Step |
Hyp |
Ref |
Expression |
1 |
|
lpolsat.a |
⊢ 𝐴 = ( LSAtoms ‘ 𝑊 ) |
2 |
|
lpolsat.h |
⊢ 𝐻 = ( LSHyp ‘ 𝑊 ) |
3 |
|
lpolsat.p |
⊢ 𝑃 = ( LPol ‘ 𝑊 ) |
4 |
|
lpolsat.w |
⊢ ( 𝜑 → 𝑊 ∈ 𝑋 ) |
5 |
|
lpolsat.o |
⊢ ( 𝜑 → ⊥ ∈ 𝑃 ) |
6 |
|
lpolsat.q |
⊢ ( 𝜑 → 𝑄 ∈ 𝐴 ) |
7 |
|
eqid |
⊢ ( Base ‘ 𝑊 ) = ( Base ‘ 𝑊 ) |
8 |
|
eqid |
⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 ) |
9 |
|
eqid |
⊢ ( 0g ‘ 𝑊 ) = ( 0g ‘ 𝑊 ) |
10 |
7 8 9 1 2 3
|
islpolN |
⊢ ( 𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 ( Base ‘ 𝑊 ) ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ ( Base ‘ 𝑊 ) ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑦 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) ) |
11 |
4 10
|
syl |
⊢ ( 𝜑 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 ( Base ‘ 𝑊 ) ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ ( Base ‘ 𝑊 ) ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑦 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) ) |
12 |
5 11
|
mpbid |
⊢ ( 𝜑 → ( ⊥ : 𝒫 ( Base ‘ 𝑊 ) ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ ( Base ‘ 𝑊 ) ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑦 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) |
13 |
|
simpr3 |
⊢ ( ( ⊥ : 𝒫 ( Base ‘ 𝑊 ) ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ ( Base ‘ 𝑊 ) ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑦 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) → ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) |
14 |
|
fveq2 |
⊢ ( 𝑥 = 𝑄 → ( ⊥ ‘ 𝑥 ) = ( ⊥ ‘ 𝑄 ) ) |
15 |
14
|
eleq1d |
⊢ ( 𝑥 = 𝑄 → ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ↔ ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ) ) |
16 |
|
2fveq3 |
⊢ ( 𝑥 = 𝑄 → ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = ( ⊥ ‘ ( ⊥ ‘ 𝑄 ) ) ) |
17 |
|
id |
⊢ ( 𝑥 = 𝑄 → 𝑥 = 𝑄 ) |
18 |
16 17
|
eqeq12d |
⊢ ( 𝑥 = 𝑄 → ( ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑄 ) ) = 𝑄 ) ) |
19 |
15 18
|
anbi12d |
⊢ ( 𝑥 = 𝑄 → ( ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ↔ ( ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑄 ) ) = 𝑄 ) ) ) |
20 |
19
|
rspcv |
⊢ ( 𝑄 ∈ 𝐴 → ( ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) → ( ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑄 ) ) = 𝑄 ) ) ) |
21 |
6 20
|
syl |
⊢ ( 𝜑 → ( ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) → ( ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑄 ) ) = 𝑄 ) ) ) |
22 |
|
simpl |
⊢ ( ( ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑄 ) ) = 𝑄 ) → ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ) |
23 |
13 21 22
|
syl56 |
⊢ ( 𝜑 → ( ( ⊥ : 𝒫 ( Base ‘ 𝑊 ) ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ ( Base ‘ 𝑊 ) ) = { ( 0g ‘ 𝑊 ) } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑦 ⊆ ( Base ‘ 𝑊 ) ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) → ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ) ) |
24 |
12 23
|
mpd |
⊢ ( 𝜑 → ( ⊥ ‘ 𝑄 ) ∈ 𝐻 ) |