Metamath Proof Explorer


Theorem cdleme35b

Description: Part of proof of Lemma E in Crawley p. 113. TODO: FIX COMMENT. (Contributed by NM, 10-Mar-2013)

Ref Expression
Hypotheses cdleme35.l = ( le ‘ 𝐾 )
cdleme35.j = ( join ‘ 𝐾 )
cdleme35.m = ( meet ‘ 𝐾 )
cdleme35.a 𝐴 = ( Atoms ‘ 𝐾 )
cdleme35.h 𝐻 = ( LHyp ‘ 𝐾 )
cdleme35.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
cdleme35.f 𝐹 = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) )
Assertion cdleme35b ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ( 𝑄 ( 𝑅 𝑈 ) ) )

Proof

Step Hyp Ref Expression
1 cdleme35.l = ( le ‘ 𝐾 )
2 cdleme35.j = ( join ‘ 𝐾 )
3 cdleme35.m = ( meet ‘ 𝐾 )
4 cdleme35.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdleme35.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdleme35.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
7 cdleme35.f 𝐹 = ( ( 𝑅 𝑈 ) ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) )
8 simp11l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝐾 ∈ HL )
9 8 hllatd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝐾 ∈ Lat )
10 simp13l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑄𝐴 )
11 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
12 11 4 atbase ( 𝑄𝐴𝑄 ∈ ( Base ‘ 𝐾 ) )
13 10 12 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
14 simp2rl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅𝐴 )
15 simp11 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
16 simp12 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
17 simp2l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑃𝑄 )
18 1 2 3 4 5 6 cdleme0a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )
19 15 16 10 17 18 syl112anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑈𝐴 )
20 11 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑅𝐴𝑈𝐴 ) → ( 𝑅 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
21 8 14 19 20 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑅 𝑈 ) ∈ ( Base ‘ 𝐾 ) )
22 11 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) → 𝑄 ( 𝑄 ( 𝑅 𝑈 ) ) )
23 9 13 21 22 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑄 ( 𝑄 ( 𝑅 𝑈 ) ) )
24 simp12l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑃𝐴 )
25 11 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
26 24 25 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
27 11 4 atbase ( 𝑅𝐴𝑅 ∈ ( Base ‘ 𝐾 ) )
28 14 27 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑅 ∈ ( Base ‘ 𝐾 ) )
29 11 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
30 9 26 28 29 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑃 𝑅 ) ∈ ( Base ‘ 𝐾 ) )
31 simp11r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑊𝐻 )
32 11 5 lhpbase ( 𝑊𝐻𝑊 ∈ ( Base ‘ 𝐾 ) )
33 31 32 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
34 11 3 latmcl ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑅 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑅 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
35 9 30 33 34 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
36 11 2 latjcl ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑅 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑅 ) 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
37 9 30 13 36 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
38 11 1 3 latmle1 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑅 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑅 ) 𝑊 ) ( 𝑃 𝑅 ) )
39 9 30 33 38 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑊 ) ( 𝑃 𝑅 ) )
40 11 1 2 latlej1 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 𝑅 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 𝑅 ) ( ( 𝑃 𝑅 ) 𝑄 ) )
