Metamath Proof Explorer

Theorem cdleme11

Description: Part of proof of Lemma E in Crawley p. 113, 1st sentence of 3rd paragraph on p. 114. F and G represent f(s) and f(t) respectively. Their proof provides no details of our cdleme11a through cdleme11 , so there may be a simpler proof that we have overlooked. (Contributed by NM, 15-Jun-2012)

Ref Expression
Hypotheses cdleme12.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme12.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme12.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme12.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme12.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme12.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
Assertion cdleme11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐹 ∨ 𝐺 ) = ( 𝑆 ∨ 𝑇 ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme12.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme12.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme12.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme12.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme12.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme12.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme12.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
9 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐾 ∈ HL )
10 9 hllatd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐾 ∈ Lat )
11 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ∈ 𝐴 )
13 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑄 ∈ 𝐴 )
14 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
15 1 2 3 4 5 6 14 cdleme0aa ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
16 11 12 13 15 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
17 14 2 latjidm ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑈 ∨ 𝑈 ) = 𝑈 )
18 10 16 17 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑈 ∨ 𝑈 ) = 𝑈 )
19 18 oveq2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝑈 ∨ 𝑈 ) ) = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) )
20 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) )
21 simp21l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑆 ∈ 𝐴 )
22 14 4 atbase ⊢ ( 𝑆 ∈ 𝐴 → 𝑆 ∈ ( Base ‘ 𝐾 ) )
23 21 22 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑆 ∈ ( Base ‘ 𝐾 ) )
24 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑇 ∈ 𝐴 )
25 14 4 atbase ⊢ ( 𝑇 ∈ 𝐴 → 𝑇 ∈ ( Base ‘ 𝐾 ) )
26 24 25 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑇 ∈ ( Base ‘ 𝐾 ) )
27 14 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑇 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
28 10 23 26 27 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
29 14 1 2 latleeqj2 ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ↔ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) = ( 𝑆 ∨ 𝑇 ) ) )
30 10 16 28 29 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ↔ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) = ( 𝑆 ∨ 𝑇 ) ) )
31 20 30 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) = ( 𝑆 ∨ 𝑇 ) )
32 19 31 eqtr2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑆 ∨ 𝑇 ) = ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝑈 ∨ 𝑈 ) ) )
33 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
34 1 2 3 4 5 6 7 cdleme1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ) → ( 𝑆 ∨ 𝐹 ) = ( 𝑆 ∨ 𝑈 ) )
35 11 12 13 33 34 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑆 ∨ 𝐹 ) = ( 𝑆 ∨ 𝑈 ) )
36 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) )
37 1 2 3 4 5 6 8 cdleme1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ) → ( 𝑇 ∨ 𝐺 ) = ( 𝑇 ∨ 𝑈 ) )
38 11 12 13 36 37 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑇 ∨ 𝐺 ) = ( 𝑇 ∨ 𝑈 ) )
39 35 38 oveq12d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∨ ( 𝑇 ∨ 𝐺 ) ) = ( ( 𝑆 ∨ 𝑈 ) ∨ ( 𝑇 ∨ 𝑈 ) ) )
40 14 2 latj4 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝑇 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝑈 ∨ 𝑈 ) ) = ( ( 𝑆 ∨ 𝑈 ) ∨ ( 𝑇 ∨ 𝑈 ) ) )
41 10 23 26 16 16 40 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝑈 ∨ 𝑈 ) ) = ( ( 𝑆 ∨ 𝑈 ) ∨ ( 𝑇 ∨ 𝑈 ) ) )
42 39 41 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∨ ( 𝑇 ∨ 𝐺 ) ) = ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝑈 ∨ 𝑈 ) ) )
43 32 42 eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑆 ∨ 𝑇 ) = ( ( 𝑆 ∨ 𝐹 ) ∨ ( 𝑇 ∨ 𝐺 ) ) )
44 1 2 3 4 5 6 7 14 cdleme1b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → 𝐹 ∈ ( Base ‘ 𝐾 ) )
45 11 12 13 21 44 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐹 ∈ ( Base ‘ 𝐾 ) )
46 1 2 3 4 5 6 8 14 cdleme1b ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ) → 𝐺 ∈ ( Base ‘ 𝐾 ) )
47 11 12 13 24 46 syl13anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐺 ∈ ( Base ‘ 𝐾 ) )
48 14 2 latj4 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∈ ( Base ‘ 𝐾 ) ∧ 𝐹 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑇 ∈ ( Base ‘ 𝐾 ) ∧ 𝐺 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∨ ( 𝑇 ∨ 𝐺 ) ) = ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝐹 ∨ 𝐺 ) ) )
49 10 23 45 26 47 48 syl122anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝐹 ) ∨ ( 𝑇 ∨ 𝐺 ) ) = ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝐹 ∨ 𝐺 ) ) )
50 43 49 eqtr2d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝐹 ∨ 𝐺 ) ) = ( 𝑆 ∨ 𝑇 ) )
51 14 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝐹 ∈ ( Base ‘ 𝐾 ) ∧ 𝐺 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ∨ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
52 10 45 47 51 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐹 ∨ 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
53 14 1 2 latleeqj2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ∨ 𝐺 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ∨ 𝐺 ) ≤ ( 𝑆 ∨ 𝑇 ) ↔ ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝐹 ∨ 𝐺 ) ) = ( 𝑆 ∨ 𝑇 ) ) )
54 10 52 28 53 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝐹 ∨ 𝐺 ) ≤ ( 𝑆 ∨ 𝑇 ) ↔ ( ( 𝑆 ∨ 𝑇 ) ∨ ( 𝐹 ∨ 𝐺 ) ) = ( 𝑆 ∨ 𝑇 ) ) )
55 50 54 mpbird ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐹 ∨ 𝐺 ) ≤ ( 𝑆 ∨ 𝑇 ) )
56 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
57 simp13 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
58 simp23l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝑃 ≠ 𝑄 )
59 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
60 1 2 3 4 5 6 7 cdleme3fa ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝐴 )
61 11 56 57 33 58 59 60 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐹 ∈ 𝐴 )
62 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )
63 1 2 3 4 5 6 8 cdleme3fa ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝐴 )
64 11 56 57 36 58 62 63 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐺 ∈ 𝐴 )
65 1 2 3 4 5 6 7 8 cdleme11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → 𝐹 ≠ 𝐺 )
66 1 2 4 ps-1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ∈ 𝐴 ∧ 𝐺 ∈ 𝐴 ∧ 𝐹 ≠ 𝐺 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ) ) → ( ( 𝐹 ∨ 𝐺 ) ≤ ( 𝑆 ∨ 𝑇 ) ↔ ( 𝐹 ∨ 𝐺 ) = ( 𝑆 ∨ 𝑇 ) ) )
67 9 61 64 65 21 24 66 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( ( 𝐹 ∨ 𝐺 ) ≤ ( 𝑆 ∨ 𝑇 ) ↔ ( 𝐹 ∨ 𝐺 ) = ( 𝑆 ∨ 𝑇 ) ) )
68 55 67 mpbid ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) → ( 𝐹 ∨ 𝐺 ) = ( 𝑆 ∨ 𝑇 ) )