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 |
|
dalem60.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
7 |
|
dalem60.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
8 |
|
dalem60.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
9 |
|
dalem60.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
10 |
|
dalem60.d |
⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) |
11 |
|
dalem60.e |
⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) ) |
12 |
|
dalem60.g |
⊢ 𝐺 = ( ( 𝑐 ∨ 𝑃 ) ∧ ( 𝑑 ∨ 𝑆 ) ) |
13 |
|
dalem60.h |
⊢ 𝐻 = ( ( 𝑐 ∨ 𝑄 ) ∧ ( 𝑑 ∨ 𝑇 ) ) |
14 |
|
dalem60.i |
⊢ 𝐼 = ( ( 𝑐 ∨ 𝑅 ) ∧ ( 𝑑 ∨ 𝑈 ) ) |
15 |
|
dalem60.b1 |
⊢ 𝐵 = ( ( ( 𝐺 ∨ 𝐻 ) ∨ 𝐼 ) ∧ 𝑌 ) |
16 |
1 2 3 4 5 6 7 8 9 10 12 13 14 15
|
dalem57 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ≤ 𝐵 ) |
17 |
1 2 3 4 5 6 7 8 9 11 12 13 14 15
|
dalem58 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐸 ≤ 𝐵 ) |
18 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
19 |
18
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ Lat ) |
20 |
1 2 3 4 6 7 8 9 10
|
dalemdea |
⊢ ( 𝜑 → 𝐷 ∈ 𝐴 ) |
21 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
22 |
21 4
|
atbase |
⊢ ( 𝐷 ∈ 𝐴 → 𝐷 ∈ ( Base ‘ 𝐾 ) ) |
23 |
20 22
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ( Base ‘ 𝐾 ) ) |
24 |
23
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐷 ∈ ( Base ‘ 𝐾 ) ) |
25 |
1 2 3 4 6 7 8 9 11
|
dalemeea |
⊢ ( 𝜑 → 𝐸 ∈ 𝐴 ) |
26 |
21 4
|
atbase |
⊢ ( 𝐸 ∈ 𝐴 → 𝐸 ∈ ( Base ‘ 𝐾 ) ) |
27 |
25 26
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ ( Base ‘ 𝐾 ) ) |
28 |
27
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐸 ∈ ( Base ‘ 𝐾 ) ) |
29 |
|
eqid |
⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 ) |
30 |
1 2 3 4 5 6 29 7 8 9 12 13 14 15
|
dalem53 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐵 ∈ ( LLines ‘ 𝐾 ) ) |
31 |
21 29
|
llnbase |
⊢ ( 𝐵 ∈ ( LLines ‘ 𝐾 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) ) |
32 |
30 31
|
syl |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐵 ∈ ( Base ‘ 𝐾 ) ) |
33 |
21 2 3
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐷 ∈ ( Base ‘ 𝐾 ) ∧ 𝐸 ∈ ( Base ‘ 𝐾 ) ∧ 𝐵 ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐷 ≤ 𝐵 ∧ 𝐸 ≤ 𝐵 ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ) ) |
34 |
19 24 28 32 33
|
syl13anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐷 ≤ 𝐵 ∧ 𝐸 ≤ 𝐵 ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ) ) |
35 |
16 17 34
|
mpbi2and |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ) |
36 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
37 |
36
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → 𝐾 ∈ HL ) |
38 |
1 2 3 4 6 7 8 9 10 11
|
dalemdnee |
⊢ ( 𝜑 → 𝐷 ≠ 𝐸 ) |
39 |
3 4 29
|
llni2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴 ) ∧ 𝐷 ≠ 𝐸 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ) |
40 |
36 20 25 38 39
|
syl31anc |
⊢ ( 𝜑 → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ) |
41 |
40
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ) |
42 |
2 29
|
llncmp |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ∧ 𝐵 ∈ ( LLines ‘ 𝐾 ) ) → ( ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ↔ ( 𝐷 ∨ 𝐸 ) = 𝐵 ) ) |
43 |
37 41 30 42
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( ( 𝐷 ∨ 𝐸 ) ≤ 𝐵 ↔ ( 𝐷 ∨ 𝐸 ) = 𝐵 ) ) |
44 |
35 43
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ 𝜓 ) → ( 𝐷 ∨ 𝐸 ) = 𝐵 ) |