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Metamath Proof Explorer

Theorem ltrncl

Description: Closure of a lattice translation. (Contributed by NM, 20-May-2012)

Ref Expression
Hypotheses ltrn1o.b 
|- B = ( Base ` K )
ltrn1o.h 
|- H = ( LHyp ` K )
ltrn1o.t 
|- T = ( ( LTrn ` K ) ` W )
Assertion ltrncl 
|- ( ( ( K e. V /\ W e. H ) /\ F e. T /\ X e. B ) -> ( F ` X ) e. B )

Proof

Step Hyp Ref Expression
1 ltrn1o.b 
 |-  B = ( Base ` K )
2 ltrn1o.h 
 |-  H = ( LHyp ` K )
3 ltrn1o.t 
 |-  T = ( ( LTrn ` K ) ` W )
4 simp1l 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ X e. B ) -> K e. V )
5 eqid 
 |-  ( LAut ` K ) = ( LAut ` K )
6 2 5 3 ltrnlaut 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. ( LAut ` K ) )
7 6 3adant3 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ X e. B ) -> F e. ( LAut ` K ) )
8 simp3 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ X e. B ) -> X e. B )
9 1 5 lautcl 
 |-  ( ( ( K e. V /\ F e. ( LAut ` K ) ) /\ X e. B ) -> ( F ` X ) e. B )
10 4 7 8 9 syl21anc 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ X e. B ) -> ( F ` X ) e. B )