Step |
Hyp |
Ref |
Expression |
1 |
|
atpsub.a |
⊢ 𝐴 = ( Atoms ‘ 𝐾 ) |
2 |
|
atpsub.s |
⊢ 𝑆 = ( PSubSp ‘ 𝐾 ) |
3 |
|
ssid |
⊢ 𝐴 ⊆ 𝐴 |
4 |
|
ax-1 |
⊢ ( 𝑟 ∈ 𝐴 → ( 𝑟 ( le ‘ 𝐾 ) ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) → 𝑟 ∈ 𝐴 ) ) |
5 |
4
|
rgen |
⊢ ∀ 𝑟 ∈ 𝐴 ( 𝑟 ( le ‘ 𝐾 ) ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) → 𝑟 ∈ 𝐴 ) |
6 |
5
|
rgen2w |
⊢ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ( le ‘ 𝐾 ) ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) → 𝑟 ∈ 𝐴 ) |
7 |
3 6
|
pm3.2i |
⊢ ( 𝐴 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ( le ‘ 𝐾 ) ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) → 𝑟 ∈ 𝐴 ) ) |
8 |
|
eqid |
⊢ ( le ‘ 𝐾 ) = ( le ‘ 𝐾 ) |
9 |
|
eqid |
⊢ ( join ‘ 𝐾 ) = ( join ‘ 𝐾 ) |
10 |
8 9 1 2
|
ispsubsp |
⊢ ( 𝐾 ∈ 𝑉 → ( 𝐴 ∈ 𝑆 ↔ ( 𝐴 ⊆ 𝐴 ∧ ∀ 𝑝 ∈ 𝐴 ∀ 𝑞 ∈ 𝐴 ∀ 𝑟 ∈ 𝐴 ( 𝑟 ( le ‘ 𝐾 ) ( 𝑝 ( join ‘ 𝐾 ) 𝑞 ) → 𝑟 ∈ 𝐴 ) ) ) ) |
11 |
7 10
|
mpbiri |
⊢ ( 𝐾 ∈ 𝑉 → 𝐴 ∈ 𝑆 ) |