Theorem ltrnj

Description: Lattice translation of a meet. TODO: change antecedent to K e. HL (Contributed by NM, 25-May-2012)

	Ref	Expression
Hypotheses	ltrnj.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	ltrnj.j	⊢ ∨ = ( join ‘ 𝐾 )
	ltrnj.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	ltrnj.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
Assertion	ltrnj	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltrnj.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		ltrnj.j	⊢ ∨ = ( join ‘ 𝐾 )
3		ltrnj.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
4		ltrnj.t	⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
5		simp1l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ HL )
6	5	hllatd	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐾 ∈ Lat )
7		eqid	⊢ ( LAut ‘ 𝐾 ) = ( LAut ‘ 𝐾 )
8	3 7 4	ltrnlaut	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
9	8	3adant3	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝐹 ∈ ( LAut ‘ 𝐾 ) )
10		simp3l	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑋 ∈ 𝐵 )
11		simp3r	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → 𝑌 ∈ 𝐵 )
12	1 2 7	lautj	⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∈ ( LAut ‘ 𝐾 ) ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) )
13	6 9 10 11 12	syl13anc	⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝐹 ‘ ( 𝑋 ∨ 𝑌 ) ) = ( ( 𝐹 ‘ 𝑋 ) ∨ ( 𝐹 ‘ 𝑌 ) ) )