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Metamath Proof Explorer

Theorem ltrncvr

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

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

Proof

Step Hyp Ref Expression
1 ltrncvr.b 
 |-  B = ( Base ` K )
2 ltrncvr.c 
 |-  C = (
3 ltrncvr.h 
 |-  H = ( LHyp ` K )
4 ltrncvr.t 
 |-  T = ( ( LTrn ` K ) ` W )
5 simp1l 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> K e. V )
6 eqid 
 |-  ( LAut ` K ) = ( LAut ` K )
7 3 6 4 ltrnlaut 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. ( LAut ` K ) )
8 7 3adant3 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> F e. ( LAut ` K ) )
9 simp3l 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> X e. B )
10 simp3r 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
11 1 2 6 lautcvr 
 |-  ( ( K e. V /\ ( F e. ( LAut ` K ) /\ X e. B /\ Y e. B ) ) -> ( X C Y <-> ( F ` X ) C ( F ` Y ) ) )
12 5 8 9 10 11 syl13anc 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> ( X C Y <-> ( F ` X ) C ( F ` Y ) ) )