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
|
dalemkelat |
⊢ ( 𝜑 → 𝐾 ∈ Lat ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝐾 ∈ Lat ) |
15 |
5 4
|
dalemcceb |
⊢ ( 𝜓 → 𝑐 ∈ ( Base ‘ 𝐾 ) ) |
16 |
15
|
adantl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑐 ∈ ( Base ‘ 𝐾 ) ) |
17 |
1 3 4
|
dalempjqeb |
⊢ ( 𝜑 → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
18 |
17
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ) |
19 |
1 4
|
dalemreb |
⊢ ( 𝜑 → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝜓 ) → 𝑅 ∈ ( Base ‘ 𝐾 ) ) |
21 |
5
|
dalem-ccly |
⊢ ( 𝜓 → ¬ 𝑐 ≤ 𝑌 ) |
22 |
8
|
breq2i |
⊢ ( 𝑐 ≤ 𝑌 ↔ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
23 |
21 22
|
sylnib |
⊢ ( 𝜓 → ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
24 |
23
|
adantl |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) |
25 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
26 |
25 2 3
|
latnlej2l |
⊢ ( ( 𝐾 ∈ Lat ∧ ( 𝑐 ∈ ( Base ‘ 𝐾 ) ∧ ( 𝑃 ∨ 𝑄 ) ∈ ( Base ‘ 𝐾 ) ∧ 𝑅 ∈ ( Base ‘ 𝐾 ) ) ∧ ¬ 𝑐 ≤ ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) ) → ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ) |
27 |
14 16 18 20 24 26
|
syl131anc |
⊢ ( ( 𝜑 ∧ 𝜓 ) → ¬ 𝑐 ≤ ( 𝑃 ∨ 𝑄 ) ) |