Step |
Hyp |
Ref |
Expression |
1 |
|
dalem62.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 |
|
dalem62.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
3 |
|
dalem62.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
4 |
|
dalem62.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
5 |
|
dalem62.m |
⊢ ∧ = ( meet ‘ 𝐾 ) |
6 |
|
dalem62.o |
⊢ 𝑂 = ( LPlanes ‘ 𝐾 ) |
7 |
|
dalem62.y |
⊢ 𝑌 = ( ( 𝑃 ∨ 𝑄 ) ∨ 𝑅 ) |
8 |
|
dalem62.z |
⊢ 𝑍 = ( ( 𝑆 ∨ 𝑇 ) ∨ 𝑈 ) |
9 |
|
dalem62.d |
⊢ 𝐷 = ( ( 𝑃 ∨ 𝑄 ) ∧ ( 𝑆 ∨ 𝑇 ) ) |
10 |
|
dalem62.e |
⊢ 𝐸 = ( ( 𝑄 ∨ 𝑅 ) ∧ ( 𝑇 ∨ 𝑈 ) ) |
11 |
|
dalem62.f |
⊢ 𝐹 = ( ( 𝑅 ∨ 𝑃 ) ∧ ( 𝑈 ∨ 𝑆 ) ) |
12 |
|
biid |
⊢ ( ( ( 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 ) ) ) ) |
13 |
1 2 3 4 12 6 7 8
|
dalem20 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ∃ 𝑐 ∃ 𝑑 ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) |
14 |
1 2 3 4 12 5 6 7 8 9 10 11
|
dalem61 |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ∧ ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) ) |
15 |
14
|
3expia |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) ) ) |
16 |
15
|
exlimdvv |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → ( ∃ 𝑐 ∃ 𝑑 ( ( 𝑐 ∈ 𝐴 ∧ 𝑑 ∈ 𝐴 ) ∧ ¬ 𝑐 ≤ 𝑌 ∧ ( 𝑑 ≠ 𝑐 ∧ ¬ 𝑑 ≤ 𝑌 ∧ 𝐶 ≤ ( 𝑐 ∨ 𝑑 ) ) ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) ) ) |
17 |
13 16
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑌 = 𝑍 ) → 𝐹 ≤ ( 𝐷 ∨ 𝐸 ) ) |