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Theorem lshpne 38604
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 2725 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 eqid 2725 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
6 lshpne.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 38601 . . . 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 231 . 2 (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
109simp2d 1140 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1084   = wceq 1533  wcel 2098  wne 2929  wrex 3059  cun 3942  {csn 4630  cfv 6549  Basecbs 17199  LModclmod 20772  LSubSpclss 20844  LSpanclspn 20884  LSHypclsh 38597
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-iota 6501  df-fun 6551  df-fv 6557  df-lshyp 38599
This theorem is referenced by:  lshpnel  38605  lshpcmp  38610  lkrshp3  38728  lkrshp4  38730  dochshpncl  41007  dochlkr  41008  dochkrshp  41009  dochsatshpb  41075
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