Theorem lpolvN

Description: The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lpolv.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lpolv.z	⊢ 0 = ( 0g ‘ 𝑊 )
	lpolv.p	⊢ 𝑃 = ( LPol ‘ 𝑊 )
	lpolv.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
	lpolv.o	⊢ ( 𝜑 → ⊥ ∈ 𝑃 )
Assertion	lpolvN	⊢ ( 𝜑 → ( ⊥ ‘ 𝑉 ) = { 0 } )

Proof

Step	Hyp	Ref	Expression
1		lpolv.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lpolv.z	⊢ 0 = ( 0g ‘ 𝑊 )
3		lpolv.p	⊢ 𝑃 = ( LPol ‘ 𝑊 )
4		lpolv.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
5		lpolv.o	⊢ ( 𝜑 → ⊥ ∈ 𝑃 )
6		eqid	⊢ ( LSubSp ‘ 𝑊 ) = ( LSubSp ‘ 𝑊 )
7		eqid	⊢ ( LSAtoms ‘ 𝑊 ) = ( LSAtoms ‘ 𝑊 )
8		eqid	⊢ ( LSHyp ‘ 𝑊 ) = ( LSHyp ‘ 𝑊 )
9	1 6 2 7 8 3	islpolN	⊢ ( 𝑊 ∈ 𝑋 → ( ⊥ ∈ 𝑃 ↔ ( ⊥ : 𝒫 𝑉 ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 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 ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 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 ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ ( LSAtoms ‘ 𝑊 ) ( ( ⊥ ‘ 𝑥 ) ∈ ( LSHyp ‘ 𝑊 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) )
12		simpr1	⊢ ( ( ⊥ : 𝒫 𝑉 ⟶ ( LSubSp ‘ 𝑊 ) ∧ ( ( ⊥ ‘ 𝑉 ) = { 0 } ∧ ∀ 𝑥 ∀ 𝑦 ( ( 𝑥 ⊆ 𝑉 ∧ 𝑦 ⊆ 𝑉 ∧ 𝑥 ⊆ 𝑦 ) → ( ⊥ ‘ 𝑦 ) ⊆ ( ⊥ ‘ 𝑥 ) ) ∧ ∀ 𝑥 ∈ ( LSAtoms ‘ 𝑊 ) ( ( ⊥ ‘ 𝑥 ) ∈ ( LSHyp ‘ 𝑊 ) ∧ ( ⊥ ‘ ( ⊥ ‘ 𝑥 ) ) = 𝑥 ) ) ) → ( ⊥ ‘ 𝑉 ) = { 0 } )
13	11 12	syl	⊢ ( 𝜑 → ( ⊥ ‘ 𝑉 ) = { 0 } )