Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lpolfN Structured version   Visualization version   GIF version

Theorem lpolfN 39499
Description: Functionality of a polarity. (Contributed by NM, 26-Nov-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
lpolf.v 𝑉 = (Base‘𝑊)
lpolf.s 𝑆 = (LSubSp‘𝑊)
lpolf.p 𝑃 = (LPol‘𝑊)
lpolf.w (𝜑𝑊𝑋)
lpolf.o (𝜑𝑃)
Assertion
Ref Expression
lpolfN (𝜑 :𝒫 𝑉𝑆)

Proof of Theorem lpolfN
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpolf.o . . 3 (𝜑𝑃)
2 lpolf.w . . . 4 (𝜑𝑊𝑋)
3 lpolf.v . . . . 5 𝑉 = (Base‘𝑊)
4 lpolf.s . . . . 5 𝑆 = (LSubSp‘𝑊)
5 eqid 2738 . . . . 5 (0g𝑊) = (0g𝑊)
6 eqid 2738 . . . . 5 (LSAtoms‘𝑊) = (LSAtoms‘𝑊)
7 eqid 2738 . . . . 5 (LSHyp‘𝑊) = (LSHyp‘𝑊)
8 lpolf.p . . . . 5 𝑃 = (LPol‘𝑊)
93, 4, 5, 6, 7, 8islpolN 39497 . . . 4 (𝑊𝑋 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
102, 9syl 17 . . 3 (𝜑 → ( 𝑃 ↔ ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥)))))
111, 10mpbid 231 . 2 (𝜑 → ( :𝒫 𝑉𝑆 ∧ (( 𝑉) = {(0g𝑊)} ∧ ∀𝑥𝑦((𝑥𝑉𝑦𝑉𝑥𝑦) → ( 𝑦) ⊆ ( 𝑥)) ∧ ∀𝑥 ∈ (LSAtoms‘𝑊)(( 𝑥) ∈ (LSHyp‘𝑊) ∧ ( ‘( 𝑥)) = 𝑥))))
1211simpld 495 1 (𝜑 :𝒫 𝑉𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086  wal 1537   = wceq 1539  wcel 2106  wral 3064  wss 3887  𝒫 cpw 4533  {csn 4561  wf 6429  cfv 6433  Basecbs 16912  0gc0g 17150  LSubSpclss 20193  LSAtomsclsa 36988  LSHypclsh 36989  LPolclpoN 39494
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-map 8617  df-lpolN 39495
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator