Theorem cdleme25cl

Description: Show closure of the unique element in cdleme25c . (Contributed by NM, 2-Feb-2013)

	Ref	Expression
Hypotheses	cdleme24.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdleme24.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdleme24.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdleme24.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdleme24.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdleme24.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdleme24.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	cdleme24.f	⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
	cdleme24.n	⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) )
	cdleme25cl.i	⊢ 𝐼 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
Assertion	cdleme25cl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐼 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cdleme24.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdleme24.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdleme24.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdleme24.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdleme24.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdleme24.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdleme24.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8		cdleme24.f	⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9		cdleme24.n	⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) )
10		cdleme25cl.i	⊢ 𝐼 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
11	1 2 3 4 5 6 7 8 9	cdleme25c	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃! 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
12		riotacl	⊢ ( ∃! 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) → ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) ∈ 𝐵 )
13	11 12	syl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) ∈ 𝐵 )
14	10 13	eqeltrid	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐼 ∈ 𝐵 )