41 9 30 13 40 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑃 𝑅 ) ( ( 𝑃 𝑅 ) 𝑄 ) )
42 11 1 9 35 30 37 39 41 lattrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑊 ) ( ( 𝑃 𝑅 ) 𝑄 ) )
43 6 oveq2i ( 𝑄 𝑈 ) = ( 𝑄 ( ( 𝑃 𝑄 ) 𝑊 ) )
44 11 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
45 8 24 10 44 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
46 1 2 4 hlatlej2 ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → 𝑄 ( 𝑃 𝑄 ) )
47 8 24 10 46 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝑄 ( 𝑃 𝑄 ) )
48 11 1 2 3 4 atmod3i1 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴 ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) ∧ 𝑄 ( 𝑃 𝑄 ) ) → ( 𝑄 ( ( 𝑃 𝑄 ) 𝑊 ) ) = ( ( 𝑃 𝑄 ) ( 𝑄 𝑊 ) ) )
49 8 10 45 33 47 48 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 ( ( 𝑃 𝑄 ) 𝑊 ) ) = ( ( 𝑃 𝑄 ) ( 𝑄 𝑊 ) ) )
50 simp13 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
51 eqid ( 1. ‘ 𝐾 ) = ( 1. ‘ 𝐾 )
52 1 2 51 4 5 lhpjat2 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) → ( 𝑄 𝑊 ) = ( 1. ‘ 𝐾 ) )
53 15 50 52 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 𝑊 ) = ( 1. ‘ 𝐾 ) )
54 53 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑄 ) ( 𝑄 𝑊 ) ) = ( ( 𝑃 𝑄 ) ( 1. ‘ 𝐾 ) ) )
55 hlol ( 𝐾 ∈ HL → 𝐾 ∈ OL )
56 8 55 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → 𝐾 ∈ OL )
57 11 3 51 olm11 ( ( 𝐾 ∈ OL ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 𝑄 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 𝑄 ) )
58 56 45 57 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑄 ) ( 1. ‘ 𝐾 ) ) = ( 𝑃 𝑄 ) )
59 49 54 58 3eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 ( ( 𝑃 𝑄 ) 𝑊 ) ) = ( 𝑃 𝑄 ) )
60 43 59 syl5eq ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 𝑈 ) = ( 𝑃 𝑄 ) )
61 60 oveq2d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑅 ( 𝑄 𝑈 ) ) = ( 𝑅 ( 𝑃 𝑄 ) ) )
62 2 4 hlatj12 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴𝑅𝐴𝑈𝐴 ) ) → ( 𝑄 ( 𝑅 𝑈 ) ) = ( 𝑅 ( 𝑄 𝑈 ) ) )
63 8 10 14 19 62 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 ( 𝑅 𝑈 ) ) = ( 𝑅 ( 𝑄 𝑈 ) ) )
64 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑅𝐴 ) → ( 𝑃 𝑅 ) = ( 𝑅 𝑃 ) )
65 8 24 14 64 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑃 𝑅 ) = ( 𝑅 𝑃 ) )
66 65 oveq1d ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑄 ) = ( ( 𝑅 𝑃 ) 𝑄 ) )
67 2 4 hlatjass ( ( 𝐾 ∈ HL ∧ ( 𝑅𝐴𝑃𝐴𝑄𝐴 ) ) → ( ( 𝑅 𝑃 ) 𝑄 ) = ( 𝑅 ( 𝑃 𝑄 ) ) )
68 8 14 24 10 67 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑅 𝑃 ) 𝑄 ) = ( 𝑅 ( 𝑃 𝑄 ) ) )
69 66 68 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑄 ) = ( 𝑅 ( 𝑃 𝑄 ) ) )
70 61 63 69 3eqtr4rd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑄 ) = ( 𝑄 ( 𝑅 𝑈 ) ) )
71 42 70 breqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑃 𝑅 ) 𝑊 ) ( 𝑄 ( 𝑅 𝑈 ) ) )
72 11 2 latjcl ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ( 𝑅 𝑈 ) ) ∈ ( Base ‘ 𝐾 ) )
73 9 13 21 72 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 ( 𝑅 𝑈 ) ) ∈ ( Base ‘ 𝐾 ) )
74 11 1 2 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 𝑅 ) 𝑊 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ( 𝑅 𝑈 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑄 ( 𝑄 ( 𝑅 𝑈 ) ) ∧ ( ( 𝑃 𝑅 ) 𝑊 ) ( 𝑄 ( 𝑅 𝑈 ) ) ) ↔ ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ( 𝑄 ( 𝑅 𝑈 ) ) ) )
75 9 13 35 73 74 syl13anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( ( 𝑄 ( 𝑄 ( 𝑅 𝑈 ) ) ∧ ( ( 𝑃 𝑅 ) 𝑊 ) ( 𝑄 ( 𝑅 𝑈 ) ) ) ↔ ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ( 𝑄 ( 𝑅 𝑈 ) ) ) )
76 23 71 75 mpbi2and ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ( 𝑅𝐴 ∧ ¬ 𝑅 𝑊 ) ) ∧ ¬ 𝑅 ( 𝑃 𝑄 ) ) → ( 𝑄 ( ( 𝑃 𝑅 ) 𝑊 ) ) ( 𝑄 ( 𝑅 𝑈 ) ) )