Step |
Hyp |
Ref |
Expression |
1 |
|
psubspset.l |
⊢ ≤ = ( le ‘ 𝐾 ) |
2 |
|
psubspset.j |
⊢ ∨ = ( join ‘ 𝐾 ) |
3 |
|
psubspset.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
4 |
|
psubspset.s |
⊢ 𝑆 = ( PSubSp ‘ 𝐾 ) |
5 |
1 2 3 4
|
ispsubsp |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) ) ) |
6 |
|
ralcom |
⊢ ( ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ∀ 𝑟 ∈ 𝑋 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
7 |
|
r19.23v |
⊢ ( ∀ 𝑟 ∈ 𝑋 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
8 |
7
|
ralbii |
⊢ ( ∀ 𝑝 ∈ 𝐴 ∀ 𝑟 ∈ 𝑋 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
9 |
6 8
|
bitri |
⊢ ( ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
10 |
9
|
ralbii |
⊢ ( ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑞 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
11 |
|
ralcom |
⊢ ( ∀ 𝑞 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝑋 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
12 |
|
r19.23v |
⊢ ( ∀ 𝑞 ∈ 𝑋 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
13 |
12
|
ralbii |
⊢ ( ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝑋 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
14 |
11 13
|
bitri |
⊢ ( ∀ 𝑞 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
15 |
10 14
|
bitri |
⊢ ( ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) |
16 |
15
|
a1i |
⊢ ( 𝐾 ∈ 𝐷 → ( ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ↔ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) ) |
17 |
16
|
anbi2d |
⊢ ( 𝐾 ∈ 𝐷 → ( ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑞 ∈ 𝑋 ∀ 𝑟 ∈ 𝑋 ∀ 𝑝 ∈ 𝐴 ( 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) ) ) |
18 |
5 17
|
bitrd |
⊢ ( 𝐾 ∈ 𝐷 → ( 𝑋 ∈ 𝑆 ↔ ( 𝑋 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝐴 ( ∃ 𝑞 ∈ 𝑋 ∃ 𝑟 ∈ 𝑋 𝑝 ≤ ( 𝑞 ∨ 𝑟 ) → 𝑝 ∈ 𝑋 ) ) ) ) |