This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

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

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 ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) )
10 simp3r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
11 simp1l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐾 ∈ HL )
12 simp1r ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑊 ∈ 𝐻 )
13 simp21 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) )
14 simp22l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑄 ∈ 𝐴 )
15 simp3l1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ≠ 𝑄 )
16 1 2 3 4 5 8 cdleme0a ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ) → 𝑈 ∈ 𝐴 )
17 11 12 13 14 15 16 syl212anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑈 ∈ 𝐴 )
18 simp1 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) )
19 simp23 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐹 ∈ 𝑇 )
20 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
21 18 19 14 20 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 )
22 1 2 4 hlatlej1 ⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
23 11 17 21 22 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
24 11 hllatd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝐾 ∈ Lat )
25 simp21l ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ∈ 𝐴 )
26 eqid ⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 )
27 26 4 atbase ⊢ ( 𝑃 ∈ 𝐴 → 𝑃 ∈ ( Base ‘ 𝐾 ) )
28 25 27 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑃 ∈ ( Base ‘ 𝐾 ) )
29 26 4 atbase ⊢ ( 𝑈 ∈ 𝐴 → 𝑈 ∈ ( Base ‘ 𝐾 ) )
30 17 29 syl ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → 𝑈 ∈ ( Base ‘ 𝐾 ) )
31 26 2 4 hlatjcl ⊢ ( ( 𝐾 ∈ HL ∧ 𝑈 ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
32 11 17 21 31 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) )
33 26 1 2 latjle12 ⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑃 ∨ 𝑈 ) ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
34 24 28 30 32 33 syl13anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ∧ 𝑈 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ↔ ( 𝑃 ∨ 𝑈 ) ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
35 10 23 34 mpbi2and ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑈 ) ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
36 1 2 3 4 5 8 cdleme0cp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ 𝑄 ∈ 𝐴 ) ) → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) )
37 11 12 13 14 36 syl22anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑈 ) = ( 𝑃 ∨ 𝑄 ) )
38 simp22 ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) )
39 5 6 1 2 4 3 8 cdlemg2kq ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) )
40 18 13 38 19 39 syl121anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) )
41 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) = ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
42 11 21 17 41 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ 𝑈 ) = ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )
43 40 42 eqtr2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
44 35 37 43 3brtr3d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
45 1 4 5 6 ltrnat ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
46 18 19 25 45 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 )
47 1 2 4 ps-1 ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) ) → ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
48 11 25 14 15 46 21 47 syl132anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ↔ ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
49 44 48 mpbid ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( 𝑃 ∨ 𝑄 ) = ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) )
50 2 4 hlatjcom ⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐹 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ 𝑄 ) ∈ 𝐴 ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
51 11 46 21 50 syl3anc ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑃 ) ∨ ( 𝐹 ‘ 𝑄 ) ) = ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) )
52 49 51 eqtr2d ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) )
53 52 3exp ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) ) )
54 53 exp4a ⊢ ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) → ( ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) → ( ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) → ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) ) ) )
55 54 3imp ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) → ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) = ( 𝑃 ∨ 𝑄 ) ) )
56 55 necon3ad ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) → ¬ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) ) )
57 9 56 mpd ⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝑃 ≠ 𝑄 ∧ ( 𝐹 ‘ 𝑃 ) ≠ 𝑄 ∧ ( ( 𝐹 ‘ 𝑄 ) ∨ ( 𝐹 ‘ 𝑃 ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ 𝑃 ≤ ( 𝑈 ∨ ( 𝐹 ‘ 𝑄 ) ) )