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 |
|
dalem10.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dalem10.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
7 |
|
dalem10.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
8 |
|
dalem10.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
9 |
|
dalem10.x |
⊢ 𝑋 = ( 𝑌 ∧ 𝑍 ) |
10 |
|
dalem10.d |
⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) |
11 |
1
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
12 |
1 3 4
|
dalempjqeb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
13 |
1 4
|
dalemreb |
⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
14 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
15 |
14 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
16 |
11 12 13 15
|
syl3anc |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
17 |
1 3 4
|
dalemsjteb |
⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
1 4
|
dalemueb |
⊢ ( 𝜑 → 𝑈 ∈ ( Base ‘ 𝐾 ) ) |
19 |
14 2 3
|
latlej1 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑈 ∈ ( Base ‘ 𝐾 ) ) → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) |
20 |
11 17 18 19
|
syl3anc |
⊢ ( 𝜑 → ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) |
21 |
1 6
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
22 |
7 21
|
eqeltrrid |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) |
23 |
1
|
dalemzeo |
⊢ ( 𝜑 → 𝑍 ∈ 𝑂 ) |
24 |
14 6
|
lplnbase |
⊢ ( 𝑍 ∈ 𝑂 → 𝑍 ∈ ( Base ‘ 𝐾 ) ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → 𝑍 ∈ ( Base ‘ 𝐾 ) ) |
26 |
8 25
|
eqeltrrid |
⊢ ( 𝜑 → ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) |
27 |
14 2 5
|
latmlem12 |
⊢ ( ( 𝐾 ∈ Lat ∧ ( ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∈ ( Base ‘ 𝐾 ) ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∈ ( Base ‘ 𝐾 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ∈ ( Base ‘ 𝐾 ) ) ) → ( ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) ) ) |
28 |
11 12 22 17 26 27
|
syl122anc |
⊢ ( 𝜑 → ( ( ( 𝑃 ∨ 𝑄 ) ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( 𝑆 ∨ 𝑇 ) ≤ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) ) ) |
29 |
16 20 28
|
mp2and |
⊢ ( 𝜑 → ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) ≤ ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) ) |
30 |
7 8
|
oveq12i |
⊢ ( 𝑌 ∧ 𝑍 ) = ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) |
31 |
9 30
|
eqtri |
⊢ 𝑋 = ( ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ∧ ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) ) |
32 |
29 10 31
|
3brtr4g |
⊢ ( 𝜑 → 𝐷 ≤ 𝑋 ) |