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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) )
6	1 2 4	ldillaut	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ ( ( LDil ‘ 𝐾 ) ‘ 𝑊 ) ) → 𝐹 ∈ 𝐼 )
7	5 6	syldan	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ 𝐼 )