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 ( 𝜑𝐾𝑉 )
meetval2.x ( 𝜑𝑋𝐵 )
meetval2.y ( 𝜑𝑌𝐵 )
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 ( 𝜑𝐾𝑉 )
5 meetval2.x ( 𝜑𝑋𝐵 )
6 meetval2.y ( 𝜑𝑌𝐵 )
7 meetlem.e ( 𝜑 → ⟨ 𝑋 , 𝑌 ⟩ ∈ dom )
8 1 2 3 4 5 6 7 meetlem ( 𝜑 → ( ( ( 𝑋 𝑌 ) 𝑋 ∧ ( 𝑋 𝑌 ) 𝑌 ) ∧ ∀ 𝑧𝐵 ( ( 𝑧 𝑋𝑧 𝑌 ) → 𝑧 ( 𝑋 𝑌 ) ) ) )
9 8 simplrd ( 𝜑 → ( 𝑋 𝑌 ) 𝑌 )