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Metamath Proof Explorer

Theorem luble

Description: The greatest lower bound is the least element. (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 )
luble.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
Assertion luble ( 𝜑 → 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) )

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 luble.x ⊢ ( 𝜑 → 𝑋 ∈ 𝑆 )
7 breq1 ⊢ ( 𝑦 = 𝑋 → ( 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ↔ 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) ) )
8 1 2 3 4 5 lubprop ⊢ ( 𝜑 → ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ 𝑆 𝑦 ≤ 𝑧 → ( 𝑈 ‘ 𝑆 ) ≤ 𝑧 ) ) )
9 8 simpld ⊢ ( 𝜑 → ∀ 𝑦 ∈ 𝑆 𝑦 ≤ ( 𝑈 ‘ 𝑆 ) )
10 7 9 6 rspcdva ⊢ ( 𝜑 → 𝑋 ≤ ( 𝑈 ‘ 𝑆 ) )