Step |
Hyp |
Ref |
Expression |
1 |
|
dalem.ph |
⊢ ( 𝜑 ↔ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑇 ∈ 𝐴 ∧ 𝑈 ∈ 𝐴 ) ) ∧ ( 𝑌 ∈ 𝑂 ∧ 𝑍 ∈ 𝑂 ) ∧ ( ( ¬ 𝐶 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝐶 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝐶 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( ¬ 𝐶 ≤ ( 𝑆 ∨ 𝑇 ) ∧ ¬ 𝐶 ≤ ( 𝑇 ∨ 𝑈 ) ∧ ¬ 𝐶 ≤ ( 𝑈 ∨ 𝑆 ) ) ∧ ( 𝐶 ≤ ( 𝑃 ∨ 𝑆 ) ∧ 𝐶 ≤ ( 𝑄 ∨ 𝑇 ) ∧ 𝐶 ≤ ( 𝑅 ∨ 𝑈 ) ) ) ) ) |
2 |
|
dalem.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalem.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalem.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem.ps |
⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
6 |
|
dalem44.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
7 |
|
dalem44.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
8 |
|
dalem44.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
9 |
|
dalem44.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
10 |
|
dalem44.g |
⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) |
11 |
|
dalem44.h |
⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) ) |
12 |
|
dalem44.i |
⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) ) |
13 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
14 |
13
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL ) |
15 |
5
|
dalemccea |
⊢ ( 𝜓 → 𝑐 ∈ 𝐴 ) |
16 |
15
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 ) |
17 |
14 16
|
jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ) ) |
18 |
1 2 3 4 5 6 7 8 9 10
|
dalem23 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 ) |
19 |
1 2 3 4 5 6 7 8 9 11
|
dalem29 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐻 ∈ 𝐴 ) |
20 |
1 2 3 4 5 6 7 8 9 12
|
dalem34 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐼 ∈ 𝐴 ) |
21 |
18 19 20
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ) |
22 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
23 |
1
|
dalemqea |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
24 |
1
|
dalemrea |
⊢ ( 𝜑 → 𝑅 ∈ 𝐴 ) |
25 |
22 23 24
|
3jca |
⊢ ( 𝜑 → ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) |
26 |
25
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) |
27 |
17 21 26
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) ) |
28 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem42 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ) |
29 |
1
|
dalemyeo |
⊢ ( 𝜑 → 𝑌 ∈ 𝑂 ) |
30 |
29
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑌 ∈ 𝑂 ) |
31 |
28 30
|
jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) ) |
32 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem45 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ) |
33 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem46 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ) |
34 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem47 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) |
35 |
32 33 34
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ) |
36 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem48 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ) |
37 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem49 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ) |
38 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem50 |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) |
39 |
36 37 38
|
3jca |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ) |
40 |
39
|
3adant2 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ) |
41 |
1 2 3 4 5 6 7 8 9 10
|
dalem27 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ) |
42 |
1 2 3 4 5 6 7 8 9 11
|
dalem32 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ) |
43 |
1 2 3 4 5 6 7 8 9 12
|
dalem36 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) |
44 |
41 42 43
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) |
45 |
35 40 44
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ∧ ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) ) |
46 |
27 31 45
|
3jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) ∧ ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) ∧ ( ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ∧ ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) ) ) |
47 |
1 2 3 4 5 6 7 8 9 10 11 12
|
dalem43 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ≠ 𝑌 ) |
48 |
46 47
|
jca |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ) ∧ ( 𝐺 ∈ 𝐴 ∧ 𝐻 ∈ 𝐴 ∧ 𝐼 ∈ 𝐴 ) ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ) ) ∧ ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∈ 𝑂 ∧ 𝑌 ∈ 𝑂 ) ∧ ( ( ¬ 𝑐 ≤ ( 𝐺 ∨ 𝐻 ) ∧ ¬ 𝑐 ≤ ( 𝐻 ∨ 𝐼 ) ∧ ¬ 𝑐 ≤ ( 𝐼 ∨ 𝐺 ) ) ∧ ( ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ∧ ¬ 𝑐 ≤ ( 𝑄 ∨ 𝑅 ) ∧ ¬ 𝑐 ≤ ( 𝑅 ∨ 𝑃 ) ) ∧ ( 𝑐 ≤ ( 𝐺 ∨ 𝑃 ) ∧ 𝑐 ≤ ( 𝐻 ∨ 𝑄 ) ∧ 𝑐 ≤ ( 𝐼 ∨ 𝑅 ) ) ) ) ∧ ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ≠ 𝑌 ) ) |