Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  islshp Structured version   Visualization version   GIF version
Theorem islshp 38451
Description: The predicate "is a hyperplane" (of a left module or left vector space). (Contributed by NM, 29-Jun-2014.)
Hypotheses
Ref Expression
lshpset.v 𝑉 = (Baseβ€˜π‘Š)
lshpset.n 𝑁 = (LSpanβ€˜π‘Š)
lshpset.s 𝑆 = (LSubSpβ€˜π‘Š)
lshpset.h 𝐻 = (LSHypβ€˜π‘Š)
Assertion
Ref Expression
islshp (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
Distinct variable groups:   𝑣,𝑉   𝑣,π‘Š   𝑣,π‘ˆ
Allowed substitution hints:   𝑆(𝑣)   𝐻(𝑣)   𝑁(𝑣)   𝑋(𝑣)
Proof of Theorem islshp
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 lshpset.v . . . 4 𝑉 = (Baseβ€˜π‘Š)
2 lshpset.n . . . 4 𝑁 = (LSpanβ€˜π‘Š)
3 lshpset.s . . . 4 𝑆 = (LSubSpβ€˜π‘Š)
4 lshpset.h . . . 4 𝐻 = (LSHypβ€˜π‘Š)
51, 2, 3, 4lshpset 38450 . . 3 (π‘Š ∈ 𝑋 β†’ 𝐻 = {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)})
65eleq2d 2815 . 2 (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐻 ↔ π‘ˆ ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)}))
7 neeq1 3000 . . . . 5 (𝑠 = π‘ˆ β†’ (𝑠 β‰  𝑉 ↔ π‘ˆ β‰  𝑉))
8 uneq1 4155 . . . . . . 7 (𝑠 = π‘ˆ β†’ (𝑠 βˆͺ {𝑣}) = (π‘ˆ βˆͺ {𝑣}))
98fveqeq2d 6905 . . . . . 6 (𝑠 = π‘ˆ β†’ ((π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉 ↔ (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
109rexbidv 3175 . . . . 5 (𝑠 = π‘ˆ β†’ (βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉 ↔ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
117, 10anbi12d 631 . . . 4 (𝑠 = π‘ˆ β†’ ((𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉) ↔ (π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
1211elrab 3682 . . 3 (π‘ˆ ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)} ↔ (π‘ˆ ∈ 𝑆 ∧ (π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
13 3anass 1093 . . 3 ((π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉) ↔ (π‘ˆ ∈ 𝑆 ∧ (π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
1412, 13bitr4i 278 . 2 (π‘ˆ ∈ {𝑠 ∈ 𝑆 ∣ (𝑠 β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(𝑠 βˆͺ {𝑣})) = 𝑉)} ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉))
156, 14bitrdi 287 1 (π‘Š ∈ 𝑋 β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  𝑉 ∧ βˆƒπ‘£ ∈ 𝑉 (π‘β€˜(π‘ˆ βˆͺ {𝑣})) = 𝑉)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099   β‰  wne 2937  βˆƒwrex 3067  {crab 3429   βˆͺ cun 3945  {csn 4629  β€˜cfv 6548  Basecbs 17180  LSubSpclss 20815  LSpanclspn 20855  LSHypclsh 38447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6500  df-fun 6550  df-fv 6556  df-lshyp 38449
This theorem is referenced by:  islshpsm  38452  lshplss  38453  lshpne  38454  lshpnel2N  38457  lkrshp  38577  lshpset2N  38591  dochsatshp  40924
  Copyright terms: Public domain W3C validator