Metamath Proof Explorer

Theorem cdleme22f2

Description: Part of proof of Lemma E in Crawley p. 113. cdleme22f with s and t swapped (this case is not mentioned by them). If s <_ t \/ v, then f(s) <_ f_s(t) \/ v. (Contributed by NM, 7-Dec-2012)

Ref Expression
Hypotheses cdleme22.l ⊢ ≤ = ( le ‘ 𝐾 )
cdleme22.j ⊢ ∨ = ( join ‘ 𝐾 )
cdleme22.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdleme22.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdleme22.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdleme22f2.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
cdleme22f2.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
cdleme22f2.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑇 ∨ 𝑆 ) ∧ 𝑊 ) ) )
Assertion cdleme22f2 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ≤ ( 𝑁 ∨ 𝑉 ) )

Proof

Step Hyp Ref Expression
1 cdleme22.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdleme22.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdleme22.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdleme22.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme22.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme22f2.u ⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 )
7 cdleme22f2.f ⊢ 𝐹 = ( ( 𝑆 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑆 ) ∧ 𝑊 ) ) )
8 cdleme22f2.n ⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑇 ∨ 𝑆 ) ∧ 𝑊 ) ) )
9 simp11 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
10 simp2l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
11 simp2r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
12 9 10 11 3jca ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) )
13 simp12 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) )
14 simp31l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑆 ∈ 𝐴 )
15 simp33 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) )
16 simp32l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑆 ≠ 𝑇 )
17 16 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑇 ≠ 𝑆 )
18 simp32r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑆 ≤ ( 𝑇 ∨ 𝑉 ) )
19 simp11l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝐾 ∈ HL )
20 hlcvl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
21 19 20 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝐾 ∈ CvLat )
22 simp12l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑇 ∈ 𝐴 )
23 simp33l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑉 ∈ 𝐴 )
24 simp33r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑉 ≤ 𝑊 )
25 simp31r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ¬ 𝑆 ≤ 𝑊 )
26 nbrne2 ⊢ ( ( 𝑉 ≤ 𝑊 ∧ ¬ 𝑆 ≤ 𝑊 ) → 𝑉 ≠ 𝑆 )
27 26 necomd ⊢ ( ( 𝑉 ≤ 𝑊 ∧ ¬ 𝑆 ≤ 𝑊 ) → 𝑆 ≠ 𝑉 )
28 24 25 27 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑆 ≠ 𝑉 )
29 1 2 4 cvlatexch2 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) ∧ 𝑆 ≠ 𝑉 ) → ( 𝑆 ≤ ( 𝑇 ∨ 𝑉 ) → 𝑇 ≤ ( 𝑆 ∨ 𝑉 ) ) )
30 21 14 22 23 28 29 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑆 ≤ ( 𝑇 ∨ 𝑉 ) → 𝑇 ≤ ( 𝑆 ∨ 𝑉 ) ) )
31 18 30 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑇 ≤ ( 𝑆 ∨ 𝑉 ) )
32 1 2 3 4 5 6 7 8 cdleme22f ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ 𝑆 ∈ 𝐴 ∧ ( 𝑉 ∈ 𝐴 ∧ 𝑉 ≤ 𝑊 ) ) ∧ ( 𝑇 ≠ 𝑆 ∧ 𝑇 ≤ ( 𝑆 ∨ 𝑉 ) ) ) → 𝑁 ≤ ( 𝐹 ∨ 𝑉 ) )
33 12 13 14 15 17 31 32 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑁 ≤ ( 𝐹 ∨ 𝑉 ) )
34 simp31 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) )
35 simp133 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑃 ≠ 𝑄 )
36 simp132 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) )
37 simp131 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) )
38 1 2 3 4 5 6 7 8 cdleme7ga ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 ∈ 𝐴 )
39 12 13 34 35 36 37 38 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑁 ∈ 𝐴 )
40 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 )
41 9 10 11 34 35 37 40 syl132anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ∈ 𝐴 )
42 1 2 3 4 5 6 7 8 cdleme7 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑇 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑁 ≤ 𝑊 )
43 12 13 34 35 36 37 42 syl123anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ¬ 𝑁 ≤ 𝑊 )
44 nbrne2 ⊢ ( ( 𝑉 ≤ 𝑊 ∧ ¬ 𝑁 ≤ 𝑊 ) → 𝑉 ≠ 𝑁 )
45 44 necomd ⊢ ( ( 𝑉 ≤ 𝑊 ∧ ¬ 𝑁 ≤ 𝑊 ) → 𝑁 ≠ 𝑉 )
46 24 43 45 syl2anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → 𝑁 ≠ 𝑉 )
47 1 2 4 cvlatexch2 ⊢ ( ( 𝐾 ∈ CvLat ∧ ( 𝑁 ∈ 𝐴 ∧ 𝐹 ∈ 𝐴 ∧ 𝑉 ∈ 𝐴 ) ∧ 𝑁 ≠ 𝑉 ) → ( 𝑁 ≤ ( 𝐹 ∨ 𝑉 ) → 𝐹 ≤ ( 𝑁 ∨ 𝑉 ) ) )
48 21 39 41 23 46 47 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ) ) ) → ( 𝑁 ≤ ( 𝐹 ∨ 𝑉 ) → 𝐹 ≤ ( 𝑁 ∨ 𝑉 ) ) )
49 33 48 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑇 ∈ 𝐴 ∧ ¬ 𝑇 ≤ 𝑊 ) ∧ ( ¬ 𝑆 ≤ ( 𝑃 ∨ 𝑄 ) ∧ 𝑇 ≤ ( 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 ≤ ( 𝑁 ∨ 𝑉 ) )