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 |
1 2 3 4 5 6 7
|
cdlemg12b |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) |
9 |
|
simp1l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐾 ∈ HL ) |
10 |
|
simp21l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑃 ∈ 𝐴 ) |
11 |
|
simp1 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
12 |
|
simp31 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐺 ∈ 𝑇 ) |
13 |
1 4 5 6
|
ltrnat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑃 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) |
14 |
11 12 10 13
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) |
15 |
|
simp23 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝐹 ∈ 𝑇 ) |
16 |
1 4 5 6
|
ltrnat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐹 ∈ 𝑇 ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) |
17 |
11 15 14 16
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) |
18 |
|
simp22l |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑄 ∈ 𝐴 ) |
19 |
1 4 5 6
|
ltrnat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ 𝐺 ∈ 𝑇 ∧ 𝑄 ∈ 𝐴 ) → ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ) |
20 |
11 12 18 19
|
syl3anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ) |
21 |
1 4 5 6
|
ltrncoat |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ) ∧ 𝑄 ∈ 𝐴 ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) |
22 |
11 15 12 18 21
|
syl121anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) |
23 |
1 2 3 4
|
dalaw |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑃 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∈ 𝐴 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ( 𝐺 ‘ 𝑄 ) ∈ 𝐴 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∈ 𝐴 ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) ) ) |
24 |
9 10 14 17 18 20 22 23
|
syl133anc |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) ) ) |
25 |
8 24
|
mpd |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑃 ≠ 𝑄 ∧ ¬ ( 𝑅 ‘ 𝐺 ) ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ( ( 𝑃 ∨ ( 𝐺 ‘ 𝑃 ) ) ∧ ( 𝑄 ∨ ( 𝐺 ‘ 𝑄 ) ) ) ≤ ( ( ( ( 𝐺 ‘ 𝑃 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ) ∧ ( ( 𝐺 ‘ 𝑄 ) ∨ ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ) ) ∨ ( ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) ∨ 𝑃 ) ∧ ( ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) ∨ 𝑄 ) ) ) ) |