Metamath Proof Explorer

Theorem islpolN

Description: The predicate "is a polarity". (Contributed by NM, 24-Nov-2014) (New usage is discouraged.)

Ref Expression
Hypotheses lpolset.v ⊢ 𝑉 = ( Base ‘ 𝑊 )
lpolset.s ⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
lpolset.z ⊢ 0 = ( 0g ‘ 𝑊 )
lpolset.a ⊢ 𝐴 = ( LSAtoms ‘ 𝑊 )
lpolset.h ⊢ 𝐻 = ( LSHyp ‘ 𝑊 )
lpolset.p ⊢ 𝑃 = ( LPol ‘ 𝑊 )
Assertion 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 ) ) ) ) )

Proof

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 1 2 3 4 5 6 lpolsetN ⊢ ( 𝑊 ∈ 𝑋 → 𝑃 = { 𝑜 ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ∣ ( ( 𝑜 ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ) } )
8 7 eleq2d ⊢ ( 𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ⊥ ∈ { 𝑜 ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ∣ ( ( 𝑜 ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ) } ) )
9 fveq1 ⊢ ( 𝑜 = ⊥ → ( 𝑜 ‘ 𝑉 ) = ( ⊥ ‘ 𝑉 ) )
10 9 eqeq1d ⊢ ( 𝑜 = ⊥ → ( ( 𝑜 ‘ 𝑉 ) = { 0 } ↔ ( ⊥ ‘ 𝑉 ) = { 0 } ) )
11 fveq1 ⊢ ( 𝑜 = ⊥ → ( 𝑜 ‘ 𝑦 ) = ( ⊥ ‘ 𝑦 ) )
12 fveq1 ⊢ ( 𝑜 = ⊥ → ( 𝑜 ‘ 𝑥 ) = ( ⊥ ‘ 𝑥 ) )
13 11 12 sseq12d ⊢ ( 𝑜 = ⊥ → ( ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ↔ ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) )
14 13 imbi2d ⊢ ( 𝑜 = ⊥ → ( ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ↔ ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ) )
15 14 2albidv ⊢ ( 𝑜 = ⊥ → ( ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ↔ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ) )
16 12 eleq1d ⊢ ( 𝑜 = ⊥ → ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ↔ ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ) )
17 id ⊢ ( 𝑜 = ⊥ → 𝑜 = ⊥ )
18 17 12 fveq12d ⊢ ( 𝑜 = ⊥ → ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) )
19 18 eqeq1d ⊢ ( 𝑜 = ⊥ → ( ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ↔ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) )
20 16 19 anbi12d ⊢ ( 𝑜 = ⊥ → ( ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ↔ ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) )
21 20 ralbidv ⊢ ( 𝑜 = ⊥ → ( ∀ 𝑥 ∈ 𝐴 ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ↔ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) )
22 10 15 21 3anbi123d ⊢ ( 𝑜 = ⊥ → ( ( ( 𝑜 ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ) ↔ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) )
23 22 elrab ⊢ ( ⊥ ∈ { 𝑜 ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ∣ ( ( 𝑜 ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ) } ↔ ( ⊥ ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) )
24 2 fvexi ⊢ 𝑆 ∈ V
25 1 fvexi ⊢ 𝑉 ∈ V
26 25 pwex ⊢ 𝒫 𝑉 ∈ V
27 24 26 elmap ⊢ ( ⊥ ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ↔ ⊥ : 𝒫 𝑉 ⟶ 𝑆 )
28 27 anbi1i ⊢ ( ( ⊥ ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) )
29 23 28 bitri ⊢ ( ⊥ ∈ { 𝑜 ∈ ( 𝑆 ↑m 𝒫 𝑉 ) ∣ ( ( 𝑜 ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( 𝑜 ‘ 𝑦 ) ⊆ ( 𝑜 ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( 𝑜 ‘ 𝑥 ) ∈ 𝐻 ∧ ( 𝑜 ‘ ( 𝑜 ‘ 𝑥 ) ) = 𝑥 ) ) } ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) )
30 8 29 bitrdi ⊢ ( 𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 𝑉 ⟶ 𝑆 ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ 𝐴 ( ( ⊥ ‘ 𝑥 ) ∈ 𝐻 ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) ) )