Metamath Proof Explorer


Theorem islpoldN

Description: Properties that determine a polarity. (Contributed by NM, 26-Nov-2014) (New usage is discouraged.)

Ref Expression
Hypotheses lpolset.v 𝑉 = ( Base ‘ 𝑊 )
lpolset.s 𝑆 = ( LSubSp ‘ 𝑊 )
lpolset.z 0 = ( 0g𝑊 )
lpolset.a 𝐴 = ( LSAtoms ‘ 𝑊 )
lpolset.h 𝐻 = ( LSHyp ‘ 𝑊 )
lpolset.p 𝑃 = ( LPol ‘ 𝑊 )
islpold.w ( 𝜑𝑊𝑋 )
islpold.1 ( 𝜑 : 𝒫 𝑉𝑆 )
islpold.2 ( 𝜑 → ( 𝑉 ) = { 0 } )
islpold.3 ( ( 𝜑 ∧ ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) ) → ( 𝑦 ) ⊆ ( 𝑥 ) )
islpold.4 ( ( 𝜑𝑥𝐴 ) → ( 𝑥 ) ∈ 𝐻 )
islpold.5 ( ( 𝜑𝑥𝐴 ) → ( ‘ ( 𝑥 ) ) = 𝑥 )
Assertion islpoldN ( 𝜑𝑃 )

Proof

Step Hyp Ref Expression
1 lpolset.v 𝑉 = ( Base ‘ 𝑊 )
2 lpolset.s 𝑆 = ( LSubSp ‘ 𝑊 )
3 lpolset.z 0 = ( 0g𝑊 )
4 lpolset.a 𝐴 = ( LSAtoms ‘ 𝑊 )
5 lpolset.h 𝐻 = ( LSHyp ‘ 𝑊 )
6 lpolset.p 𝑃 = ( LPol ‘ 𝑊 )
7 islpold.w ( 𝜑𝑊𝑋 )
8 islpold.1 ( 𝜑 : 𝒫 𝑉𝑆 )
9 islpold.2 ( 𝜑 → ( 𝑉 ) = { 0 } )
10 islpold.3 ( ( 𝜑 ∧ ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) ) → ( 𝑦 ) ⊆ ( 𝑥 ) )
11 islpold.4 ( ( 𝜑𝑥𝐴 ) → ( 𝑥 ) ∈ 𝐻 )
12 islpold.5 ( ( 𝜑𝑥𝐴 ) → ( ‘ ( 𝑥 ) ) = 𝑥 )
13 10 ex ( 𝜑 → ( ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) → ( 𝑦 ) ⊆ ( 𝑥 ) ) )
14 13 alrimivv ( 𝜑 → ∀ 𝑥𝑦 ( ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) → ( 𝑦 ) ⊆ ( 𝑥 ) ) )
15 11 12 jca ( ( 𝜑𝑥𝐴 ) → ( ( 𝑥 ) ∈ 𝐻 ∧ ( ‘ ( 𝑥 ) ) = 𝑥 ) )
16 15 ralrimiva ( 𝜑 → ∀ 𝑥𝐴 ( ( 𝑥 ) ∈ 𝐻 ∧ ( ‘ ( 𝑥 ) ) = 𝑥 ) )
17 9 14 16 3jca ( 𝜑 → ( ( 𝑉 ) = { 0 } ∧ ∀ 𝑥𝑦 ( ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) → ( 𝑦 ) ⊆ ( 𝑥 ) ) ∧ ∀ 𝑥𝐴 ( ( 𝑥 ) ∈ 𝐻 ∧ ( ‘ ( 𝑥 ) ) = 𝑥 ) ) )
18 1 2 3 4 5 6 islpolN ( 𝑊𝑋 → ( 𝑃 ↔ ( : 𝒫 𝑉𝑆 ∧ ( ( 𝑉 ) = { 0 } ∧ ∀ 𝑥𝑦 ( ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) → ( 𝑦 ) ⊆ ( 𝑥 ) ) ∧ ∀ 𝑥𝐴 ( ( 𝑥 ) ∈ 𝐻 ∧ ( ‘ ( 𝑥 ) ) = 𝑥 ) ) ) ) )
19 7 18 syl ( 𝜑 → ( 𝑃 ↔ ( : 𝒫 𝑉𝑆 ∧ ( ( 𝑉 ) = { 0 } ∧ ∀ 𝑥𝑦 ( ( 𝑥𝑉𝑦𝑉𝑥𝑦 ) → ( 𝑦 ) ⊆ ( 𝑥 ) ) ∧ ∀ 𝑥𝐴 ( ( 𝑥 ) ∈ 𝐻 ∧ ( ‘ ( 𝑥 ) ) = 𝑥 ) ) ) ) )
20 8 17 19 mpbir2and ( 𝜑𝑃 )