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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 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 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐵 ) → 𝐾 ∈ 𝑉 )
5		eqid	⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
6	2 5 3	ltrnlaut	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
7	6	3adant3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐵 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
8		simp3	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ∈ 𝐵 )
9	1 5	lautcl	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝐹 ∈ ( LAut ‘ 𝐾 ) ) ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )
10	4 7 8 9	syl21anc	⊢ ( ( ( 𝐾 ∈ 𝑉 ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑋 ∈ 𝐵 ) → ( 𝐹 ‘ 𝑋 ) ∈ 𝐵 )