Metamath Proof Explorer


Theorem cdleme19d

Description: Part of proof of Lemma E in Crawley p. 113, 5th paragraph on p. 114. D , F , G represent s_2, f(s), f(t). We prove f(s) \/ s_2 = f(s) \/ f(t). (Contributed by NM, 14-Nov-2012)

Ref Expression
Hypotheses cdleme19.l = ( le ‘ 𝐾 )
cdleme19.j = ( join ‘ 𝐾 )
cdleme19.m = ( meet ‘ 𝐾 )
cdleme19.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme19.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme19.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme19.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme19.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
cdleme19.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
cdleme19.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
Assertion cdleme19d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝐹 𝐷 ) = ( 𝐹 𝐺 ) )

Proof

Step Hyp Ref Expression
1 cdleme19.l = ( le ‘ 𝐾 )
2 cdleme19.j = ( join ‘ 𝐾 )
3 cdleme19.m = ( meet ‘ 𝐾 )
4 cdleme19.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme19.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme19.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme19.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme19.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
9 cdleme19.d 𝐷 = ( ( 𝑅 𝑆 ) 𝑊 )
10 cdleme19.y 𝑌 = ( ( 𝑅 𝑇 ) 𝑊 )
11 1 2 3 4 5 6 7 8 9 10 cdleme19b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐷 ( 𝐹 𝐺 ) )
12 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐾 ∈ HL )
13 hlcvl ( 𝐾 ∈ HL → 𝐾 ∈ CvLat )
14 12 13 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐾 ∈ CvLat )
15 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑊𝐻 )
16 simp21l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑆𝐴 )
17 simp21r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ¬ 𝑆 𝑊 )
18 simp23 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑅𝐴 )
19 simp33l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑅 ( 𝑃 𝑄 ) )
20 simp32l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
21 1 2 3 4 5 9 cdlemeda ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑅𝐴𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐷𝐴 )
22 12 15 16 17 18 19 20 21 syl223anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐷𝐴 )
23 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
24 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
25 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
26 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) )
27 simp31l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝑃𝑄 )
28 simp32r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ¬ 𝑇 ( 𝑃 𝑄 ) )
29 1 2 3 4 5 6 8 cdleme3fa ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) → 𝐺𝐴 )
30 23 24 25 26 27 28 29 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐺𝐴 )
31 simp21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
32 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 )
33 23 24 25 31 27 20 32 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐹𝐴 )
34 1 2 3 4 5 6 7 8 9 10 cdleme19c ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑅𝐴𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐹𝐷 )
35 12 15 24 25 31 18 27 20 34 syl233anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐹𝐷 )
36 35 necomd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → 𝐷𝐹 )
37 1 2 4 cvlatexchb1 ( ( 𝐾 ∈ CvLat ∧ ( 𝐷𝐴𝐺𝐴𝐹𝐴 ) ∧ 𝐷𝐹 ) → ( 𝐷 ( 𝐹 𝐺 ) ↔ ( 𝐹 𝐷 ) = ( 𝐹 𝐺 ) ) )
38 14 22 30 33 36 37 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝐷 ( 𝐹 𝐺 ) ↔ ( 𝐹 𝐷 ) = ( 𝐹 𝐺 ) ) )
39 11 38 mpbid ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ 𝑅𝐴 ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ ( 𝑅 ( 𝑃 𝑄 ) ∧ 𝑅 ( 𝑆 𝑇 ) ) ) ) → ( 𝐹 𝐷 ) = ( 𝐹 𝐺 ) )