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 |
|
dalem19.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
6 |
|
dalem19.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
7 |
|
dalem19.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
8 |
1
|
dalemkehl |
⊢ ( 𝜑 → 𝐾 ∈ HL ) |
9 |
8
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝐾 ∈ HL ) |
10 |
1 2 3 4 5 6
|
dalemcea |
⊢ ( 𝜑 → 𝐶 ∈ 𝐴 ) |
11 |
10
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝐶 ∈ 𝐴 ) |
12 |
|
simplr |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝑐 ∈ 𝐴 ) |
13 |
1 5
|
dalemyeb |
⊢ ( 𝜑 → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
14 |
13
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝑌 ∈ ( Base ‘ 𝐾 ) ) |
15 |
1 2 3 4 5 6 7
|
dalem17 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐶 ≤ 𝑌 ) |
16 |
15
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → 𝐶 ≤ 𝑌 ) |
17 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → ¬ 𝑐 ≤ 𝑌 ) |
18 |
|
eqid |
⊢ ( Base ‘ 𝐾 ) = ( Base ‘ 𝐾 ) |
19 |
18 2 3 4
|
atbtwnex |
⊢ ( ( ( 𝐾 ∈ HL ∧ 𝐶 ∈ 𝐴 ∧ 𝑐 ∈ 𝐴 ) ∧ ( 𝑌 ∈ ( Base ‘ 𝐾 ) ∧ 𝐶 ≤ 𝑌 ∧ ¬ 𝑐 ≤ 𝑌 ) ) → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) |
20 |
9 11 12 14 16 17 19
|
syl33anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) |