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Metamath Proof Explorer

Theorem glbprdm

Description: The set of two comparable elements in a poset has GLB. (Contributed by Zhi Wang, 26-Sep-2024)

Ref Expression
Hypotheses lubpr.k ⊢ ( 𝜑 → 𝐾 ∈ Poset )
lubpr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lubpr.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
lubpr.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
lubpr.l ⊢ ≤ = ( le ‘ 𝐾 )
lubpr.c ⊢ ( 𝜑 → 𝑋 ≤ 𝑌 )
lubpr.s ⊢ ( 𝜑 → 𝑆 = { 𝑋 , 𝑌 } )
glbpr.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
Assertion glbprdm ( 𝜑 → 𝑆 ∈ dom 𝐺 )

Proof

Step Hyp Ref Expression
1 lubpr.k ⊢ ( 𝜑 → 𝐾 ∈ Poset )
2 lubpr.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
3 lubpr.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
4 lubpr.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
5 lubpr.l ⊢ ≤ = ( le ‘ 𝐾 )
6 lubpr.c ⊢ ( 𝜑 → 𝑋 ≤ 𝑌 )
7 lubpr.s ⊢ ( 𝜑 → 𝑆 = { 𝑋 , 𝑌 } )
8 glbpr.g ⊢ 𝐺 = ( glb ‘ 𝐾 )
9 1 2 3 4 5 6 7 8 glbprlem ⊢ ( 𝜑 → ( 𝑆 ∈ dom 𝐺 ∧ ( 𝐺 ‘ 𝑆 ) = 𝑋 ) )
10 9 simpld ⊢ ( 𝜑 → 𝑆 ∈ dom 𝐺 )