Metamath Proof Explorer


Theorem trlval4

Description: The value of the trace of a lattice translation in terms of 2 atoms. (Contributed by NM, 3-May-2013)

Ref Expression
Hypotheses trlval3.l = ( le ‘ 𝐾 )
trlval3.j = ( join ‘ 𝐾 )
trlval3.m = ( meet ‘ 𝐾 )
trlval3.a 𝐴 = ( Atoms ‘ 𝐾 )
trlval3.h 𝐻 = ( LHyp ‘ 𝐾 )
trlval3.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
trlval3.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion trlval4 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( 𝑅𝐹 ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( 𝑄 ( 𝐹𝑄 ) ) ) )

Proof

Step Hyp Ref Expression
1 trlval3.l = ( le ‘ 𝐾 )
2 trlval3.j = ( join ‘ 𝐾 )
3 trlval3.m = ( meet ‘ 𝐾 )
4 trlval3.a 𝐴 = ( Atoms ‘ 𝐾 )
5 trlval3.h 𝐻 = ( LHyp ‘ 𝐾 )
6 trlval3.t 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 trlval3.r 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
9 simp21 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → 𝐹𝑇 )
10 simp22 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
11 simp23 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) )
12 simp3r ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) )
13 simpl1l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝐾 ∈ HL )
14 simp23l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → 𝑄𝐴 )
15 14 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝑄𝐴 )
16 simpl1 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
17 simpl21 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝐹𝑇 )
18 1 4 5 6 ltrnat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇𝑄𝐴 ) → ( 𝐹𝑄 ) ∈ 𝐴 )
19 16 17 15 18 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝐹𝑄 ) ∈ 𝐴 )
20 1 2 4 hlatlej1 ( ( 𝐾 ∈ HL ∧ 𝑄𝐴 ∧ ( 𝐹𝑄 ) ∈ 𝐴 ) → 𝑄 ( 𝑄 ( 𝐹𝑄 ) ) )
21 13 15 19 20 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝑄 ( 𝑄 ( 𝐹𝑄 ) ) )
22 simpl22 ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
23 1 2 4 5 6 7 trljat1 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ) → ( 𝑃 ( 𝑅𝐹 ) ) = ( 𝑃 ( 𝐹𝑃 ) ) )
24 16 17 22 23 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑃 ( 𝑅𝐹 ) ) = ( 𝑃 ( 𝐹𝑃 ) ) )
25 simpr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) )
26 24 25 eqtrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑃 ( 𝑅𝐹 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) )
27 21 26 breqtrrd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝑄 ( 𝑃 ( 𝑅𝐹 ) ) )
28 simpl3r ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) )
29 simpll1 ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) )
30 22 adantr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) )
31 17 adantr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → 𝐹𝑇 )
32 simpr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝐹𝑃 ) = 𝑃 )
33 eqid ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
34 1 33 4 5 6 7 trl0 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) = 𝑃 ) ) → ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) )
35 29 30 31 32 34 syl112anc ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝑅𝐹 ) = ( 0. ‘ 𝐾 ) )
36 hlatl ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
37 13 36 syl ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝐾 ∈ AtLat )
38 simp22l ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → 𝑃𝐴 )
39 38 adantr ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝑃𝐴 )
40 eqid ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
41 40 2 4 hlatjcl ( ( 𝐾 ∈ HL ∧ 𝑃𝐴𝑄𝐴 ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
42 13 39 15 41 syl3anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
43 40 1 33 atl0le ( ( 𝐾 ∈ AtLat ∧ ( 𝑃 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) → ( 0. ‘ 𝐾 ) ( 𝑃 𝑄 ) )
44 37 42 43 syl2anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 0. ‘ 𝐾 ) ( 𝑃 𝑄 ) )
45 44 adantr ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 0. ‘ 𝐾 ) ( 𝑃 𝑄 ) )
46 35 45 eqbrtrd ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) ∧ ( 𝐹𝑃 ) = 𝑃 ) → ( 𝑅𝐹 ) ( 𝑃 𝑄 ) )
47 46 ex ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( ( 𝐹𝑃 ) = 𝑃 → ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )
48 47 necon3bd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) → ( 𝐹𝑃 ) ≠ 𝑃 ) )
49 28 48 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝐹𝑃 ) ≠ 𝑃 )
50 1 4 5 6 7 trlat ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝐹𝑇 ∧ ( 𝐹𝑃 ) ≠ 𝑃 ) ) → ( 𝑅𝐹 ) ∈ 𝐴 )
51 16 22 17 49 50 syl112anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑅𝐹 ) ∈ 𝐴 )
52 simpl3l ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝑃𝑄 )
53 52 necomd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → 𝑄𝑃 )
54 1 2 4 hlatexch1 ( ( 𝐾 ∈ HL ∧ ( 𝑄𝐴 ∧ ( 𝑅𝐹 ) ∈ 𝐴𝑃𝐴 ) ∧ 𝑄𝑃 ) → ( 𝑄 ( 𝑃 ( 𝑅𝐹 ) ) → ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )
55 13 15 51 39 53 54 syl131anc ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑄 ( 𝑃 ( 𝑅𝐹 ) ) → ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )
56 27 55 mpd ( ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) ) → ( 𝑅𝐹 ) ( 𝑃 𝑄 ) )
57 56 ex ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( ( 𝑃 ( 𝐹𝑃 ) ) = ( 𝑄 ( 𝐹𝑄 ) ) → ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) )
58 57 necon3bd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) → ( 𝑃 ( 𝐹𝑃 ) ) ≠ ( 𝑄 ( 𝐹𝑄 ) ) ) )
59 12 58 mpd ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( 𝑃 ( 𝐹𝑃 ) ) ≠ ( 𝑄 ( 𝐹𝑄 ) ) )
60 1 2 3 4 5 6 7 trlval3 ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ 𝐹𝑇 ∧ ( ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ∧ ( 𝑃 ( 𝐹𝑃 ) ) ≠ ( 𝑄 ( 𝐹𝑄 ) ) ) ) → ( 𝑅𝐹 ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( 𝑄 ( 𝐹𝑄 ) ) ) )
61 8 9 10 11 59 60 syl113anc ( ( ( 𝐾 ∈ HL ∧ 𝑊𝐻 ) ∧ ( 𝐹𝑇 ∧ ( 𝑃𝐴 ∧ ¬ 𝑃 𝑊 ) ∧ ( 𝑄𝐴 ∧ ¬ 𝑄 𝑊 ) ) ∧ ( 𝑃𝑄 ∧ ¬ ( 𝑅𝐹 ) ( 𝑃 𝑄 ) ) ) → ( 𝑅𝐹 ) = ( ( 𝑃 ( 𝐹𝑃 ) ) ( 𝑄 ( 𝐹𝑄 ) ) ) )