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Metamath Proof Explorer

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 )