glbprlem

Description: Lemma for glbprdm and glbpr . (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	glbprlem	⊢ ( 𝜑 → ( 𝑆 ∈ dom 𝐺 ∧ ( 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		eqid	⊢ ( ODual ‘ 𝐾 ) = ( ODual ‘ 𝐾 )
10	9	odupos	⊢ ( 𝐾 ∈ Poset → ( ODual ‘ 𝐾 ) ∈ Poset )
11	1 10	syl	⊢ ( 𝜑 → ( ODual ‘ 𝐾 ) ∈ Poset )
12	9 2	odubas	⊢ 𝐵 = ( Base ‘ ( ODual ‘ 𝐾 ) )
13	9 5	oduleval	⊢ ^◡ ≤ = ( le ‘ ( ODual ‘ 𝐾 ) )
14		brcnvg	⊢ ( ( 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑌 ^◡ ≤ 𝑋 ↔ 𝑋 ≤ 𝑌 ) )
15	4 3 14	syl2anc	⊢ ( 𝜑 → ( 𝑌 ^◡ ≤ 𝑋 ↔ 𝑋 ≤ 𝑌 ) )
16	6 15	mpbird	⊢ ( 𝜑 → 𝑌 ^◡ ≤ 𝑋 )
17		prcom	⊢ { 𝑋 , 𝑌 } = { 𝑌 , 𝑋 }
18	7 17	eqtrdi	⊢ ( 𝜑 → 𝑆 = { 𝑌 , 𝑋 } )
19		eqid	⊢ ( lub ‘ ( ODual ‘ 𝐾 ) ) = ( lub ‘ ( ODual ‘ 𝐾 ) )
20	11 12 4 3 13 16 18 19	lubprdm	⊢ ( 𝜑 → 𝑆 ∈ dom ( lub ‘ ( ODual ‘ 𝐾 ) ) )
21	9 8	odulub	⊢ ( 𝐾 ∈ Poset → 𝐺 = ( lub ‘ ( ODual ‘ 𝐾 ) ) )
22	1 21	syl	⊢ ( 𝜑 → 𝐺 = ( lub ‘ ( ODual ‘ 𝐾 ) ) )
23	22	dmeqd	⊢ ( 𝜑 → dom 𝐺 = dom ( lub ‘ ( ODual ‘ 𝐾 ) ) )
24	20 23	eleqtrrd	⊢ ( 𝜑 → 𝑆 ∈ dom 𝐺 )
25	22	fveq1d	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) = ( ( lub ‘ ( ODual ‘ 𝐾 ) ) ‘ 𝑆 ) )
26	11 12 4 3 13 16 18 19	lubpr	⊢ ( 𝜑 → ( ( lub ‘ ( ODual ‘ 𝐾 ) ) ‘ 𝑆 ) = 𝑋 )
27	25 26	eqtrd	⊢ ( 𝜑 → ( 𝐺 ‘ 𝑆 ) = 𝑋 )
28	24 27	jca	⊢ ( 𝜑 → ( 𝑆 ∈ dom 𝐺 ∧ ( 𝐺 ‘ 𝑆 ) = 𝑋 ) )

Metamath Proof Explorer

Theorem glbprlem

Proof