Theorem ltrnlaut

Description: A lattice translation is a lattice automorphism. (Contributed by NM, 20-May-2012)

	Ref	Expression
Hypotheses	ltrnlaut.h	|- H = ( LHyp ` K )
	ltrnlaut.i	|- I = ( LAut ` K )
	ltrnlaut.t	|- T = ( ( LTrn ` K ) ` W )
Assertion	ltrnlaut	|- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. I )

Proof

Step	Hyp	Ref	Expression
1		ltrnlaut.h	|- H = ( LHyp ` K )
2		ltrnlaut.i	|- I = ( LAut ` K )
3		ltrnlaut.t	|- T = ( ( LTrn ` K ) ` W )
4		eqid	|- ( ( LDil ` K ) ` W ) = ( ( LDil ` K ) ` W )
5	1 4 3	ltrnldil	|- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. ( ( LDil ` K ) ` W ) )
6	1 2 4	ldillaut	|- ( ( ( K e. V /\ W e. H ) /\ F e. ( ( LDil ` K ) ` W ) ) -> F e. I )
7	5 6	syldan	|- ( ( ( K e. V /\ W e. H ) /\ F e. T ) -> F e. I )