Step |
Hyp |
Ref |
Expression |
1 |
|
dalema.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 |
|
dalemc.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalemc.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalemc.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem16.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dalem16.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
7 |
|
dalem16.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
8 |
|
dalem16.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
9 |
|
dalem16.d |
⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) |
10 |
|
dalem16.e |
⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) ) |
11 |
|
dalem16.f |
⊢ 𝐹 = ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑆 ) ) |
12 |
|
eqid |
⊢ ( 𝑌 ∧ 𝑍 ) = ( 𝑌 ∧ 𝑍 ) |
13 |
1 2 3 4 5 6 7 8 12 11
|
dalem12 |
⊢ ( 𝜑 → 𝐹 ≤ ( 𝑌 ∧ 𝑍 ) ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐹 ≤ ( 𝑌 ∧ 𝑍 ) ) |
15 |
1 2 3 4 5 6 7 8 12 9
|
dalem10 |
⊢ ( 𝜑 → 𝐷 ≤ ( 𝑌 ∧ 𝑍 ) ) |
16 |
1 2 3 4 5 6 7 8 12 10
|
dalem11 |
⊢ ( 𝜑 → 𝐸 ≤ ( 𝑌 ∧ 𝑍 ) ) |
17 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
18 |
1 2 3 4 5 6 7 8 9
|
dalemdea |
⊢ ( 𝜑 → 𝐷 ∈ 𝐴 ) |
19 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
20 |
19 4
|
atbase |
⊢ ( 𝐷 ∈ 𝐴 → 𝐷 ∈ ( Base ‘ 𝐾 ) ) |
21 |
18 20
|
syl |
⊢ ( 𝜑 → 𝐷 ∈ ( Base ‘ 𝐾 ) ) |
22 |
1 2 3 4 5 6 7 8 10
|
dalemeea |
⊢ ( 𝜑 → 𝐸 ∈ 𝐴 ) |
23 |
19 4
|
atbase |
⊢ ( 𝐸 ∈ 𝐴 → 𝐸 ∈ ( Base ‘ 𝐾 ) ) |
24 |
22 23
|
syl |
⊢ ( 𝜑 → 𝐸 ∈ ( Base ‘ 𝐾 ) ) |
25 |
1 6
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
26 |
1
|
dalemzeo |
⊢ ( 𝜑 → 𝑍 ∈ 𝑂 ) |
27 |
19 6
|
lplnbase |
⊢ ( 𝑍 ∈ 𝑂 → 𝑍 ∈ ( Base ‘ 𝐾 ) ) |
28 |
26 27
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐾 ) ) |
29 |
19 5
|
latmcl |
⊢ ( ( 𝐾 ∈ Lat ∧ 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝑍 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑌 ∧ 𝑍 ) ∈ ( Base ‘ 𝐾 ) ) |
30 |
17 25 28 29
|
syl3anc |
⊢ ( 𝜑 → ( 𝑌 ∧ 𝑍 ) ∈ ( Base ‘ 𝐾 ) ) |
31 |
19 2 3
|
latjle12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝐷 ∈ ( Base ‘ 𝐾 ) ∧ 𝐸 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑌 ∧ 𝑍 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( 𝐷 ≤ ( 𝑌 ∧ 𝑍 ) ∧ 𝐸 ≤ ( 𝑌 ∧ 𝑍 ) ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ) ) |
32 |
17 21 24 30 31
|
syl13anc |
⊢ ( 𝜑 → ( ( 𝐷 ≤ ( 𝑌 ∧ 𝑍 ) ∧ 𝐸 ≤ ( 𝑌 ∧ 𝑍 ) ) ↔ ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ) ) |
33 |
15 16 32
|
mpbi2and |
⊢ ( 𝜑 → ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ) |
34 |
33
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ) |
35 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
36 |
35
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐾 ∈ HL ) |
37 |
1 2 3 4 5 6 7 8 9 10
|
dalemdnee |
⊢ ( 𝜑 → 𝐷 ≠ 𝐸 ) |
38 |
|
eqid |
⊢ ( LLines ‘ 𝐾 ) = ( LLines ‘ 𝐾 ) |
39 |
3 4 38
|
llni2 |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐷 ∈ 𝐴 ∧ 𝐸 ∈ 𝐴 ) ∧ 𝐷 ≠ 𝐸 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ) |
40 |
35 18 22 37 39
|
syl31anc |
⊢ ( 𝜑 → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ) |
41 |
40
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ) |
42 |
1 2 3 4 5 38 6 7 8 12
|
dalem15 |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝑌 ∧ 𝑍 ) ∈ ( LLines ‘ 𝐾 ) ) |
43 |
2 38
|
llncmp |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝐷 ∨ 𝐸 ) ∈ ( LLines ‘ 𝐾 ) ∧ ( 𝑌 ∧ 𝑍 ) ∈ ( LLines ‘ 𝐾 ) ) → ( ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ↔ ( 𝐷 ∨ 𝐸 ) = ( 𝑌 ∧ 𝑍 ) ) ) |
44 |
36 41 42 43
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( ( 𝐷 ∨ 𝐸 ) ≤ ( 𝑌 ∧ 𝑍 ) ↔ ( 𝐷 ∨ 𝐸 ) = ( 𝑌 ∧ 𝑍 ) ) ) |
45 |
34 44
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → ( 𝐷 ∨ 𝐸 ) = ( 𝑌 ∧ 𝑍 ) ) |
46 |
14 45
|
breqtrrd |
⊢ ( ( 𝜑 ∧ 𝑌 ≠ 𝑍 ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) ) |