Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemh.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdlemh.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdlemh.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdlemh.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdlemh.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdlemh.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdlemh.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
8 |
|
cdlemh.r |
⊢ 𝑅 = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
9 |
|
cdlemh.s |
⊢ 𝑆 = ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
10 |
|
simp1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ) |
11 |
|
simp21l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑃 ∈ 𝐴 ) |
12 |
|
simp22l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑄 ∈ 𝐴 ) |
13 |
|
simp23 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) |
14 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) |
15 |
1 2 3 4 5 6 7 8 9
|
cdlemh1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ) ∧ ( 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
16 |
10 11 12 13 14 15
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
17 |
|
oveq1 |
⊢ ( 𝑆 = ( 0. ‘ 𝐾 ) → ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( ( 0. ‘ 𝐾 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
18 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐾 ∈ HL ) |
19 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
20 |
18 19
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐾 ∈ OL ) |
21 |
|
simp11r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑊 ∈ 𝐻 ) |
22 |
18 21
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
23 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐺 ∈ 𝑇 ) |
24 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐹 ∈ 𝑇 ) |
25 |
6 7
|
ltrncnv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ◡ 𝐹 ∈ 𝑇 ) |
26 |
22 24 25
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ◡ 𝐹 ∈ 𝑇 ) |
27 |
23 26
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐺 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) ) |
28 |
14
|
necomd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ 𝐹 ) ) |
29 |
6 7 8
|
trlcnv |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( 𝑅 ‘ 𝐹 ) ) |
30 |
22 24 29
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ ◡ 𝐹 ) = ( 𝑅 ‘ 𝐹 ) ) |
31 |
28 30
|
neeqtrrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ ◡ 𝐹 ) ) |
32 |
5 6 7 8
|
trlcoat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) ∧ ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ ◡ 𝐹 ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 ) |
33 |
22 27 31 32
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 ) |
34 |
1 5
|
atbase |
⊢ ( ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐵 ) |
35 |
33 34
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐵 ) |
36 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
37 |
1 3 36
|
olj02 |
⊢ ( ( 𝐾 ∈ OL ∧ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐵 ) → ( ( 0. ‘ 𝐾 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) |
38 |
20 35 37
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 0. ‘ 𝐾 ) ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) |
39 |
17 38
|
sylan9eqr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) → ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) |
40 |
6 7
|
ltrnco |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) → ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 ) |
41 |
22 23 26 40
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 ) |
42 |
2 6 7 8
|
trlle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∘ ◡ 𝐹 ) ∈ 𝑇 ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≤ 𝑊 ) |
43 |
22 41 42
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≤ 𝑊 ) |
44 |
|
simp22r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ¬ 𝑄 ≤ 𝑊 ) |
45 |
|
nbrne2 |
⊢ ( ( ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ 𝑄 ) |
46 |
45
|
necomd |
⊢ ( ( ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≤ 𝑊 ∧ ¬ 𝑄 ≤ 𝑊 ) → 𝑄 ≠ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) |
47 |
43 44 46
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑄 ≠ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) |
48 |
|
eqid |
⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 ) |
49 |
3 5 48
|
llni2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 ) ∧ 𝑄 ≠ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) → ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ∈ ( LLines ‘ 𝐾 ) ) |
50 |
18 12 33 47 49
|
syl31anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ∈ ( LLines ‘ 𝐾 ) ) |
51 |
5 48
|
llnneat |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ∈ ( LLines ‘ 𝐾 ) ) → ¬ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ∈ 𝐴 ) |
52 |
18 50 51
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ¬ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ∈ 𝐴 ) |
53 |
|
nelne2 |
