Metamath Proof Explorer


Theorem cdleme20g

Description: Part of proof of Lemma E in Crawley p. 113, last paragraph on p. 114, antepenultimate line. D , F , Y , G represent s_2, f(s), t_2, f(t). (Contributed by NM, 18-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 )
cdleme20.v 𝑉 = ( ( 𝑆 𝑇 ) 𝑊 )
Assertion cdleme20g ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( ( 𝐷 𝑆 ) ( 𝑌 𝑇 ) ) ( ( 𝑆 𝐹 ) ( 𝑇 𝐺 ) ) ) = ( ( ( 𝑆 𝑅 ) ( 𝑇 𝑅 ) ) ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) ) )

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 cdleme20.v 𝑉 = ( ( 𝑆 𝑇 ) 𝑊 )
12 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝐾 ∈ HL )
13 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑊𝐻 )
14 simp21l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑆𝐴 )
15 simp21r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑆 𝑊 )
16 simp23l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑅𝐴 )
17 simp33 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑅 ( 𝑃 𝑄 ) )
18 simp32l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
19 1 2 3 4 5 9 cdlemeda ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑅𝐴𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → 𝐷𝐴 )
20 12 13 14 15 16 17 18 19 syl223anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝐷𝐴 )
21 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝐷𝐴𝑆𝐴 ) → ( 𝐷 𝑆 ) = ( 𝑆 𝐷 ) )
22 12 20 14 21 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝐷 𝑆 ) = ( 𝑆 𝐷 ) )
23 1 2 3 4 5 9 cdleme10 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑅𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) → ( 𝑆 𝐷 ) = ( 𝑆 𝑅 ) )
24 12 13 16 14 15 23 syl212anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑆 𝐷 ) = ( 𝑆 𝑅 ) )
25 22 24 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝐷 𝑆 ) = ( 𝑆 𝑅 ) )
26 simp22l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑇𝐴 )
27 simp22r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑇 𝑊 )
28 simp32r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ¬ 𝑇 ( 𝑃 𝑄 ) )
29 1 2 3 4 5 10 cdlemeda ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴𝑅 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ) → 𝑌𝐴 )
30 12 13 26 27 16 17 28 29 syl223anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑌𝐴 )
31 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑌𝐴𝑇𝐴 ) → ( 𝑌 𝑇 ) = ( 𝑇 𝑌 ) )
32 12 30 26 31 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑌 𝑇 ) = ( 𝑇 𝑌 ) )
33 1 2 3 4 5 10 cdleme10 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝑅𝐴 ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) → ( 𝑇 𝑌 ) = ( 𝑇 𝑅 ) )
34 12 13 16 26 27 33 syl212anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑇 𝑌 ) = ( 𝑇 𝑅 ) )
35 32 34 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑌 𝑇 ) = ( 𝑇 𝑅 ) )
36 25 35 oveq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝐷 𝑆 ) ( 𝑌 𝑇 ) ) = ( ( 𝑆 𝑅 ) ( 𝑇 𝑅 ) ) )
37 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑃𝐴 )
38 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → 𝑄𝐴 )
39 simp21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
40 1 2 3 4 5 6 7 cdleme1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝑆 𝐹 ) = ( 𝑆 𝑈 ) )
41 12 13 37 38 39 40 syl23anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑆 𝐹 ) = ( 𝑆 𝑈 ) )
42 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) )
43 1 2 3 4 5 6 8 cdleme1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ) → ( 𝑇 𝐺 ) = ( 𝑇 𝑈 ) )
44 12 13 37 38 42 43 syl23anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( 𝑇 𝐺 ) = ( 𝑇 𝑈 ) )
45 41 44 oveq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( 𝑆 𝐹 ) ( 𝑇 𝐺 ) ) = ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) )
46 36 45 oveq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ( ( 𝑃𝑄𝑆𝑇 ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ) ∧ 𝑅 ( 𝑃 𝑄 ) ) ) → ( ( ( 𝐷 𝑆 ) ( 𝑌 𝑇 ) ) ( ( 𝑆 𝐹 ) ( 𝑇 𝐺 ) ) ) = ( ( ( 𝑆 𝑅 ) ( 𝑇 𝑅 ) ) ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) ) )