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Metamath Proof Explorer

Theorem meetcl

Description: Closure of meet of elements in the domain. (Contributed by NM, 12-Sep-2018)

Ref Expression
Hypotheses meetcl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
meetcl.m ⊢ ∧ = ( meet ‘ 𝐾 )
meetcl.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
meetcl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
meetcl.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
meetcl.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
Assertion meetcl ( 𝜑 → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 meetcl.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 meetcl.m ⊢ ∧ = ( meet ‘ 𝐾 )
3 meetcl.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
4 meetcl.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
5 meetcl.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
6 meetcl.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
7 eqid ⊢ ( glb ‘ 𝐾 ) = ( glb ‘ 𝐾 )
8 7 2 3 4 5 meetval ⊢ ( 𝜑 → ( 𝑋 ∧ 𝑌 ) = ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) )
9 7 2 3 4 5 meetdef ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ↔ { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) ) )
10 6 9 mpbid ⊢ ( 𝜑 → { 𝑋 , 𝑌 } ∈ dom ( glb ‘ 𝐾 ) )
11 1 7 3 10 glbcl ⊢ ( 𝜑 → ( ( glb ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) ∈ 𝐵 )
12 8 11 eqeltrd ⊢ ( 𝜑 → ( 𝑋 ∧ 𝑌 ) ∈ 𝐵 )