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Theorem lpolvN 36594
 Description: The polarity of the whole space is the zero subspace. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolv.v 𝑉 = (Base‘𝑊)
lpolv.z 0 = (0g𝑊)
lpolv.p 𝑃 = (LPol‘𝑊)
lpolv.w (𝜑𝑊𝑋)
lpolv.o (𝜑𝑃)
Assertion
Ref Expression
lpolvN (𝜑 → ( 𝑉) = { 0 })

Proof of Theorem lpolvN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolv.o . . 3 (𝜑𝑃)
2 lpolv.w . . . 4 (𝜑𝑊𝑋)
3 lpolv.v . . . . 5 𝑉 = (Base‘𝑊)
4 eqid 2620 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 lpolv.z . . . . 5 0 = (0g𝑊)
6 eqid 2620 . . . . 5 (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7 eqid 2620 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolv.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 36591 . . . 4 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
102, 9syl 17 . . 3 (𝜑 → ( 𝑃 ↔ ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
111, 10mpbid 222 . 2 (𝜑 → ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))))
12 simpr1 1065 . 2 (( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))) → ( 𝑉) = { 0 })
1311, 12syl 17 1 (𝜑 → ( 𝑉) = { 0 })
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1036  ∀wal 1479   = wceq 1481   ∈ wcel 1988  ∀wral 2909   ⊆ wss 3567  𝒫 cpw 4149  {csn 4168  ⟶wf 5872  ‘cfv 5876  Basecbs 15838  0gc0g 16081  LSubSpclss 18913  LSAtomsclsa 34080  LSHypclsh 34081  LPolclpoN 36588 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-map 7844  df-lpolN 36589 This theorem is referenced by: (None)
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