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Theorem lpolvN 41489
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 2736 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
5 lpolv.z . . . . 5 0 = (0g𝑊)
6 eqid 2736 . . . . 5 (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7 eqid 2736 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolv.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 41486 . . . 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 232 . 2 (𝜑 → ( :𝒫 𝑉⟶(LSubSp‘𝑊) ∧ (( 𝑉) = { 0 } ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))))
12 simpr1 1194 . 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 206  wa 395  w3a 1086  wal 1537   = wceq 1539  wcel 2107  wral 3060  wss 3950  𝒫 cpw 4599  {csn 4625  wf 6556  cfv 6560  Basecbs 17248  0gc0g 17485  LSubSpclss 20930  LSAtomsclsa 38976  LSHypclsh 38977  LPolclpoN 41483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-fv 6568  df-ov 7435  df-oprab 7436  df-mpo 7437  df-map 8869  df-lpolN 41484
This theorem is referenced by: (None)
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