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Metamath Proof Explorer

Theorem cdleme26ee

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph, 4th line on p. 115. F , N , O represent f(z), f_z(s), f_z(t) respectively. When t \/ v = p \/ q, f_z(s) <_ f_z(t) \/ v. TODO: FIX COMMENT. (Contributed by NM, 2-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 )
cdleme26e.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme26e.f ⊢ 𝐹 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme26e.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑆 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme26e.o ⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑇 ∨ 𝑧 ) ∧ 𝑊 ) ) )
cdleme26e.i ⊢ 𝐼 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
cdleme26e.e ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
Assertion 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 ) ) ) → 𝐼 ≤ ( 𝐸 ∨ 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 cdleme26e.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
8 cdleme26e.f ⊢ 𝐹 = ( ( 𝑧 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑧 ) ∧ 𝑊 ) ) )
9 cdleme26e.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑆 ∨ 𝑧 ) ∧ 𝑊 ) ) )
10 cdleme26e.o ⊢ 𝑂 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑇 ∨ 𝑧 ) ∧ 𝑊 ) ) )
11 cdleme26e.i ⊢ 𝐼 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
12 cdleme26e.e ⊢ 𝐸 = ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
13 simp11l ⊢ ( ( ( ( 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 ) ) ) → 𝐾 ∈ HL )
14 simp11r ⊢ ( ( ( ( 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 ) ) ) → 𝑊 ∈ 𝐻 )
15 simp12 ⊢ ( ( ( ( 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 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
16 simp13 ⊢ ( ( ( ( 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 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
17 simp3l1 ⊢ ( ( ( ( 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 ) ) ) → 𝑃 ≠ 𝑄 )
18 2 3 5 6 cdlemb2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝑃 ≠ 𝑄 ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) )
19 13 14 15 16 17 18 syl221anc ⊢ ( ( ( ( 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 ) ) ) → ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) )
20 nfv ⊢ Ⅎ 𝑧 ( ( ( 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 ) ) )
21 nfra1 ⊢ Ⅎ 𝑧 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 )
22 nfcv ⊢ Ⅎ 𝑧 𝐵
23 21 22 nfriota ⊢ Ⅎ 𝑧 ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) )
24 11 23 nfcxfr ⊢ Ⅎ 𝑧 𝐼
25 nfcv ⊢ Ⅎ 𝑧 ≤
26 nfra1 ⊢ Ⅎ 𝑧 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 )
27 26 22 nfriota ⊢ Ⅎ 𝑧 ( ℩ 𝑢 ∈ 𝐵 ∀ 𝑧 ∈ 𝐴 ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑂 ) )
28 12 27 nfcxfr ⊢ Ⅎ 𝑧 𝐸
29 nfcv ⊢ Ⅎ 𝑧 ∨
30 nfcv ⊢ Ⅎ 𝑧 𝑉
31 28 29 30 nfov ⊢ Ⅎ 𝑧 ( 𝐸 ∨ 𝑉 )
32 24 25 31 nfbr ⊢ Ⅎ 𝑧 𝐼 ≤ ( 𝐸 ∨ 𝑉 )
33 simp111 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
34 simp112 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
35 simp113 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
36 simp121 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
37 simp122 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) )
38 simp123 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
39 simp13l ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ≠ 𝑄 ∧ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ) )
40 simp13r ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑇 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) )
41 simp3r ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) )
42 40 41 jca ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑇 ∨ 𝑉 ) = ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) )
43 simp2 ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑧 ∈ 𝐴 )
44 simp3l ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑧 ≤ 𝑊 )
45 43 44 jca ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) )
46 1 2 3 4 5 6 7 8 9 10 11 12 cdleme26e ⊢ ( ( ( ( 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 ) ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑧 ∈ 𝐴 ∧ ¬ 𝑧 ≤ 𝑊 ) ) ) → 𝐼 ≤ ( 𝐸 ∨ 𝑉 ) )
47 33 34 35 36 37 38 39 42 45 46 syl333anc ⊢ ( ( ( ( ( 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 ) ) ) ∧ 𝑧 ∈ 𝐴 ∧ ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐼 ≤ ( 𝐸 ∨ 𝑉 ) )
48 47 3exp ⊢ ( ( ( ( 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 ) ) ) → ( 𝑧 ∈ 𝐴 → ( ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐼 ≤ ( 𝐸 ∨ 𝑉 ) ) ) )
49 20 32 48 rexlimd ⊢ ( ( ( ( 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 ) ) ) → ( ∃ 𝑧 ∈ 𝐴 ( ¬ 𝑧 ≤ 𝑊 ∧ ¬ 𝑧 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝐼 ≤ ( 𝐸 ∨ 𝑉 ) ) )
50 19 49 mpd ⊢ ( ( ( ( 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 ) ) ) → 𝐼 ≤ ( 𝐸 ∨ 𝑉 ) )