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Theorem lshpne 35003
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 2799 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 eqid 2799 . . . . 5 (LSubSp‘𝑊) = (LSubSp‘𝑊)
6 lshpne.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 35000 . . . 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 224 . 2 (𝜑 → (𝑈 ∈ (LSubSp‘𝑊) ∧ 𝑈𝑉 ∧ ∃𝑣𝑉 ((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = 𝑉))
109simp2d 1174 1 (𝜑𝑈𝑉)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  w3a 1108   = wceq 1653  wcel 2157  wne 2971  wrex 3090  cun 3767  {csn 4368  cfv 6101  Basecbs 16184  LModclmod 19181  LSubSpclss 19250  LSpanclspn 19292  LSHypclsh 34996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-iota 6064  df-fun 6103  df-fv 6109  df-lshyp 34998
This theorem is referenced by:  lshpnel  35004  lshpcmp  35009  lkrshp3  35127  lkrshp4  35129  dochshpncl  37405  dochlkr  37406  dochkrshp  37407  dochsatshpb  37473
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