Metamath Proof Explorer


Theorem cdleme16c

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, 2nd part of 3rd sentence. F and G represent f(s) and f(t) respectively. We show, in their notation, s \/ t \/ f(s) \/ f(t)=s \/ t \/ u. (Contributed by NM, 11-Oct-2012)

Ref Expression
Hypotheses cdleme12.l = ( le ‘ 𝐾 )
cdleme12.j = ( join ‘ 𝐾 )
cdleme12.m = ( meet ‘ 𝐾 )
cdleme12.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme12.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme12.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme12.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
cdleme12.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
Assertion cdleme16c ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) = ( ( 𝑆 𝑇 ) 𝑈 ) )

Proof

Step Hyp Ref Expression
1 cdleme12.l = ( le ‘ 𝐾 )
2 cdleme12.j = ( join ‘ 𝐾 )
3 cdleme12.m = ( meet ‘ 𝐾 )
4 cdleme12.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme12.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme12.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme12.f 𝐹 = ( ( 𝑆 𝑈 ) ( 𝑄 ( ( 𝑃 𝑆 ) 𝑊 ) ) )
8 cdleme12.g 𝐺 = ( ( 𝑇 𝑈 ) ( 𝑄 ( ( 𝑃 𝑇 ) 𝑊 ) ) )
9 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝐾 ∈ HL )
10 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑊𝐻 )
11 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑃𝐴 )
12 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑄𝐴 )
13 simp21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) )
14 1 2 3 4 5 6 7 cdleme1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ) → ( 𝑆 𝐹 ) = ( 𝑆 𝑈 ) )
15 9 10 11 12 13 14 syl23anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑆 𝐹 ) = ( 𝑆 𝑈 ) )
16 simp22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) )
17 1 2 3 4 5 6 8 cdleme1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴𝑄𝐴 ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ) ) → ( 𝑇 𝐺 ) = ( 𝑇 𝑈 ) )
18 9 10 11 12 16 17 syl23anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑇 𝐺 ) = ( 𝑇 𝑈 ) )
19 15 18 oveq12d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝐹 ) ( 𝑇 𝐺 ) ) = ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) )
20 simp21l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑆𝐴 )
21 simp22l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑇𝐴 )
22 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
23 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
24 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
25 simp23l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑃𝑄 )
26 simp31 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ¬ 𝑆 ( 𝑃 𝑄 ) )
27 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 )
28 22 23 24 13 25 26 27 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝐹𝐴 )
29 simp32 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ¬ 𝑇 ( 𝑃 𝑄 ) )
30 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 )
31 22 23 24 16 25 29 30 syl132anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝐺𝐴 )
32 2 4 hlatj4 ( ( 𝐾 ∈ HL ∧ ( 𝑆𝐴𝑇𝐴 ) ∧ ( 𝐹𝐴𝐺𝐴 ) ) → ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) = ( ( 𝑆 𝐹 ) ( 𝑇 𝐺 ) ) )
33 9 20 21 28 31 32 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) = ( ( 𝑆 𝐹 ) ( 𝑇 𝐺 ) ) )
34 simp12r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ¬ 𝑃 𝑊 )
35 1 2 3 4 5 6 lhpat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )
36 9 10 11 34 12 25 35 syl222anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → 𝑈𝐴 )
37 2 4 hlatjidm ( ( 𝐾 ∈ HL ∧ 𝑈𝐴 ) → ( 𝑈 𝑈 ) = 𝑈 )
38 9 36 37 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( 𝑈 𝑈 ) = 𝑈 )
39 38 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) ( 𝑈 𝑈 ) ) = ( ( 𝑆 𝑇 ) 𝑈 ) )
40 2 4 hlatj4 ( ( 𝐾 ∈ HL ∧ ( 𝑆𝐴𝑇𝐴 ) ∧ ( 𝑈𝐴𝑈𝐴 ) ) → ( ( 𝑆 𝑇 ) ( 𝑈 𝑈 ) ) = ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) )
41 9 20 21 36 36 40 syl122anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) ( 𝑈 𝑈 ) ) = ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) )
42 39 41 eqtr3d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) 𝑈 ) = ( ( 𝑆 𝑈 ) ( 𝑇 𝑈 ) ) )
43 19 33 42 3eqtr4d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ∧ ( 𝑇𝐴 ∧ ¬ 𝑇 𝑊 ) ∧ ( 𝑃𝑄𝑆𝑇 ) ) ∧ ( ¬ 𝑆 ( 𝑃 𝑄 ) ∧ ¬ 𝑇 ( 𝑃 𝑄 ) ∧ ¬ 𝑈 ( 𝑆 𝑇 ) ) ) → ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) = ( ( 𝑆 𝑇 ) 𝑈 ) )