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Metamath Proof Explorer

Theorem lemeet2

Description: A meet's second argument is less than or equal to the meet. (Contributed by NM, 16-Sep-2011) (Revised by NM, 12-Sep-2018)

Ref Expression
Hypotheses meetval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
meetval2.l ⊢ ≤ = ( le ‘ 𝐾 )
meetval2.m ⊢ ∧ = ( meet ‘ 𝐾 )
meetval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
meetval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
meetval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
meetlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
Assertion lemeet2 ( 𝜑 → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 )

Proof

Step Hyp Ref Expression
1 meetval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 meetval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 meetval2.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 meetval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 meetval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 meetval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 meetlem.e ⊢ ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom ∧ )
8 1 2 3 4 5 6 7 meetlem ⊢ ( 𝜑 → ( ( ( 𝑋 ∧ 𝑌 ) ≤ 𝑋 ∧ ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑧 ≤ 𝑋 ∧ 𝑧 ≤ 𝑌 ) → 𝑧 ≤ ( 𝑋 ∧ 𝑌 ) ) ) )
9 8 simplrd ⊢ ( 𝜑 → ( 𝑋 ∧ 𝑌 ) ≤ 𝑌 )