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Metamath Proof Explorer

Theorem joinval2

Description: Value of join for a poset with LUB expanded. (Contributed by NM, 16-Sep-2011) (Revised by NM, 11-Sep-2018)

Ref Expression
Hypotheses joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion joinval2 ( 𝜑 → ( 𝑋 ∨ 𝑌 ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )

Proof

Step Hyp Ref Expression
1 joinval2.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 joinval2.l ⊢ ≤ = ( le ‘ 𝐾 )
3 joinval2.j ⊢ ∨ = ( join ‘ 𝐾 )
4 joinval2.k ⊢ ( 𝜑 → 𝐾 ∈ 𝑉 )
5 joinval2.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 joinval2.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 eqid ⊢ ( lub ‘ 𝐾 ) = ( lub ‘ 𝐾 )
8 7 3 4 5 6 joinval ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑌 ) = ( ( lub ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) )
9 biid ⊢ ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) )
10 5 6 prssd ⊢ ( 𝜑 → { 𝑋 , 𝑌 } ⊆ 𝐵 )
11 1 2 7 9 4 10 lubval ⊢ ( 𝜑 → ( ( lub ‘ 𝐾 ) ‘ { 𝑋 , 𝑌 } ) = ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) )
12 1 2 3 4 5 6 joinval2lem ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ↔ ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
13 12 riotabidv ⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
14 5 6 13 syl2anc ⊢ ( 𝜑 → ( ℩ 𝑥 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑥 ∧ ∀ 𝑧 ∈ 𝐵 ( ∀ 𝑦 ∈ { 𝑋 , 𝑌 } 𝑦 ≤ 𝑧 → 𝑥 ≤ 𝑧 ) ) ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )
15 8 11 14 3eqtrd ⊢ ( 𝜑 → ( 𝑋 ∨ 𝑌 ) = ( ℩ 𝑥 ∈ 𝐵 ( ( 𝑋 ≤ 𝑥 ∧ 𝑌 ≤ 𝑥 ) ∧ ∀ 𝑧 ∈ 𝐵 ( ( 𝑋 ≤ 𝑧 ∧ 𝑌 ≤ 𝑧 ) → 𝑥 ≤ 𝑧 ) ) ) )