Theorem hlatjcl

Description: Closure of join operation. Frequently-used special case of latjcl for atoms. (Contributed by NM, 15-Jun-2012)

	Ref	Expression
Hypotheses	hlatjcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	hlatjcl.j	⊢ ∨ = ( join ‘ 𝐾 )
	hlatjcl.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
Assertion	hlatjcl	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		hlatjcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		hlatjcl.j	⊢ ∨ = ( join ‘ 𝐾 )
3		hlatjcl.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4		hllat	⊢ ( 𝐾 ∈ HL → 𝐾 ∈ Lat )
5	1 3	atbase	⊢ ( 𝑋 ∈ 𝐴 → 𝑋 ∈ 𝐵 )
6	1 3	atbase	⊢ ( 𝑌 ∈ 𝐴 → 𝑌 ∈ 𝐵 )
7	1 2	latjcl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )
8	4 5 6 7	syl3an	⊢ ( ( 𝐾 ∈ HL ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )