Step |
Hyp |
Ref |
Expression |
1 |
|
islpln2a.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
islpln2a.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
islpln2a.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
islpln2a.p |
⊢ 𝑃 = ( LPlanes ‘ 𝐾 ) |
5 |
|
islpln2a.y |
⊢ 𝑌 = ( ( 𝑄 ∨ 𝑅 ) ∨ 𝑆 ) |
6 |
1 2 3
|
3noncolr1N |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ( 𝑆 ≠ 𝑄 ∧ ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) ) |
7 |
6
|
simprd |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) → ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) |
8 |
7
|
3expia |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) → ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) ) |
9 |
1 2 3 4 5
|
islpln2ah |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 ↔ ( 𝑄 ≠ 𝑅 ∧ ¬ 𝑆 ≤ ( 𝑄 ∨ 𝑅 ) ) ) ) |
10 |
2 3
|
hlatjcom |
⊢ ( ( 𝐾 ∈ HL ∧ 𝑄 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑄 ) ) |
11 |
10
|
3adant3r2 |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑄 ∨ 𝑆 ) = ( 𝑆 ∨ 𝑄 ) ) |
12 |
11
|
breq2d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ↔ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) ) |
13 |
12
|
notbid |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ↔ ¬ 𝑅 ≤ ( 𝑆 ∨ 𝑄 ) ) ) |
14 |
8 9 13
|
3imtr4d |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ) → ( 𝑌 ∈ 𝑃 → ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ) ) |
15 |
14
|
3impia |
⊢ ( ( 𝐾 ∈ HL ∧ ( 𝑄 ∈ 𝐴 ∧ 𝑅 ∈ 𝐴 ∧ 𝑆 ∈ 𝐴 ) ∧ 𝑌 ∈ 𝑃 ) → ¬ 𝑅 ≤ ( 𝑄 ∨ 𝑆 ) ) |