Theorem lpolfN

Description: Functionality of a polarity. (Contributed by NM, 26-Nov-2014) (New usage is discouraged.)

	Ref	Expression
Hypotheses	lpolf.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lpolf.s	⊢ 𝑆 = ( LSubSp ‘ 𝑊 )
	lpolf.p	⊢ 𝑃 = ( LPol ‘ 𝑊 )
	lpolf.w	⊢ ( 𝜑 → 𝑊 ∈ 𝑋 )
	lpolf.o	⊢ ( 𝜑 → ⊥ ∈ 𝑃 )
Assertion	lpolfN	⊢ ( 𝜑 → ⊥ : 𝒫 𝑉 ⟶ 𝑆 )

Proof

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 )