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 |
|
simp11l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL ) |
9 |
|
hlop |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OP ) |
10 |
8 9
|
syl |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ OP ) |
11 |
8
|
hllatd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ Lat ) |
12 |
|
simp12l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 ) |
13 |
|
simp11 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
14 |
|
simp21 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 ) |
15 |
|
simp22 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 ) |
16 |
1 4 5 6
|
ltrncoat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑃 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) |
17 |
13 14 15 12 16
|
syl121anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
19 |
18 2 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
20 |
8 12 17 19
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
21 |
|
simp13l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 ) |
22 |
1 4 5 6
|
ltrncoat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) |
23 |
13 14 15 21 22
|
syl121anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) |
24 |
18 2 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
25 |
8 21 23 24
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
26 |
18 3
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ∈ ( Base ‘ 𝐾 ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
11 20 25 26
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ ( Base ‘ 𝐾 ) ) |
28 |
|
simp12 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
29 |
|
simp13 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
30 |
|
simp33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) |
31 |
1 2 3 4 5 6
|
cdlemg11a |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ≠ 𝑃 ) |
32 |
31
|
necomd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) |
33 |
13 28 29 14 15 30 32
|
syl123anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) |
34 |
1 2 3 4 5
|
lhpat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ∧ 𝑃 ≠ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ∈ 𝐴 ) |
35 |
13 28 17 33 34
|
syl112anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ∈ 𝐴 ) |
36 |
2 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑃 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ) |
37 |
8 12 17 36
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ) |
38 |
2 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) |
39 |
8 21 23 38
|
syl3anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) = ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) |
40 |
37 39
|
oveq12d |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) |
41 |
|
simp1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ) |
42 |
|
simp2 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ) |
43 |
|
simp31l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
44 |
|
simp31r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) |
45 |
|
simp32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) |
46 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
47 |
1 2 3 4 5 6 7 46
|
cdlemg12e |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ) ) → ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ≠ ( 0. ‘ 𝐾 ) ) |
48 |
41 42 43 44 45 47
|
syl113anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ≠ ( 0. ‘ 𝐾 ) ) |
49 |
40 48
|
eqnetrd |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≠ ( 0. ‘ 𝐾 ) ) |
50 |
1 2 3 4 5 6 7
|
cdlemg12f |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) |
51 |
18 1 46 4
|
leat2 |
⊢ ( ( ( 𝐾 ∈ OP ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ∈ 𝐴 ) ∧ ( ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≠ ( 0. ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ≤ ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) |
52 |
10 27 35 49 50 51
|
syl32anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ) ∧ ( ( ¬ ( 𝑅 ‘ 𝐹 ) ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( 𝑅 ‘ 𝐹 ) ≠ ( 𝑅 ‘ 𝐺 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ≠ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( 𝑄 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) = ( ( 𝑃 ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ 𝑊 ) ) |