Step |
Hyp |
Ref |
Expression |
1 |
|
cdlemg6.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
cdlemg6.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
3 |
|
cdlemg6.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
4 |
|
cdlemg6.t |
⊢ 𝑇 = ( ( LTrn ‘ 𝐾 ) ‘ 𝑊 ) |
5 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
6 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
7 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
8 |
|
simpl31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → 𝐹 ∈ 𝑇 ) |
9 |
|
simpl32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → 𝐺 ∈ 𝑇 ) |
10 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) |
11 |
|
simpl33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) |
12 |
|
eqid |
⊢ ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) = ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) |
13 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
14 |
|
eqid |
⊢ ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) = ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) |
15 |
1 2 3 4 12 13 14
|
cdlemg6e |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) |
16 |
5 6 7 8 9 10 11 15
|
syl133anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) |
17 |
|
simpl1 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ) |
18 |
|
simpl2l |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ) |
19 |
|
simpl2r |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) |
20 |
|
simpl31 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → 𝐹 ∈ 𝑇 ) |
21 |
|
simpl32 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → 𝐺 ∈ 𝑇 ) |
22 |
|
simpr |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) |
23 |
|
simpl33 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) |
24 |
1 2 3 4 12 13 14
|
cdlemg4 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ∧ 𝐹 ∈ 𝑇 ) ∧ ( 𝐺 ∈ 𝑇 ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) |
25 |
17 18 19 20 21 22 23 24
|
syl133anc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) ∧ ¬ 𝑄 ≤ ( 𝑃 ( join ‘ 𝐾 ) ( ( ( trL ‘ 𝐾 ) ‘ 𝑊 ) ‘ 𝐺 ) ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) |
26 |
16 25
|
pm2.61dan |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝐹 ∈ 𝑇 ∧ 𝐺 ∈ 𝑇 ∧ ( 𝐹 ‘ ( 𝐺 ‘ 𝑃 ) ) = 𝑃 ) ) → ( 𝐹 ‘ ( 𝐺 ‘ 𝑄 ) ) = 𝑄 ) |