⊢ ( ( ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 ∧ ¬ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ∈ 𝐴 ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
54 |
33 52 53
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
55 |
54
|
adantr |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
56 |
39 55
|
eqnetrd |
⊢ ( ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) ∧ 𝑆 = ( 0. ‘ 𝐾 ) ) → ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
57 |
56
|
ex |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 = ( 0. ‘ 𝐾 ) → ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) |
58 |
57
|
necon2d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝑆 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) = ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) → 𝑆 ≠ ( 0. ‘ 𝐾 ) ) ) |
59 |
16 58
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑆 ≠ ( 0. ‘ 𝐾 ) ) |
60 |
|
simp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐺 ≠ ( I ↾ 𝐵 ) ) |
61 |
1 5 6 7 8
|
trlnidat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ) → ( 𝑅 ‘ 𝐺 ) ∈ 𝐴 ) |
62 |
22 23 60 61
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐺 ) ∈ 𝐴 ) |
63 |
2 3 5
|
hlatlej2 |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝑅 ‘ 𝐺 ) ∈ 𝐴 ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ) |
64 |
18 11 62 63
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ) |
65 |
|
simp22 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
66 |
|
simp31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝐹 ≠ ( I ↾ 𝐵 ) ) |
67 |
1 6 7
|
ltrncnvnid |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐹 ≠ ( I ↾ 𝐵 ) ) → ◡ 𝐹 ≠ ( I ↾ 𝐵 ) ) |
68 |
22 24 66 67
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ◡ 𝐹 ≠ ( I ↾ 𝐵 ) ) |
69 |
1 6 7 8
|
trlcone |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ ◡ 𝐹 ) ∧ ◡ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) |
70 |
69
|
necomd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ◡ 𝐹 ∈ 𝑇 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ≠ ( 𝑅 ‘ ◡ 𝐹 ) ∧ ◡ 𝐹 ≠ ( I ↾ 𝐵 ) ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ ( 𝑅 ‘ 𝐺 ) ) |
71 |
22 23 26 31 68 70
|
syl122anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ ( 𝑅 ‘ 𝐺 ) ) |
72 |
2 6 7 8
|
trlle |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ) → ( 𝑅 ‘ 𝐺 ) ≤ 𝑊 ) |
73 |
22 23 72
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑅 ‘ 𝐺 ) ≤ 𝑊 ) |
74 |
2 3 5 6
|
lhp2atnle |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≠ ( 𝑅 ‘ 𝐺 ) ) ∧ ( ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 ∧ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ≤ 𝑊 ) ∧ ( ( 𝑅 ‘ 𝐺 ) ∈ 𝐴 ∧ ( 𝑅 ‘ 𝐺 ) ≤ 𝑊 ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
75 |
22 65 71 33 43 62 73 74
|
syl322anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
76 |
|
nbrne1 |
⊢ ( ( ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
77 |
64 75 76
|
syl2anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) |
78 |
3 4 36 5
|
2atmat0 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝑅 ‘ 𝐺 ) ∈ 𝐴 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ∈ 𝐴 ∧ ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ≠ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ) → ( ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) = ( 0. ‘ 𝐾 ) ) ) |
79 |
18 11 62 12 33 77 78
|
syl33anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) = ( 0. ‘ 𝐾 ) ) ) |
80 |
9
|
eleq1i |
⊢ ( 𝑆 ∈ 𝐴 ↔ ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ∈ 𝐴 ) |
81 |
9
|
eqeq1i |
⊢ ( 𝑆 = ( 0. ‘ 𝐾 ) ↔ ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) = ( 0. ‘ 𝐾 ) ) |
82 |
80 81
|
orbi12i |
⊢ ( ( 𝑆 ∈ 𝐴 ∨ 𝑆 = ( 0. ‘ 𝐾 ) ) ↔ ( ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) ∈ 𝐴 ∨ ( ( 𝑃 ∨ ( 𝑅 ‘ 𝐺 ) ) ∧ ( 𝑄 ∨ ( 𝑅 ‘ ( 𝐺 ∘ ◡ 𝐹 ) ) ) ) = ( 0. ‘ 𝐾 ) ) ) |
83 |
79 82
|
sylibr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ∈ 𝐴 ∨ 𝑆 = ( 0. ‘ 𝐾 ) ) ) |
84 |
83
|
ord |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ¬ 𝑆 ∈ 𝐴 → 𝑆 = ( 0. ‘ 𝐾 ) ) ) |
85 |
84
|
necon1ad |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ≠ ( 0. ‘ 𝐾 ) → 𝑆 ∈ 𝐴 ) ) |
86 |
59 85
|
mpd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → 𝑆 ∈ 𝐴 ) |
87 |
|
simp21 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
88 |
87 65
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) |
89 |
1 2 3 4 5 6 7 8 9 36
|
cdlemh2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
90 |
88 89
|
syld3an2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) |
91 |
2 4 36 5 6
|
lhpmatb |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝑆 ∈ 𝐴 ) → ( ¬ 𝑆 ≤ 𝑊 ↔ ( 𝑆 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) ) |
92 |
18 21 86 91
|
syl21anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ¬ 𝑆 ≤ 𝑊 ↔ ( 𝑆 ∧ 𝑊 ) = ( 0. ‘ 𝐾 ) ) ) |
93 |
90 92
|
mpbird |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ¬ 𝑆 ≤ 𝑊 ) |
94 |
86 93
|
jca |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝑄 ≤ ( 𝑃 ∨ ( 𝑅 ‘ 𝐹 ) ) ) ∧ ( 𝐹 ≠ ( I ↾ 𝐵 ) ∧ 𝐺 ≠ ( I ↾ 𝐵 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( 𝑆 ∈ 𝐴 ∧ ¬ 𝑆 ≤ 𝑊 ) ) |