Metamath Proof Explorer

Theorem 4atexlemex6

Description: Lemma for 4atexlem7 . (Contributed by NM, 25-Nov-2012)

Ref Expression
Hypotheses 4thatleme.l ⊢ ≤ = ( le ‘ 𝐾 )
4thatleme.j ⊢ ∨ = ( join ‘ 𝐾 )
4thatleme.m ⊢ ∧ = ( meet ‘ 𝐾 )
4thatleme.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
4thatleme.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
Assertion 4atexlemex6 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )

Proof

Step Hyp Ref Expression
1 4thatleme.l ⊢ ≤ = ( le ‘ 𝐾 )
2 4thatleme.j ⊢ ∨ = ( join ‘ 𝐾 )
3 4thatleme.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 4thatleme.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 4thatleme.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL )
7 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
8 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
9 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
10 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ 𝑄 )
11 1 2 3 4 5 lhpat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 )
12 7 8 9 10 11 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 )
13 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑆 ∈ 𝐴 )
14 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
15 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
16 1 2 4 atnlej1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑆 ≠ 𝑃 )
17 6 13 14 9 15 16 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑆 ≠ 𝑃 )
18 17 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ 𝑆 )
19 1 2 3 4 5 lhpat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑃 ≠ 𝑆 ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ 𝐴 )
20 7 8 13 18 19 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ 𝐴 )
21 2 4 hlsupr2 ⊢ ( ( 𝐾 ∈ HL ∧ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 ∧ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∈ 𝐴 ) → ∃ 𝑡 ∈ 𝐴 ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) )
22 6 12 20 21 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑡 ∈ 𝐴 ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) )
23 simp111 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
24 simp112 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
25 simp113 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
26 simp12r ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → 𝑆 ∈ 𝐴 )
27 simp2ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑅 ∈ 𝐴 )
28 27 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → 𝑅 ∈ 𝐴 )
29 simp2lr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑅 ≤ 𝑊 )
30 29 3ad2ant1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ¬ 𝑅 ≤ 𝑊 )
31 simp131 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) )
32 28 30 31 3jca ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) )
33 3simpc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ( 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) )
34 simp132 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → 𝑃 ≠ 𝑄 )
35 simp133 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
36 biid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ↔ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) )
37 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
38 eqid ⊢ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 )
39 eqid ⊢ ( ( 𝑄 ∨ 𝑡 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑄 ∨ 𝑡 ) ∧ ( 𝑃 ∨ 𝑆 ) )
40 eqid ⊢ ( ( 𝑅 ∨ 𝑡 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = ( ( 𝑅 ∨ 𝑡 ) ∧ ( 𝑃 ∨ 𝑆 ) )
41 36 1 2 3 4 5 37 38 39 40 4atexlemex4 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( ( 𝑄 ∨ 𝑡 ) ∧ ( 𝑃 ∨ 𝑆 ) ) = 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )
42 36 1 2 3 4 5 37 38 39 4atexlemex2 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( ( 𝑄 ∨ 𝑡 ) ∧ ( 𝑃 ∨ 𝑆 ) ) ≠ 𝑆 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )
43 41 42 pm2.61dane ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )
44 23 24 25 26 32 33 34 35 43 syl332anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ 𝑡 ∈ 𝐴 ∧ ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )
45 44 rexlimdv3a ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ∃ 𝑡 ∈ 𝐴 ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∨ 𝑡 ) = ( ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ∨ 𝑡 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) ) )
46 22 45 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ) ∧ ( ( 𝑃 ∨ 𝑅 ) = ( 𝑄 ∨ 𝑅 ) ∧ 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ( 𝑃 ∨ 𝑧 ) = ( 𝑆 ∨ 𝑧 ) ) )