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Theorem lshpne 38983
Description: A hyperplane is not equal to the vector space. (Contributed by NM, 4-Jul-2014.)
Hypotheses
Ref Expression
lshpne.v 𝑉 = (Base‘𝑊)
lshpne.h 𝐻 = (LSHyp‘𝑊)
lshpne.w (𝜑𝑊 ∈ LMod)
lshpne.u (𝜑𝑈𝐻)
Assertion
Ref Expression
lshpne (𝜑𝑈𝑉)

Proof of Theorem lshpne
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 lshpne.u . . 3 (𝜑𝑈𝐻)
2 lshpne.w . . . 4 (𝜑𝑊 ∈ LMod)
3 lshpne.v . . . . 5 𝑉 = (Base‘𝑊)
4 eqid 2737 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 eqid 2737 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
6 lshpne.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 38980 . . . 4 (𝑊 ∈ LMod → (𝑈𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)))
82, 7syl 17 . . 3 (𝜑 → (𝑈𝐻 ↔ (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉)))
91, 8mpbid 232 . 2 (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
109simp2d 1144 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wrex 3070  cun 3949  {csn 4626  cfv 6561  Basecbs 17247  LModclmod 20858  LSubSpclss 20929  LSpanclspn 20969  LSHypclsh 38976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-lshyp 38978
This theorem is referenced by:  lshpnel  38984  lshpcmp  38989  lkrshp3  39107  lkrshp4  39109  dochshpncl  41386  dochlkr  41387  dochkrshp  41388  dochsatshpb  41454
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