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Metamath Proof Explorer

Theorem cdleme20i

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, antepenultimate line. D , F , Y , G represent s_2, f(s), t_2, f(t). We show (f(s) \/ s_2) /\ (f(t) \/ t_2) <_ p \/ q. (Contributed by NM, 18-Nov-2012)

Ref Expression
Hypotheses cdleme19.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme19.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme19.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme19.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme19.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme19.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme19.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme19.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
cdleme19.d ⊢ 𝐷 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
cdleme19.y ⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
cdleme20.v ⊢ 𝑉 = ( ( 𝑆 ∨ 𝑇 ) ∧ 𝑊 )
Assertion cdleme20i ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝐹 ∨ 𝐷 ) ∧ ( 𝐺 ∨ 𝑌 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )

Proof

Step Hyp Ref Expression
1 cdleme19.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme19.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme19.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme19.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme19.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme19.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme19.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme19.g ⊢ 𝐺 = ( ( 𝑇 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑇 ) ∧ 𝑊 ) ) )
9 cdleme19.d ⊢ 𝐷 = ( ( 𝑅 ∨ 𝑆 ) ∧ 𝑊 )
10 cdleme19.y ⊢ 𝑌 = ( ( 𝑅 ∨ 𝑇 ) ∧ 𝑊 )
11 cdleme20.v ⊢ 𝑉 = ( ( 𝑆 ∨ 𝑇 ) ∧ 𝑊 )
12 simp1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
13 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
14 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) )
15 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) )
16 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) )
17 simp321 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
18 simp322 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )
19 17 18 jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) )
20 simp323 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) )
21 16 19 20 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) )
22 1 2 3 4 5 6 7 8 9 10 11 cdleme20f ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ∨ 𝐷 ) ∧ ( 𝐺 ∨ 𝑌 ) ) ≤ ( ( ( 𝐷 ∨ 𝑆 ) ∧ ( 𝑌 ∨ 𝑇 ) ) ∨ ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ) )
23 12 13 14 15 21 22 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝐹 ∨ 𝐷 ) ∧ ( 𝐺 ∨ 𝑌 ) ) ≤ ( ( ( 𝐷 ∨ 𝑆 ) ∧ ( 𝑌 ∨ 𝑇 ) ) ∨ ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ) )
24 1 2 3 4 5 6 7 8 9 10 11 cdleme20h ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( ( 𝑆 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑅 ) ) ∨ ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑇 ∨ 𝑈 ) ) ) = ( 𝑅 ∨ 𝑈 ) )
25 1 2 3 4 5 6 7 8 9 10 11 cdleme20g ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐷 ∨ 𝑆 ) ∧ ( 𝑌 ∨ 𝑇 ) ) ∨ ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ) = ( ( ( 𝑆 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑅 ) ) ∨ ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑇 ∨ 𝑈 ) ) ) )
26 12 13 14 15 21 25 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( ( 𝐷 ∨ 𝑆 ) ∧ ( 𝑌 ∨ 𝑇 ) ) ∨ ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ) = ( ( ( 𝑆 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑅 ) ) ∨ ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑇 ∨ 𝑈 ) ) ) )
27 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
28 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑃 ∈ 𝐴 )
29 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → 𝑄 ∈ 𝐴 )
30 1 2 3 4 5 6 cdleme4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑈 ) )
31 27 28 29 15 20 30 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( 𝑅 ∨ 𝑈 ) )
32 24 26 31 3eqtr4d ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( ( 𝐷 ∨ 𝑆 ) ∧ ( 𝑌 ∨ 𝑇 ) ) ∨ ( ( 𝑆 ∨ 𝐹 ) ∧ ( 𝑇 ∨ 𝐺 ) ) ) = ( 𝑃 ∨ 𝑄 ) )
33 23 32 breqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≠ 𝑇 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝑈 ≤ ( 𝑆 ∨ 𝑇 ) ) ) ) → ( ( 𝐹 ∨ 𝐷 ) ∧ ( 𝐺 ∨ 𝑌 ) ) ≤ ( 𝑃 ∨ 𝑄 ) )