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Metamath Proof Explorer

Theorem cdleme27a

Description: Part of proof of Lemma E in Crawley p. 113. cdleme26f with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 3-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 )
cdleme27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme27.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
cdleme27.z ⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.d ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
cdleme27.c ⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
cdleme27.g ⊢ 𝐺 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
cdleme27.o ⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme27.e ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
cdleme27.y ⊢ 𝑌 = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 )
Assertion cdleme27a ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 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 cdleme27.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme27.f ⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) )
9 cdleme27.z ⊢ 𝑍 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
10 cdleme27.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑠 ∨ 𝑧 ) ∧ 𝑊 ) ) )
11 cdleme27.d ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
12 cdleme27.c ⊢ 𝐶 = if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 )
13 cdleme27.g ⊢ 𝐺 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) )
14 cdleme27.o ⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑍 ∨ ( ( 𝑡 ∨ 𝑧 ) ∧ 𝑊 ) ) )
15 cdleme27.e ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
16 cdleme27.y ⊢ 𝑌 = if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 )
17 simp211 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
18 simp221 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
19 simp222 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
20 simp213 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
21 simp223 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) )
22 simp23r ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
23 simp212 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
24 simp1l ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
25 simp1r ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
26 23 24 25 3jca ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
27 simp3 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) )
28 1 2 3 4 5 6 7 9 10 14 11 15 cdleme26ee ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) )
29 17 18 19 20 21 22 26 27 28 syl332anc ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) )
30 29 3expia ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( ( 𝑡 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) )
31 simp1r ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
32 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐾 ∈ HL )
33 32 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝐾 ∈ HL )
34 simp13l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑠 ∈ 𝐴 )
35 34 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ∈ 𝐴 )
36 simp23l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑡 ∈ 𝐴 )
37 36 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑡 ∈ 𝐴 )
38 simp3ll ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑠 ≠ 𝑡 )
39 38 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≠ 𝑡 )
40 35 37 39 3jca ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴 ∧ 𝑠 ≠ 𝑡 ) )
41 simp21l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑃 ∈ 𝐴 )
42 41 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ∈ 𝐴 )
43 simp22l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑄 ∈ 𝐴 )
44 43 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑄 ∈ 𝐴 )
45 simp212 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑃 ≠ 𝑄 )
46 simp3rl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑉 ∈ 𝐴 )
47 46 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑉 ∈ 𝐴 )
48 simp3 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) )
49 simp3lr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) )
50 49 3ad2ant2 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) )
51 simp1l ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
52 48 50 51 3jca ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) )
53 2 3 4 5 6 cdleme22b ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑠 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴 ∧ 𝑠 ≠ 𝑡 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( 𝑉 ∈ 𝐴 ∧ ( ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
54 33 40 42 44 45 47 52 53 syl232anc ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
55 31 54 pm2.21dd ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ∧ ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) )
56 55 3expia ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( ( 𝑡 ∨ 𝑉 ) ≠ ( 𝑃 ∨ 𝑄 ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) ) )
57 30 56 pm2.61dne ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐷 ≤ ( 𝐸 ∨ 𝑉 ) )
58 iftrue ⊢ ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) = 𝐷 )
59 12 58 eqtrid ⊢ ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → 𝐶 = 𝐷 )
60 59 ad2antrr ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐷 )
61 iftrue ⊢ ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 ) = 𝐸 )
62 16 61 eqtrid ⊢ ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → 𝑌 = 𝐸 )
63 62 oveq1d ⊢ ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐸 ∨ 𝑉 ) )
64 63 ad2antlr ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐸 ∨ 𝑉 ) )
