Metamath Proof Explorer

Theorem cdlemefr29clN

Description: Show closure of the unique element in cdleme29c . TODO fix comment. TODO Not needed? (Contributed by NM, 29-Mar-2013) (New usage is discouraged.)

Ref Expression
Hypotheses cdlemefr27.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdlemefr27.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemefr27.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemefr27.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemefr27.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemefr27.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemefr27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdlemefr27.c ⊢ 𝐶 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
cdlemefr27.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
cdlemefr29cl.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) → 𝑧 = ( 𝑁 ∨ ( 𝑅 ∧ 𝑊 ) ) ) )
Assertion cdlemefr29clN ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑂 ∈ 𝐵 )

Proof

Step Hyp Ref Expression
1 cdlemefr27.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdlemefr27.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdlemefr27.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdlemefr27.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdlemefr27.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdlemefr27.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdlemefr27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdlemefr27.c ⊢ 𝐶 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9 cdlemefr27.n ⊢ 𝑁 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐼 , 𝐶 )
10 cdlemefr29cl.o ⊢ 𝑂 = ( ℩ 𝑧 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) → 𝑧 = ( 𝑁 ∨ ( 𝑅 ∧ 𝑊 ) ) ) )
11 breq1 ⊢ ( 𝑠 = 𝑅 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
12 11 notbid ⊢ ( 𝑠 = 𝑅 → ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
13 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
14 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ∈ 𝐴 )
15 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑄 ∈ 𝐴 )
16 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑠 ∈ 𝐴 )
17 simp3rr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
18 simp2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑄 )
19 1 2 3 4 5 6 7 8 9 cdlemefr27cl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑁 ∈ 𝐵 )
20 13 14 15 16 17 18 19 syl33anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑁 ∈ 𝐵 )
21 1 2 3 4 5 6 7 8 9 cdlemefr32snb ⊢ ( ( ( ( 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 )
22 1 2 3 4 5 6 12 20 21 10 cdlemefrs29clN ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ¬ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑂 ∈ 𝐵 )