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 ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 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 ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
10 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 simp2r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
12 9 10 11 3jca ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) )
13 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) )
14 simp31l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑆𝐴 )
15 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑉𝐴𝑉 𝑊 ) )
16 simp32l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑆𝑇 )
17 16 necomd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑇𝑆 )
18 simp32r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑆 ( 𝑇 𝑉 ) )
19 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐾 ∈ HL )
20 hlcvl ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
21 19 20 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐾 ∈ CvLat )
22 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑇𝐴 )
23 simp33l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑉𝐴 )
24 simp33r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑉 𝑊 )
25 simp31r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ¬ 𝑆 𝑊 )
26 nbrne2 ( ( 𝑉 𝑊 ∧ ¬ 𝑆 𝑊 ) → 𝑉𝑆 )
27 26 necomd ( ( 𝑉 𝑊 ∧ ¬ 𝑆 𝑊 ) → 𝑆𝑉 )
28 24 25 27 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑆𝑉 )
29 1 2 4 cvlatexch2 ( ( 𝐾 ∈ CvLat ∧ ( 𝑆𝐴𝑇𝐴𝑉𝐴 ) ∧ 𝑆𝑉 ) → ( 𝑆 ( 𝑇 𝑉 ) → 𝑇 ( 𝑆 𝑉 ) ) )
30 21 14 22 23 28 29 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑆 ( 𝑇 𝑉 ) → 𝑇 ( 𝑆 𝑉 ) ) )
31 18 30 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑇 ( 𝑆 𝑉 ) )
32 1 2 3 4 5 6 7 8 cdleme22f ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑆𝐴 ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ∧ ( 𝑇𝑆𝑇 ( 𝑆 𝑉 ) ) ) → 𝑁 ( 𝐹 𝑉 ) )
33 12 13 14 15 17 31 32 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑁 ( 𝐹 𝑉 ) )
34 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
35 simp133 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑃𝑄 )
36 simp132 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑇 ( 𝑃 𝑄 ) )
37 simp131 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
38 1 2 3 4 5 6 7 8 cdleme7ga ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝑁𝐴 )
39 12 13 34 35 36 37 38 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑁𝐴 )
40 1 2 3 4 5 6 7 cdleme3fa ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐹𝐴 )
41 9 10 11 34 35 37 40 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐹𝐴 )
42 1 2 3 4 5 6 7 8 cdleme7 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝑁 𝑊 )
43 12 13 34 35 36 37 42 syl123anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ¬ 𝑁 𝑊 )
44 nbrne2 ( ( 𝑉 𝑊 ∧ ¬ 𝑁 𝑊 ) → 𝑉𝑁 )
45 44 necomd ( ( 𝑉 𝑊 ∧ ¬ 𝑁 𝑊 ) → 𝑁𝑉 )
46 24 43 45 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝑁𝑉 )
47 1 2 4 cvlatexch2 ( ( 𝐾 ∈ CvLat ∧ ( 𝑁𝐴𝐹𝐴𝑉𝐴 ) ∧ 𝑁𝑉 ) → ( 𝑁 ( 𝐹 𝑉 ) → 𝐹 ( 𝑁 𝑉 ) ) )
48 21 39 41 23 46 47 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → ( 𝑁 ( 𝐹 𝑉 ) → 𝐹 ( 𝑁 𝑉 ) ) )
49 33 48 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ 𝑇 ( 𝑃 𝑄 ) ∧ 𝑃𝑄 ) ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑆𝑇𝑆 ( 𝑇 𝑉 ) ) ∧ ( 𝑉𝐴𝑉 𝑊 ) ) ) → 𝐹 ( 𝑁 𝑉 ) )