Metamath Proof Explorer

Theorem cdlemefs29bpre0N

Description: TODO: FIX COMMENT. (Contributed by NM, 26-Mar-2013) (New usage is discouraged.)

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 cdlemefs29bpre0N ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ∀ 𝑠 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) → 𝑧 = ( 𝑁 ∨ ( 𝑅 ∧ 𝑊 ) ) ) ↔ 𝑧 = ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ) )

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 breq1 ⊢ ( 𝑠 = 𝑅 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
13 simp1 ⊢ ( ( ( ( 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 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
14 simp3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → 𝑠 ∈ 𝐴 )
15 simp3rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑠 ≤ 𝑊 )
16 14 15 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
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 10 11 cdlemefs27cl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) → 𝑁 ∈ 𝐵 )
20 13 16 17 18 19 syl13anc ⊢ ( ( ( ( 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 12 20 cdlemefrs29bpre0 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ∀ 𝑠 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑠 ∨ ( 𝑅 ∧ 𝑊 ) ) = 𝑅 ) → 𝑧 = ( 𝑁 ∨ ( 𝑅 ∧ 𝑊 ) ) ) ↔ 𝑧 = ⦋ 𝑅 / 𝑠 ⦌ 𝑁 ) )