Theorem cdleme29cl

Description: Show closure of the unique element in cdleme28c . (Contributed by NM, 8-Feb-2013)

	Ref	Expression
Hypotheses	cdleme26.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdleme26.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme26.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme26.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme26.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme26.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme27.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	cdleme27.f	⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
	cdleme27.z	⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
	cdleme27.n	⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
	cdleme27.d	⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
	cdleme27.c	⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
	cdleme29cl.i	⊢ 𝐼 = ( ℩ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) )
Assertion	cdleme29cl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝐼 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cdleme26.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdleme26.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdleme26.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdleme26.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdleme26.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdleme26.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdleme27.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8		cdleme27.f	⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9		cdleme27.z	⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
10		cdleme27.n	⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
11		cdleme27.d	⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
12		cdleme27.c	⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
13		cdleme29cl.i	⊢ 𝐼 = ( ℩ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) )
14	1 2 3 4 5 6 7 8 9 10 11 12	cdleme29c	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ∃! 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) )
15		riotacl	⊢ ( ∃! 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) → ( ℩ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) ∈ 𝐵 )
16	14 15	syl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → ( ℩ 𝑣 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑋 ∧ 𝑊 ) ) = 𝑋 ) → 𝑣 = ( 𝐶 ∨ ( 𝑋 ∧ 𝑊 ) ) ) ) ∈ 𝐵 )
17	13 16	eqeltrid	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑋 ∈ 𝐵 ∧ ¬ 𝑋 ≤ 𝑊 ) ) → 𝐼 ∈ 𝐵 )