Metamath Proof Explorer


Theorem cdlemg18b

Description: Lemma for cdlemg18c . TODO: fix comment. (Contributed by NM, 15-May-2013)

Ref Expression
Hypotheses cdlemg12.l = ( le ‘ 𝐾 )
cdlemg12.j = ( join ‘ 𝐾 )
cdlemg12.m = ( meet ‘ 𝐾 )
cdlemg12.a 𝐴 = ( Atoms ‘ 𝐾 )
cdlemg12.h 𝐻 = ( LHyp ‘ 𝐾 )
cdlemg12.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemg12b.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
cdlemg18b.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
Assertion cdlemg18b ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ) → ¬ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) )

Proof

Step Hyp Ref Expression
1 cdlemg12.l = ( le ‘ 𝐾 )
2 cdlemg12.j = ( join ‘ 𝐾 )
3 cdlemg12.m = ( meet ‘ 𝐾 )
4 cdlemg12.a 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemg12.h 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemg12.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemg12b.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 cdlemg18b.u 𝑈 = ( ( 𝑃 𝑄 ) 𝑊 )
9 simp33 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) )
10 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) )
11 simp1l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝐾 ∈ HL )
12 simp1r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑊𝐻 )
13 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
14 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑄𝐴 )
15 simp3l1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑃𝑄 )
16 1 2 3 4 5 8 cdleme0a ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴𝑃𝑄 ) ) → 𝑈𝐴 )
17 11 12 13 14 15 16 syl212anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑈𝐴 )
18 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
19 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝐹𝑇 )
20 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑄𝐴 ) → ( 𝐹𝑄 ) ∈ 𝐴 )
21 18 19 14 20 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝐹𝑄 ) ∈ 𝐴 )
22 1 2 4 hlatlej1 ( ( 𝐾 ∈ HL ∧ 𝑈𝐴 ∧ ( 𝐹𝑄 ) ∈ 𝐴 ) → 𝑈 ( 𝑈 ( 𝐹𝑄 ) ) )
23 11 17 21 22 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑈 ( 𝑈 ( 𝐹𝑄 ) ) )
24 11 hllatd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝐾 ∈ Lat )
25 simp21l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑃𝐴 )
26 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
27 26 4 atbase ( 𝑃𝐴𝑃 ∈ ( Base ‘ 𝐾 ) )
28 25 27 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
29 26 4 atbase ( 𝑈𝐴𝑈 ∈ ( Base ‘ 𝐾 ) )
30 17 29 syl ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
31 26 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑈𝐴 ∧ ( 𝐹𝑄 ) ∈ 𝐴 ) → ( 𝑈 ( 𝐹𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
32 11 17 21 31 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑈 ( 𝐹𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
33 26 1 2 latjle12 ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑈 ( 𝐹𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ∧ 𝑈 ( 𝑈 ( 𝐹𝑄 ) ) ) ↔ ( 𝑃 𝑈 ) ( 𝑈 ( 𝐹𝑄 ) ) ) )
34 24 28 30 32 33 syl13anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( ( 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ∧ 𝑈 ( 𝑈 ( 𝐹𝑄 ) ) ) ↔ ( 𝑃 𝑈 ) ( 𝑈 ( 𝐹𝑄 ) ) ) )
35 10 23 34 mpbi2and ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑃 𝑈 ) ( 𝑈 ( 𝐹𝑄 ) ) )
36 1 2 3 4 5 8 cdleme0cp ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ 𝑄𝐴 ) ) → ( 𝑃 𝑈 ) = ( 𝑃 𝑄 ) )
37 11 12 13 14 36 syl22anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑃 𝑈 ) = ( 𝑃 𝑄 ) )
38 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
39 5 6 1 2 4 3 8 cdlemg2kq ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ 𝐹𝑇 ) → ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) = ( ( 𝐹𝑄 ) 𝑈 ) )
40 18 13 38 19 39 syl121anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) = ( ( 𝐹𝑄 ) 𝑈 ) )
41 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ ( 𝐹𝑄 ) ∈ 𝐴𝑈𝐴 ) → ( ( 𝐹𝑄 ) 𝑈 ) = ( 𝑈 ( 𝐹𝑄 ) ) )
42 11 21 17 41 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( ( 𝐹𝑄 ) 𝑈 ) = ( 𝑈 ( 𝐹𝑄 ) ) )
43 40 42 eqtr2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑈 ( 𝐹𝑄 ) ) = ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) )
44 35 37 43 3brtr3d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑃 𝑄 ) ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) )
45 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑃𝐴 ) → ( 𝐹𝑃 ) ∈ 𝐴 )
46 18 19 25 45 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝐹𝑃 ) ∈ 𝐴 )
47 1 2 4 ps-1 ( ( 𝐾 ∈ HL ∧ ( 𝑃𝐴𝑄𝐴𝑃𝑄 ) ∧ ( ( 𝐹𝑃 ) ∈ 𝐴 ∧ ( 𝐹𝑄 ) ∈ 𝐴 ) ) → ( ( 𝑃 𝑄 ) ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) ↔ ( 𝑃 𝑄 ) = ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) ) )
48 11 25 14 15 46 21 47 syl132anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( ( 𝑃 𝑄 ) ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) ↔ ( 𝑃 𝑄 ) = ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) ) )
49 44 48 mpbid ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( 𝑃 𝑄 ) = ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) )
50 2 4 hlatjcom ( ( 𝐾 ∈ HL ∧ ( 𝐹𝑃 ) ∈ 𝐴 ∧ ( 𝐹𝑄 ) ∈ 𝐴 ) → ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) = ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) )
51 11 46 21 50 syl3anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( ( 𝐹𝑃 ) ( 𝐹𝑄 ) ) = ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) )
52 49 51 eqtr2d ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) ) → ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) = ( 𝑃 𝑄 ) )
53 52 3exp ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) → ( ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ∧ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) → ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) = ( 𝑃 𝑄 ) ) ) )
54 53 exp4a ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) → ( ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) → ( ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) → ( 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) → ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) = ( 𝑃 𝑄 ) ) ) ) )
55 54 3imp ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) → ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) = ( 𝑃 𝑄 ) ) )
56 55 necon3ad ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ) → ( ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) → ¬ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) ) )
57 9 56 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ 𝐹𝑇 ) ∧ ( 𝑃𝑄 ∧ ( 𝐹𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹𝑄 ) ( 𝐹𝑃 ) ) ≠ ( 𝑃 𝑄 ) ) ) → ¬ 𝑃 ( 𝑈 ( 𝐹𝑄 ) ) )