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Metamath Proof Explorer

Theorem latcl2

Description: The join and meet of any two elements exist. (Contributed by NM, 14-Sep-2018)

Ref Expression
Hypotheses latcl2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
latcl2.j ⊢ ∨ = ( join ‘ 𝐾 )
latcl2.m ⊢ ∧ = ( meet ‘ 𝐾 )
latcl2.k ⊢ ( 𝜑 → 𝐾 ∈ Lat )
latcl2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
latcl2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion latcl2 ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ) )

Proof

Step Hyp Ref Expression
1 latcl2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 latcl2.j ⊢ ∨ = ( join ‘ 𝐾 )
3 latcl2.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 latcl2.k ⊢ ( 𝜑 → 𝐾 ∈ Lat )
5 latcl2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 latcl2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 5 6 opelxpd ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ ( 𝐵 × 𝐵 ) )
8 1 2 3 islat ⊢ ( 𝐾 ∈ Lat ↔ ( 𝐾 ∈ Poset ∧ ( dom ∨ = ( 𝐵 × 𝐵 ) ∧ dom ∧ = ( 𝐵 × 𝐵 ) ) ) )
9 4 8 sylib ⊢ ( 𝜑 → ( 𝐾 ∈ Poset ∧ ( dom ∨ = ( 𝐵 × 𝐵 ) ∧ dom ∧ = ( 𝐵 × 𝐵 ) ) ) )
10 9 simprld ⊢ ( 𝜑 → dom ∨ = ( 𝐵 × 𝐵 ) )
11 7 10 eleqtrrd ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ )
12 9 simprrd ⊢ ( 𝜑 → dom ∧ = ( 𝐵 × 𝐵 ) )
13 7 12 eleqtrrd ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
14 11 13 jca ⊢ ( 𝜑 → ( ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∨ ∧ ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ ) )