Metamath Proof Explorer

Theorem cdleme26f

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 6th and 7th lines on p. 115. F , N represent f(t), f_t(s) respectively. If t <_ t \/ v, then f_t(s) <_ f(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 1-Feb-2013)

Ref Expression
Hypotheses cdleme26.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
cdleme26.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme26.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme26.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme26.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme26.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme26f.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme26f.f ⊢ 𝐹 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme26f.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑆 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme26f.i ⊢ 𝐼 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
Assertion cdleme26f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐼 ≤ ( 𝐹 ∨ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 cdleme26.b ⊢ 𝐵 = ( Base ‘ 𝐾 )
2 cdleme26.l ⊢ ≤ = ( le ‘ 𝐾 )
3 cdleme26.j ⊢ ∨ = ( join ‘ 𝐾 )
4 cdleme26.m ⊢ ∧ = ( meet ‘ 𝐾 )
5 cdleme26.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
6 cdleme26.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
7 cdleme26f.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme26f.f ⊢ 𝐹 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
9 cdleme26f.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑆 ∨ 𝑡 ) ∧ 𝑊 ) ) )
10 cdleme26f.i ⊢ 𝐼 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
11 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
12 simp21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
13 simp22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
14 simp23l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑆 ∈ 𝐴 )
15 simp23r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ¬ 𝑆 ≤ 𝑊 )
16 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑃 ≠ 𝑄 )
17 simp12r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
18 1 2 3 4 5 6 7 8 9 10 cdleme25cl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐼 ∈ 𝐵 )
19 11 12 13 14 15 16 17 18 syl322anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐼 ∈ 𝐵 )
20 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑡 ∈ 𝐴 )
21 simp13r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ¬ 𝑡 ≤ 𝑊 )
22 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
23 1 fvexi ⊢ 𝐵 ∈ V
24 23 10 riotasv ⊢ ( ( 𝐼 ∈ 𝐵 ∧ 𝑡 ∈ 𝐴 ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐼 = 𝑁 )
25 19 20 21 22 24 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐼 = 𝑁 )
26 simp23 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
27 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
28 simp32 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) )
29 2 3 4 5 6 7 8 9 cdleme22f ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ∧ 𝑡 ∈ 𝐴 ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ) → 𝑁 ≤ ( 𝐹 ∨ 𝑉 ) )
30 11 12 13 26 20 27 28 29 syl331anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑁 ≤ ( 𝐹 ∨ 𝑉 ) )
31 25 30 eqbrtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ≠ 𝑡 ∧ 𝑆 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐼 ≤ ( 𝐹 ∨ 𝑉 ) )