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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )

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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
9 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 )
10 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
11 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
12 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) )
13 simpl1l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝐾 ∈ HL )
14 simp23l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
15 14 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 )
16 simpl1 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
17 simpl21 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 )
18 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
19 16 17 15 18 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
20 1 2 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
21 13 15 19 20 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝑄 ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
22 simpl22 ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
23 1 2 4 5 6 7 trljat1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
24 16 17 22 23 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
25 simpr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
26 24 25 eqtrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
27 21 26 breqtrrd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) )
28 simpl3r ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) )
29 simpll1 ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
30 22 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
31 17 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → 𝐹 ∈ 𝑇 )
32 simpr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝐹 ‘ 𝑃 ) = 𝑃 )
33 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
34 1 33 4 5 6 7 trl0 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) )
35 29 30 31 32 34 syl112anc ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑅 ‘ 𝐹 ) = ( 0. ‘ 𝐾 ) )
36 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
37 13 36 syl ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝐾 ∈ AtLat )
38 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
39 38 adantr ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 )
40 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
41 40 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
42 13 39 15 41 syl3anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 0. ‘ 𝐾 ) ≤ ( 𝑃 ∨ 𝑄 ) )
45 44 adantr ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 0. ‘ 𝐾 ) ≤ ( 𝑃 ∨ 𝑄 ) )
46 35 45 eqbrtrd ⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ∧ ( 𝐹 ‘ 𝑃 ) = 𝑃 ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) )
47 46 ex ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( ( 𝐹 ‘ 𝑃 ) = 𝑃 → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
48 47 necon3bd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) )
49 28 48 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 )
50 1 4 5 6 7 trlat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
51 16 22 17 49 50 syl112anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
52 simpl3l ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝑃 ≠ 𝑄 )
53 52 necomd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → 𝑄 ≠ 𝑃 )
54 1 2 4 hlatexch1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) ∧ 𝑄 ≠ 𝑃 ) → ( 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
55 13 15 51 39 53 54 syl131anc ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
56 27 55 mpd ⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) )
57 56 ex ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) )
58 57 necon3bd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
59 12 58 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
60 1 2 3 4 5 6 7 trlval3 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
61 8 9 10 11 59 60 syl113anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) = ( ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )