Theorem latjcl

Description: Closure of join operation in a lattice. ( chjcom analog.) (Contributed by NM, 14-Sep-2011)

	Ref	Expression
Hypotheses	latjcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	latjcl.j	⊢ ∨ = ( join ‘ 𝐾 )
Assertion	latjcl	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		latjcl.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		latjcl.j	⊢ ∨ = ( join ‘ 𝐾 )
3		eqid	⊢ ( meet ‘ 𝐾 ) = ( meet ‘ 𝐾 )
4	1 2 3	latlem	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 ∧ ( 𝑋 ( meet ‘ 𝐾 ) 𝑌 ) ∈ 𝐵 ) )
5	4	simpld	⊢ ( ( 𝐾 ∈ Lat ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ∨ 𝑌 ) ∈ 𝐵 )