Theorem glbpr

Description: The GLB of the set of two comparable elements in a poset is the less one of the two. (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	glbpr	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) = 𝑋 )

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	simprd	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) = 𝑋 )