Step |
Hyp |
Ref |
Expression |
1 |
|
cdleme24.b |
⊢ 𝐵 = ( Base ‘ 𝐾 ) |
2 |
|
cdleme24.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
cdleme24.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
cdleme24.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
5 |
|
cdleme24.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
6 |
|
cdleme24.h |
⊢ 𝐻 = ( LHyp ‘ 𝐾 ) |
7 |
|
cdleme24.u |
⊢ 𝑈 = ( ( 𝑃 ∨ 𝑄 ) ∧ 𝑊 ) |
8 |
|
cdleme24.f |
⊢ 𝐹 = ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
9 |
|
cdleme24.n |
⊢ 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) ) |
10 |
1 2 3 4 5 6 7 8 9
|
cdleme25a |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑁 ∈ 𝐵 ) ) |
11 |
|
eqid |
⊢ ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
12 |
|
eqid |
⊢ ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
13 |
1 2 3 4 5 6 7 8 9 11 12
|
cdleme24 |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ) |
14 |
|
breq1 |
⊢ ( 𝑠 = 𝑡 → ( 𝑠 ≤ 𝑊 ↔ 𝑡 ≤ 𝑊 ) ) |
15 |
14
|
notbid |
⊢ ( 𝑠 = 𝑡 → ( ¬ 𝑠 ≤ 𝑊 ↔ ¬ 𝑡 ≤ 𝑊 ) ) |
16 |
|
breq1 |
⊢ ( 𝑠 = 𝑡 → ( 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
17 |
16
|
notbid |
⊢ ( 𝑠 = 𝑡 → ( ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ↔ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) |
18 |
15 17
|
anbi12d |
⊢ ( 𝑠 = 𝑡 → ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ↔ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) ) |
19 |
|
oveq1 |
⊢ ( 𝑠 = 𝑡 → ( 𝑠 ∨ 𝑈 ) = ( 𝑡 ∨ 𝑈 ) ) |
20 |
|
oveq2 |
⊢ ( 𝑠 = 𝑡 → ( 𝑃 ∨ 𝑠 ) = ( 𝑃 ∨ 𝑡 ) ) |
21 |
20
|
oveq1d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) = ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) |
22 |
21
|
oveq2d |
⊢ ( 𝑠 = 𝑡 → ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) = ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
23 |
19 22
|
oveq12d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝑠 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) |
24 |
8 23
|
syl5eq |
⊢ ( 𝑠 = 𝑡 → 𝐹 = ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) |
25 |
|
oveq2 |
⊢ ( 𝑠 = 𝑡 → ( 𝑅 ∨ 𝑠 ) = ( 𝑅 ∨ 𝑡 ) ) |
26 |
25
|
oveq1d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) = ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) |
27 |
24 26
|
oveq12d |
⊢ ( 𝑠 = 𝑡 → ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) = ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) |
28 |
27
|
oveq2d |
⊢ ( 𝑠 = 𝑡 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝐹 ∨ ( ( 𝑅 ∨ 𝑠 ) ∧ 𝑊 ) ) ) = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) |
29 |
9 28
|
syl5eq |
⊢ ( 𝑠 = 𝑡 → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) |
30 |
18 29
|
reusv3 |
⊢ ( ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑁 ∈ 𝐵 ) → ( ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) ↔ ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) ) |
31 |
30
|
biimpd |
⊢ ( ∃ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ 𝑁 ∈ 𝐵 ) → ( ∀ 𝑠 ∈ 𝐴 ∀ 𝑡 ∈ 𝐴 ( ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) ∧ ( ¬ 𝑡 ≤ 𝑊 ∧ ¬ 𝑡 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → 𝑁 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( ( ( 𝑡 ∨ 𝑈 ) ∧ ( 𝑄 ∨ ( ( 𝑃 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ∨ ( ( 𝑅 ∨ 𝑡 ) ∧ 𝑊 ) ) ) ) → ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) ) |
32 |
10 13 31
|
sylc |
⊢ ( ( ( ( 𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻 ) ∧ ( 𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊 ) ∧ ( 𝑄 ∈ 𝐴 ∧ ¬ 𝑄 ≤ 𝑊 ) ) ∧ ( 𝑅 ∈ 𝐴 ∧ ¬ 𝑅 ≤ 𝑊 ) ∧ ( 𝑃 ≠ 𝑄 ∧ 𝑅 ≤ ( 𝑃 ∨ 𝑄 ) ) ) → ∃ 𝑢 ∈ 𝐵 ∀ 𝑠 ∈ 𝐴 ( ( ¬ 𝑠 ≤ 𝑊 ∧ ¬ 𝑠 ≤ ( 𝑃 ∨ 𝑄 ) ) → 𝑢 = 𝑁 ) ) |