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 ) ) ) )