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 |
|
dalem23.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
7 |
|
dalem23.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
8 |
|
dalem23.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
9 |
|
dalem23.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
10 |
|
dalem23.g |
⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) |
11 |
10
|
oveq1i |
⊢ ( 𝐺 ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ∧ 𝑌 ) |
12 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
13 |
|
hlol |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ OL ) |
14 |
12 13
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ OL ) |
15 |
14
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ OL ) |
16 |
12
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL ) |
17 |
5
|
dalemccea |
⊢ ( 𝜓 → 𝑐 ∈ 𝐴 ) |
18 |
17
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑐 ∈ 𝐴 ) |
19 |
1
|
dalempea |
⊢ ( 𝜑 → 𝑃 ∈ 𝐴 ) |
20 |
19
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ∈ 𝐴 ) |
21 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
22 |
21 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
16 18 20 22
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ) |
24 |
5
|
dalemddea |
⊢ ( 𝜓 → 𝑑 ∈ 𝐴 ) |
25 |
24
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑑 ∈ 𝐴 ) |
26 |
1
|
dalemsea |
⊢ ( 𝜑 → 𝑆 ∈ 𝐴 ) |
27 |
26
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ∈ 𝐴 ) |
28 |
21 3 4
|
hlatjcl |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
29 |
16 25 27 28
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ) |
30 |
1 7
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
31 |
30
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
32 |
21 6
|
latmmdir |
⊢ ( ( 𝐾 ∈ OL ∧ ( ( 𝑐 ∨ 𝑃 ) ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑑 ∨ 𝑆 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) ) |
33 |
15 23 29 31 32
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) ) |
34 |
11 33
|
syl5eq |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∧ 𝑌 ) = ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) ) |
35 |
3 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑐 ∈ 𝐴 ∧ 𝑃 ∈ 𝐴 ) → ( 𝑐 ∨ 𝑃 ) = ( 𝑃 ∨ 𝑐 ) ) |
36 |
16 18 20 35
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑐 ∨ 𝑃 ) = ( 𝑃 ∨ 𝑐 ) ) |
37 |
36
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) = ( ( 𝑃 ∨ 𝑐 ) ∧ 𝑌 ) ) |
38 |
1 2 3 4 7 8
|
dalemply |
⊢ ( 𝜑 → 𝑃 ≤ 𝑌 ) |
39 |
38
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑃 ≤ 𝑌 ) |
40 |
5
|
dalem-ccly |
⊢ ( 𝜓 → ¬ 𝑐 ≤ 𝑌 ) |
41 |
40
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑐 ≤ 𝑌 ) |
42 |
21 2 3 6 4
|
2atjm |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑃 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑃 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌 ) ) → ( ( 𝑃 ∨ 𝑐 ) ∧ 𝑌 ) = 𝑃 ) |
43 |
16 20 18 31 39 41 42
|
syl132anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑃 ∨ 𝑐 ) ∧ 𝑌 ) = 𝑃 ) |
44 |
37 43
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) = 𝑃 ) |
45 |
3 4
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑑 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑑 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑑 ) ) |
46 |
16 25 27 45
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑑 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑑 ) ) |
47 |
46
|
oveq1d |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) = ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) ) |
48 |
1 2 3 4 9
|
dalemsly |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝑆 ≤ 𝑌 ) |
49 |
48
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝑆 ≤ 𝑌 ) |
50 |
5
|
dalem-ddly |
⊢ ( 𝜓 → ¬ 𝑑 ≤ 𝑌 ) |
51 |
50
|
3ad2ant3 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝑑 ≤ 𝑌 ) |
52 |
21 2 3 6 4
|
2atjm |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑆 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) ∧ ( 𝑆 ≤ 𝑌 ∧ ¬ 𝑑 ≤ 𝑌 ) ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 ) |
53 |
16 27 25 31 49 51 52
|
syl132anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑆 ∨ 𝑑 ) ∧ 𝑌 ) = 𝑆 ) |
54 |
47 53
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) = 𝑆 ) |
55 |
44 54
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( ( 𝑐 ∨ 𝑃 ) ∧ 𝑌 ) ∧ ( ( 𝑑 ∨ 𝑆 ) ∧ 𝑌 ) ) = ( 𝑃 ∧ 𝑆 ) ) |
56 |
1 2 3 4 7 8
|
dalempnes |
⊢ ( 𝜑 → 𝑃 ≠ 𝑆 ) |
57 |
|
hlatl |
⊢ ( 𝐾 ∈ HL → 𝐾 ∈ AtLat ) |
58 |
12 57
|
syl |
⊢ ( 𝜑 → 𝐾 ∈ AtLat ) |
59 |
|
eqid |
⊢ ( 0. ‘ 𝐾 ) = ( 0. ‘ 𝐾 ) |
60 |
6 59 4
|
atnem0 |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝑃 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑃 ≠ 𝑆 ↔ ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) ) ) |
61 |
58 19 26 60
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ≠ 𝑆 ↔ ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) ) ) |
62 |
56 61
|
mpbid |
⊢ ( 𝜑 → ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) ) |
63 |
62
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝑃 ∧ 𝑆 ) = ( 0. ‘ 𝐾 ) ) |
64 |
34 55 63
|
3eqtrd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐺 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) |
65 |
58
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ AtLat ) |
66 |
1 2 3 4 5 6 7 8 9 10
|
dalem23 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐺 ∈ 𝐴 ) |
67 |
21 2 6 59 4
|
atnle |
⊢ ( ( 𝐾 ∈ AtLat ∧ 𝐺 ∈ 𝐴 ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ) → ( ¬ 𝐺 ≤ 𝑌 ↔ ( 𝐺 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) ) |
68 |
65 66 31 67
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ¬ 𝐺 ≤ 𝑌 ↔ ( 𝐺 ∧ 𝑌 ) = ( 0. ‘ 𝐾 ) ) ) |
69 |
64 68
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ¬ 𝐺 ≤ 𝑌 ) |