Metamath Proof Explorer


Theorem cdleme16f

Description: Part of proof of Lemma E in Crawley p. 113, 3rd paragraph on p. 114, 3rd 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))=(f(s) \/ f(t)) /\ w. (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 cdleme16f ( ( ( ( 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 ) ) = ( ( 𝐹 𝐺 ) 𝑊 ) )

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 9 hllatd ( ( ( ( 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 ∈ Lat )
11 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 )
12 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 )
13 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
14 13 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑆𝐴𝑇𝐴 ) → ( 𝑆 𝑇 ) ∈ ( Base ‘ 𝐾 ) )
15 9 11 12 14 syl3anc ( ( ( ( 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 ) ∈ ( Base ‘ 𝐾 ) )
16 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 ) )
17 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 ) )
18 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 ) )
19 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 ) )
20 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 )
21 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 ) )
22 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 )
23 16 17 18 19 20 21 22 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 )
24 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 ) )
25 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 ) )
26 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 )
27 16 17 18 24 20 25 26 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 )
28 13 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝐹𝐴𝐺𝐴 ) → ( 𝐹 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
29 9 23 27 28 syl3anc ( ( ( ( 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 ) ) ) → ( 𝐹 𝐺 ) ∈ ( Base ‘ 𝐾 ) )
30 13 1 3 latmle2 ( ( 𝐾 ∈ Lat ∧ ( 𝑆 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 𝐺 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ( 𝐹 𝐺 ) )
31 10 15 29 30 syl3anc ( ( ( ( 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 ) ) ( 𝐹 𝐺 ) )
32 1 2 3 4 5 6 7 8 cdleme15 ( ( ( ( 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 ) ) 𝑊 )
33 13 3 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑆 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 𝐺 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ∈ ( Base ‘ 𝐾 ) )
34 10 15 29 33 syl3anc ( ( ( ( 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 ) ) ∈ ( Base ‘ 𝐾 ) )
35 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 )
36 13 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
37 35 36 syl ( ( ( ( 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 ) ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
38 13 1 3 latlem12 ( ( 𝐾 ∈ Lat ∧ ( ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 𝐺 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ( 𝐹 𝐺 ) ∧ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) 𝑊 ) ↔ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ( ( 𝐹 𝐺 ) 𝑊 ) ) )
39 10 34 29 37 38 syl13anc ( ( ( ( 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 ) ) ( 𝐹 𝐺 ) ∧ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) 𝑊 ) ↔ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ( ( 𝐹 𝐺 ) 𝑊 ) ) )
40 31 32 39 mpbi2and ( ( ( ( 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 ) ) ( ( 𝐹 𝐺 ) 𝑊 ) )
41 hlatl ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
42 9 41 syl ( ( ( ( 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 ∈ AtLat )
43 1 2 3 4 5 6 7 8 cdleme16d ( ( ( ( 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 ) ) ∈ 𝐴 )
44 1 2 3 4 5 6 7 cdleme3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑆𝐴 ∧ ¬ 𝑆 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ 𝑆 ( 𝑃 𝑄 ) ) ) → ¬ 𝐹 𝑊 )
45 16 17 18 19 20 21 44 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 ) ) ) → ¬ 𝐹 𝑊 )
46 1 2 3 4 5 6 7 8 cdleme16b ( ( ( ( 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 ) ) ) → 𝐹𝐺 )
47 1 2 3 4 5 lhpat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝐴 ∧ ¬ 𝐹 𝑊 ) ∧ ( 𝐺𝐴𝐹𝐺 ) ) → ( ( 𝐹 𝐺 ) 𝑊 ) ∈ 𝐴 )
48 16 23 45 27 46 47 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 ) ) ) → ( ( 𝐹 𝐺 ) 𝑊 ) ∈ 𝐴 )
49 1 4 atcmp ( ( 𝐾 ∈ AtLat ∧ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ∈ 𝐴 ∧ ( ( 𝐹 𝐺 ) 𝑊 ) ∈ 𝐴 ) → ( ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) ( ( 𝐹 𝐺 ) 𝑊 ) ↔ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) = ( ( 𝐹 𝐺 ) 𝑊 ) ) )
50 42 43 48 49 syl3anc ( ( ( ( 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 ) ) ( ( 𝐹 𝐺 ) 𝑊 ) ↔ ( ( 𝑆 𝑇 ) ( 𝐹 𝐺 ) ) = ( ( 𝐹 𝐺 ) 𝑊 ) ) )
51 40 50 mpbid ( ( ( ( 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 ) ) = ( ( 𝐹 𝐺 ) 𝑊 ) )