Theorem cdlemefs32snb

Description: Show closure of [_ R / s ]_ N . (Contributed by NM, 24-Mar-2013)

	Ref	Expression
Hypotheses	cdlemefs32.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	cdlemefs32.l	⊢ ≤ = ( le ‘ 𝐾 )
	cdlemefs32.j	⊢ ∨ = ( join ‘ 𝐾 )
	cdlemefs32.m	⊢ ∧ = ( meet ‘ 𝐾 )
	cdlemefs32.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
	cdlemefs32.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
	cdlemefs32.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
	cdlemefs32.d	⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
	cdlemefs32.e	⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
	cdlemefs32.i	⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) )
	cdlemefs32.n	⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
Assertion	cdlemefs32snb	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		cdlemefs32.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		cdlemefs32.l	⊢ ≤ = ( le ‘ 𝐾 )
3		cdlemefs32.j	⊢ ∨ = ( join ‘ 𝐾 )
4		cdlemefs32.m	⊢ ∧ = ( meet ‘ 𝐾 )
5		cdlemefs32.a	⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6		cdlemefs32.h	⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7		cdlemefs32.u	⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8		cdlemefs32.d	⊢ 𝐷 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9		cdlemefs32.e	⊢ 𝐸 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10		cdlemefs32.i	⊢ 𝐼 = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = 𝐸 ) )
11		cdlemefs32.n	⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
12		eqid	⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) )
13		eqid	⊢ ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑦 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑦 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐷 ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) )
14	1 2 3 4 5 6 7 8 9 10 11 12 13	cdlemefs32sn1aw	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 ∧ ¬ ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ≤ 𝑊 ) )
15	14	simpld	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 )
16	1 5	atbase	⊢ ( ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐴 → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 )
17	15 16	syl	⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ∈ 𝐵 )