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 |
|
dalem20.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
7 |
|
dalem20.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
8 |
|
dalem20.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
9 |
1 2 3 4 7
|
dalem18 |
⊢ ( 𝜑 → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 ) |
10 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 ) |
11 |
1 2 3 4 6 7 8
|
dalem19 |
⊢ ( ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ) → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) |
12 |
11
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) → ( ¬ 𝑐 ≤ 𝑌 → ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
13 |
12
|
ancld |
⊢ ( ( ( 𝜑 ∧ 𝑌 = 𝑍 ) ∧ 𝑐 ∈ 𝐴 ) → ( ¬ 𝑐 ≤ 𝑌 → ( ¬ 𝑐 ≤ 𝑌 ∧ ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) ) |
14 |
13
|
reximdva |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ∃ 𝑐 ∈ 𝐴 ¬ 𝑐 ≤ 𝑌 → ∃ 𝑐 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) ) |
15 |
10 14
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ∃ 𝑐 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
16 |
|
3anass |
⊢ ( ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) ) |
17 |
5 16
|
bitri |
⊢ ( 𝜓 ↔ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) ) |
18 |
17
|
2exbii |
⊢ ( ∃ 𝑐 ∃ 𝑑 𝜓 ↔ ∃ 𝑐 ∃ 𝑑 ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) ) |
19 |
|
r2ex |
⊢ ( ∃ 𝑐 ∈ 𝐴 ∃ 𝑑 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ↔ ∃ 𝑐 ∃ 𝑑 ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) ) |
20 |
|
r19.42v |
⊢ ( ∃ 𝑑 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ↔ ( ¬ 𝑐 ≤ 𝑌 ∧ ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
21 |
20
|
rexbii |
⊢ ( ∃ 𝑐 ∈ 𝐴 ∃ 𝑑 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ↔ ∃ 𝑐 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
22 |
18 19 21
|
3bitr2ri |
⊢ ( ∃ 𝑐 ∈ 𝐴 ( ¬ 𝑐 ≤ 𝑌 ∧ ∃ 𝑑 ∈ 𝐴 ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ↔ ∃ 𝑐 ∃ 𝑑 𝜓 ) |
23 |
15 22
|
sylib |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ∃ 𝑐 ∃ 𝑑 𝜓 ) |