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 ) ) ) |