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Theorem lshpne 34587
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 2651 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 eqid 2651 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
6 lshpne.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 34584 . . . 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 222 . 2 (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
109simp2d 1094 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wrex 2942  cun 3605  {csn 4210  cfv 5926  Basecbs 15904  LModclmod 18911  LSubSpclss 18980  LSpanclspn 19019  LSHypclsh 34580
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-iota 5889  df-fun 5928  df-fv 5934  df-lshyp 34582
This theorem is referenced by:  lshpnel  34588  lshpcmp  34593  lkrshp3  34711  lkrshp4  34713  dochshpncl  36990  dochlkr  36991  dochkrshp  36992  dochsatshpb  37058
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