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Theorem lshpne 33085
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 2604 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 eqid 2604 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
6 lshpne.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 33082 . . . 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 220 . 2 (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
109simp2d 1066 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030   = wceq 1474  wcel 1975  wne 2774  wrex 2891  cun 3532  {csn 4119  cfv 5785  Basecbs 15636  LModclmod 18627  LSubSpclss 18694  LSpanclspn 18733  LSHypclsh 33078
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-sbc 3397  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-uni 4362  df-br 4573  df-opab 4633  df-mpt 4634  df-id 4938  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-iota 5749  df-fun 5787  df-fv 5793  df-lshyp 33080
This theorem is referenced by:  lshpnel  33086  lshpcmp  33091  lkrshp3  33209  lkrshp4  33211  dochshpncl  35489  dochlkr  35490  dochkrshp  35491  dochsatshpb  35557
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