Step |
Hyp |
Ref |
Expression |
1 |
|
lpolf.v |
⊢ 𝑉 = ( Base ‘ 𝑊 ) |
2 |
|
lpolf.s |
⊢ 𝑆 = ( LSubSp ‘ 𝑊 ) |
3 |
|
lpolf.p |
⊢ 𝑃 = ( LPol ‘ 𝑊 ) |
4 |
|
lpolf.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝑋 ) |
5 |
|
lpolf.o |
⊢ ( 𝜑 → ⊥ ∈ 𝑃 ) |
6 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
7 |
|
eqid |
⊢ ( LSAtoms ‘ 𝑊 ) = ( LSAtoms ‘ 𝑊 ) |
8 |
|
eqid |
⊢ ( LSHyp ‘ 𝑊 ) = ( LSHyp ‘ 𝑊 ) |
9 |
1 2 6 7 8 3
|
islpolN |
⊢ ( 𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ ( LSAtoms ‘ 𝑊 ) ( ( ⊥ ‘ 𝑥 ) ∈ ( LSHyp ‘ 𝑊 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) ) |
10 |
4 9
|
syl |
⊢ ( 𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ ( LSAtoms ‘ 𝑊 ) ( ( ⊥ ‘ 𝑥 ) ∈ ( LSHyp ‘ 𝑊 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) ) |
11 |
5 10
|
mpbid |
⊢ ( 𝜑 → ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ ( LSAtoms ‘ 𝑊 ) ( ( ⊥ ‘ 𝑥 ) ∈ ( LSHyp ‘ 𝑊 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) |
12 |
11
|
simpld |
⊢ ( 𝜑 → ⊥ : 𝒫 𝑉 ⟶ 𝑆 ) |