Metamath Proof Explorer

Theorem cdlemc5

Description: Lemma for cdlemc . (Contributed by NM, 26-May-2012)

Ref Expression
Hypotheses cdlemc3.l ⊢ ≤ = ( le ‘ 𝐾 )
cdlemc3.j ⊢ ∨ = ( join ‘ 𝐾 )
cdlemc3.m ⊢ ∧ = ( meet ‘ 𝐾 )
cdlemc3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
cdlemc3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
cdlemc3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
cdlemc3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
Assertion cdlemc5 ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )

Proof

Step Hyp Ref Expression
1 cdlemc3.l ⊢ ≤ = ( le ‘ 𝐾 )
2 cdlemc3.j ⊢ ∨ = ( join ‘ 𝐾 )
3 cdlemc3.m ⊢ ∧ = ( meet ‘ 𝐾 )
4 cdlemc3.a ⊢ 𝐴 = ( Atoms ‘ 𝐾 )
5 cdlemc3.h ⊢ 𝐻 = ( LHyp ‘ 𝐾 )
6 cdlemc3.t ⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 )
7 cdlemc3.r ⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 )
8 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐾 ∈ HL )
9 simp23l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑄 ∈ 𝐴 )
10 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
11 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐹 ∈ 𝑇 )
12 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
13 10 11 9 12 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
14 1 2 4 hlatlej2 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
15 8 9 13 14 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
16 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
17 1 2 4 5 6 7 trljat1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
18 10 11 16 17 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) = ( 𝑄 ∨ ( 𝐹 ‘ 𝑄 ) ) )
19 15 18 breqtrrd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) )
20 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
21 1 2 3 4 5 6 cdlemc2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) )
22 10 11 20 16 21 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) )
23 8 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐾 ∈ Lat )
24 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
25 24 4 atbase ⊢ ( 𝑄 ∈ 𝐴 → 𝑄 ∈ ( Base ‘ 𝐾 ) )
26 9 25 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑄 ∈ ( Base ‘ 𝐾 ) )
27 24 5 6 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
28 10 11 26 27 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
29 24 5 6 7 trlcl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
30 10 11 29 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) )
31 24 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ 𝑄 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑅 ‘ 𝐹 ) ∈ ( Base ‘ 𝐾 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) )
32 23 26 30 31 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) )
33 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ∈ 𝐴 )
34 24 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
35 33 34 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
36 24 5 6 ltrncl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ ( Base ‘ 𝐾 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
37 10 11 35 36 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) )
38 24 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
39 8 33 9 38 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) )
40 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑊 ∈ 𝐻 )
41 24 5 lhpbase ⊢ ( 𝑊 ∈ 𝐻 → 𝑊 ∈ ( Base ‘ 𝐾 ) )
42 40 41 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑊 ∈ ( Base ‘ 𝐾 ) )
43 24 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
44 23 39 42 43 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) )
45 24 2 latjcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐹 ‘ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
46 23 37 44 45 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) )
47 24 1 3 latlem12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝐹 ‘ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝐹 ‘ 𝑄 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ↔ ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) )
48 23 28 32 46 47 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( ( 𝐹 ‘ 𝑄 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ↔ ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) )
49 19 22 48 mpbi2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )
50 hlatl ⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat )
51 8 50 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝐾 ∈ AtLat )
52 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 )
53 1 4 5 6 7 trlat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
54 10 20 11 52 53 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 )
55 1 5 6 7 trlle ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )
56 10 11 55 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 )
57 simp23r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ¬ 𝑄 ≤ 𝑊 )
58 nbrne2 ⊢ ( ( ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝑅 ‘ 𝐹 ) ≠ 𝑄 )
59 58 necomd ⊢ ( ( ( 𝑅 ‘ 𝐹 ) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝑄 ≠ ( 𝑅 ‘ 𝐹 ) )
60 56 57 59 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑄 ≠ ( 𝑅 ‘ 𝐹 ) )
61 eqid ⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 )
62 2 4 61 llni2 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑅 ‘ 𝐹 ) ∈ 𝐴 ) ∧ 𝑄 ≠ ( 𝑅 ‘ 𝐹 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( LLines ‘ 𝐾 ) )
63 8 9 54 60 62 syl31anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( LLines ‘ 𝐾 ) )
64 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
65 10 11 33 64 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
66 1 2 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
67 8 33 65 66 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
68 simp3l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) )
69 nbrne2 ⊢ ( ( 𝑃 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → 𝑃 ≠ 𝑄 )
70 67 68 69 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → 𝑃 ≠ 𝑄 )
71 1 2 3 4 5 lhpat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 )
72 10 20 9 70 71 syl112anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 )
73 24 1 3 latmle2 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑊 ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
74 23 39 42 73 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 )
75 1 4 5 6 ltrnel ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) )
76 75 simprd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) → ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 )
77 10 11 20 76 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 )
78 nbrne2 ⊢ ( ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) → ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≠ ( 𝐹 ‘ 𝑃 ) )
79 78 necomd ⊢ ( ( ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ≤ 𝑊 ∧ ¬ ( 𝐹 ‘ 𝑃 ) ≤ 𝑊 ) → ( 𝐹 ‘ 𝑃 ) ≠ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) )
80 74 77 79 syl2anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑃 ) ≠ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) )
81 2 4 61 llni2 ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( LLines ‘ 𝐾 ) )
82 8 65 72 80 81 syl31anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( LLines ‘ 𝐾 ) )
83 1 2 3 4 5 6 7 cdlemc4 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ≠ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) )
84 83 3adant3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ≠ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) )
85 24 3 latmcl ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ∈ ( Base ‘ 𝐾 ) )
86 23 32 46 85 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ∈ ( Base ‘ 𝐾 ) )
87 eqid ⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 )
88 24 1 87 4 atlen0 ⊢ ( ( ( 𝐾 ∈ AtLat ∧ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) ∧ ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) → ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ≠ ( 0. ‘ 𝐾 ) )
89 51 86 13 49 88 syl31anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ≠ ( 0. ‘ 𝐾 ) )
90 3 87 4 61 2llnmat ⊢ ( ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∈ ( LLines ‘ 𝐾 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∈ ( LLines ‘ 𝐾 ) ) ∧ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ≠ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ∧ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ≠ ( 0. ‘ 𝐾 ) ) ) → ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ∈ 𝐴 )
91 8 63 82 84 89 90 syl32anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ∈ 𝐴 )
92 1 4 atcmp ⊢ ( ( 𝐾 ∈ AtLat ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ∧ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ↔ ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) )
93 51 13 91 92 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( ( 𝐹 ‘ 𝑄 ) ≤ ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ↔ ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) ) )
94 49 93 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( ¬ 𝑄 ≤ ( 𝑃 ∨ ( 𝐹 ‘ 𝑃 ) ) ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑃 ) ) → ( 𝐹 ‘ 𝑄 ) = ( ( 𝑄 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∨ ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) ) ) )