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Theorem islpoldN 35791
 Description: Properties that determine a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
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
Ref Expression
islpoldN (𝜑𝑃)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝑊   𝑥, ,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   𝐴(𝑦)   𝑃(𝑥,𝑦)   𝑆(𝑥,𝑦)   𝐻(𝑥,𝑦)   𝑉(𝑥,𝑦)   𝑋(𝑥,𝑦)   0 (𝑥,𝑦)

Proof of Theorem islpoldN
StepHypRef Expression
1 islpold.1 . 2 (𝜑 :𝒫 𝑉𝑆)
2 islpold.2 . . 3 (𝜑 → ( 𝑉) = { 0 })
3 islpold.3 . . . . 5 ((𝜑 ∧ (𝑥𝑉𝑦𝑉𝑥𝑦)) → ( 𝑦) ⊆ ( 𝑥))
43ex 449 . . . 4 (𝜑 → ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)))
54alrimivv 1843 . . 3 (𝜑 → ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)))
6 islpold.4 . . . . 5 ((𝜑𝑥𝐴) → ( 𝑥) ∈ 𝐻)
7 islpold.5 . . . . 5 ((𝜑𝑥𝐴) → ( ‘( 𝑥)) = 𝑥)
86, 7jca 553 . . . 4 ((𝜑𝑥𝐴) → (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))
98ralrimiva 2949 . . 3 (𝜑 → ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))
102, 5, 93jca 1235 . 2 (𝜑 → (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))
11 islpold.w . . 3 (𝜑𝑊𝑋)
12 lpolset.v . . . 4 𝑉 = (Base‘𝑊)
13 lpolset.s . . . 4 𝑆 = (LSubSp‘𝑊)
14 lpolset.z . . . 4 0 = (0g𝑊)
15 lpolset.a . . . 4 𝐴 = (LSAtoms‘𝑊)
16 lpolset.h . . . 4 𝐻 = (LSHyp‘𝑊)
17 lpolset.p . . . 4 𝑃 = (LPol‘𝑊)
1812, 13, 14, 15, 16, 17islpolN 35790 . . 3 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))))
1911, 18syl 17 . 2 (𝜑 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥)))))
201, 10, 19mpbir2and 959 1 (𝜑𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540  𝒫 cpw 4108  {csn 4125  ⟶wf 5800  ‘cfv 5804  Basecbs 15695  0gc0g 15923  LSubSpclss 18753  LSAtomsclsa 33279  LSHypclsh 33280  LPolclpoN 35787 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-map 7746  df-lpolN 35788 This theorem is referenced by:  dochpolN  35797
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