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Metamath Proof Explorer

Theorem ltrn11

Description: One-to-one property 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 ltrn11 
|- ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) )

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 /\ Y 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 /\ Y e. B ) ) -> F e. ( LAut ` K ) )
8 simp3l 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> X e. B )
9 simp3r 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> Y e. B )
10 1 5 laut11 
 |-  ( ( ( K e. V /\ F e. ( LAut ` K ) ) /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) )
11 4 7 8 9 10 syl22anc 
 |-  ( ( ( K e. V /\ W e. H ) /\ F e. T /\ ( X e. B /\ Y e. B ) ) -> ( ( F ` X ) = ( F ` Y ) <-> X = Y ) )