65 57 60 64 3brtr4d ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) )
66 65 ex ⊢ ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) )
67 simpr11 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
68 simpr12 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑃 ≠ 𝑄 )
69 simpll ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
70 68 69 jca ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) )
71 simpr23 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) )
72 simpr21 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
73 simpr22 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
74 simpr13 ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
75 simplr ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
76 simpr3l ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) )
77 simpr3r ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
78 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) )
79 eqid ⊢ ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) )
80 9 10 13 78 11 79 cdleme25cv ⊢ 𝐷 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑡 ∈ 𝐴 ( ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐺 ∨ ( ( 𝑠 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) )
81 1 2 3 4 5 6 7 13 78 80 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 ) ) ) → 𝐷 ≤ ( 𝐺 ∨ 𝑉 ) )
82 67 70 71 72 73 74 75 76 77 81 syl333anc ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐷 ≤ ( 𝐺 ∨ 𝑉 ) )
83 59 ad2antrr ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐷 )
84 iffalse ⊢ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐸 , 𝐺 ) = 𝐺 )
85 16 84 eqtrid ⊢ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → 𝑌 = 𝐺 )
86 85 oveq1d ⊢ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐺 ∨ 𝑉 ) )
87 86 ad2antlr ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐺 ∨ 𝑉 ) )
88 82 83 87 3brtr4d ⊢ ( ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) )
89 88 ex ⊢ ( ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) )
90 simpr11 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
91 simpr12 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑃 ≠ 𝑄 )
92 simplr ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
93 91 92 jca ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) )
94 simpr13 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
95 simpr21 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
96 simpr22 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
97 simpr23 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) )
98 simpll ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
99 simpr3l ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) )
100 simpr3r ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
101 eqid ⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) )
102 eqid ⊢ ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) ) ) = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) ) )
103 9 14 8 101 15 102 cdleme25cv ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑡 ∨ 𝑠 ) ∧ 𝑊 ) ) ) ) )
104 1 2 3 4 5 6 7 8 101 103 cdleme26f2 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐹 ≤ ( 𝐸 ∨ 𝑉 ) )
105 90 93 94 95 96 97 98 99 100 104 syl333anc ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐹 ≤ ( 𝐸 ∨ 𝑉 ) )
106 iffalse ⊢ ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → if ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) , 𝐷 , 𝐹 ) = 𝐹 )
107 12 106 eqtrid ⊢ ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) → 𝐶 = 𝐹 )
108 107 ad2antrr ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐹 )
109 63 ad2antlr ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐸 ∨ 𝑉 ) )
110 105 108 109 3brtr4d ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) )
111 110 ex ⊢ ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) )
112 simpr11 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
113 simpr23 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) )
114 simplr ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) )
115 simpll ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) )
116 simpr12 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝑃 ≠ 𝑄 )
117 114 115 116 3jca ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) )
118 simpr21 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
119 simpr22 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
120 simpr13 ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) )
121 simpr3l ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) )
122 simpr3r ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
123 2 3 4 5 6 7 8 13 cdleme22g ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ∧ ( ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑃 ≠ 𝑄 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ∧ ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐹 ≤ ( 𝐺 ∨ 𝑉 ) )
124 112 113 117 118 119 120 121 122 123 syl323anc ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐹 ≤ ( 𝐺 ∨ 𝑉 ) )
125 107 ad2antrr ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 = 𝐹 )
126 86 ad2antlr ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → ( 𝑌 ∨ 𝑉 ) = ( 𝐺 ∨ 𝑉 ) )
127 124 125 126 3brtr4d ⊢ ( ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) )
128 127 ex ⊢ ( ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) ) )
129 66 89 111 128 4cases ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑃 ≠ 𝑄 ∧ ( 𝑠 ∈ 𝐴 ∧ ¬ 𝑠 ≤ 𝑊 ) ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑡 ∈ 𝐴 ∧ ¬ 𝑡 ≤ 𝑊 ) ) ∧ ( ( 𝑠 ≠ 𝑡 ∧ 𝑠 ≤ ( 𝑡 ∨ 𝑉 ) ) ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ) → 𝐶 ≤ ( 𝑌 ∨ 𝑉 ) )