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Theorem islpoldN 39086
 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 416 . . . 4 (𝜑 → ((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)))
54alrimivv 1929 . . 3 (𝜑 → ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)))
6 islpold.4 . . . . 5 ((𝜑𝑥𝐴) → ( 𝑥) ∈ 𝐻)
7 islpold.5 . . . . 5 ((𝜑𝑥𝐴) → ( ‘( 𝑥)) = 𝑥)
86, 7jca 515 . . . 4 ((𝜑𝑥𝐴) → (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))
98ralrimiva 3113 . . 3 (𝜑 → ∀𝑥𝐴 (( 𝑥) ∈ 𝐻 ∧ ( ‘( 𝑥)) = 𝑥))
102, 5, 93jca 1125 . 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 39085 . . 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 712 1 (𝜑𝑃)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3070   ⊆ wss 3860  𝒫 cpw 4497  {csn 4525  ⟶wf 6335  ‘cfv 6339  Basecbs 16546  0gc0g 16776  LSubSpclss 19776  LSAtomsclsa 36576  LSHypclsh 36577  LPolclpoN 39082 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-ov 7158  df-oprab 7159  df-mpo 7160  df-map 8423  df-lpolN 39083 This theorem is referenced by:  dochpolN  39092
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