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Metamath Proof Explorer

Theorem lubprop

Description: Properties of greatest lower bound of a poset. (Contributed by NM, 22-Oct-2011) (Revised by NM, 7-Sep-2018)

Ref Expression
Hypotheses lubprop.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
lubprop.l ⊢ ≤ = ( le ‘ 𝐾 )
lubprop.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
lubprop.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
lubprop.s ⊢ ( 𝜑 → 𝑆 ∈ dom 𝑈 )
Assertion lubprop ( 𝜑 → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 lubprop.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 lubprop.l ⊢ ≤ = ( le ‘ 𝐾 )
3 lubprop.u ⊢ 𝑈 = ( lub ‘ 𝐾 )
4 lubprop.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 lubprop.s ⊢ ( 𝜑 → 𝑆 ∈ dom 𝑈 )
6 biid ⊢ ( ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
7 1 2 3 4 5 lubelss ⊢ ( 𝜑 → 𝑆 ⊆ 𝐵 )
8 1 2 3 6 4 7 lubval ⊢ ( 𝜑 → ( 𝑈 ‘ 𝑆 ) = ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) )
9 8 eqcomd ⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) = ( 𝑈 ‘ 𝑆 ) )
10 1 3 4 5 lubcl ⊢ ( 𝜑 → ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 )
11 1 2 3 6 4 5 lubeu ⊢ ( 𝜑 → ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
12 breq2 ⊢ ( 𝑥 = ( 𝑈 ‘ 𝑆 ) → ( 𝑦 ≤ 𝑥 ↔ 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ) )
13 12 ralbidv ⊢ ( 𝑥 = ( 𝑈 ‘ 𝑆 ) → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ↔ ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ) )
14 breq1 ⊢ ( 𝑥 = ( 𝑈 ‘ 𝑆 ) → ( 𝑥 ≤ 𝑧 ↔ ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) )
15 14 imbi2d ⊢ ( 𝑥 = ( 𝑈 ‘ 𝑆 ) → ( ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ↔ ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )
16 15 ralbidv ⊢ ( 𝑥 = ( 𝑈 ‘ 𝑆 ) → ( ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ↔ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )
17 13 16 anbi12d ⊢ ( 𝑥 = ( 𝑈 ‘ 𝑆 ) → ( ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) ) )
18 17 riota2 ⊢ ( ( ( 𝑈 ‘ 𝑆 ) ∈ 𝐵 ∧ ∃! 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) → ( ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) ↔ ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) = ( 𝑈 ‘ 𝑆 ) ) )
19 10 11 18 syl2anc ⊢ ( 𝜑 → ( ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) ↔ ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) = ( 𝑈 ‘ 𝑆 ) ) )
20 9 19 mpbird ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )