anc
 |-  ( ( ( K e. HL /\ W e. H ) /\ ( F e. T /\ G e. T ) /\ ( P e. A /\ -. P .<_ W ) ) -> ( ( F ` ( G ` P ) ) e. A /\ -. ( F ` ( G ` P ) ) .<_ W ) )