Step |
Hyp |
Ref |
Expression |
1 |
|
lpolset.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lpolset.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lpolset.z |
⊢ 0 = ( 0g ‘ 𝑊 ) |
4 |
|
lpolset.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
5 |
|
lpolset.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
6 |
|
lpolset.p |
⊢ 𝑃 = ( LPol ‘ 𝑊 ) |
7 |
|
islpold.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝑋 ) |
8 |
|
islpold.1 |
⊢ ( 𝜑 → ⊥ : 𝒫 𝑉 ⟶ 𝑆 ) |
9 |
|
islpold.2 |
⊢ ( 𝜑 → ( ⊥ ‘ 𝑉 ) = { 0 } ) |
10 |
|
islpold.3 |
⊢ ( ( 𝜑 ∧ ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) |
11 |
|
islpold.4 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ) |
12 |
|
islpold.5 |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) |
13 |
10
|
ex |
⊢ ( 𝜑 → ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ) |
14 |
13
|
alrimivv |
⊢ ( 𝜑 → ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ) |
15 |
11 12
|
jca |
⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) |
16 |
15
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) |
17 |
9 14 16
|
3jca |
⊢ ( 𝜑 → ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) |
18 |
1 2 3 4 5 6
|
islpolN |
⊢ ( 𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) ) |
19 |
7 18
|
syl |
⊢ ( 𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) ) |
20 |
8 17 19
|
mpbir2and |
⊢ ( 𝜑 → ⊥ ∈ 𝑃 ) |