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Theorem lshplss 38963
Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshplss.s 𝑆 = (LSubSp‘𝑊)
lshplss.h 𝐻 = (LSHyp‘𝑊)
lshplss.w (𝜑𝑊 ∈ LMod)
lshplss.u (𝜑𝑈𝐻)
Assertion
Ref Expression
lshplss (𝜑𝑈𝑆)

Proof of Theorem lshplss
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 lshplss.u . . 3 (𝜑𝑈𝐻)
2 lshplss.w . . . 4 (𝜑𝑊 ∈ LMod)
3 eqid 2735 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2735 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lshplss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
6 lshplss.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 38961 . . . 4 (𝑊 ∈ LMod → (𝑈𝐻 ↔ (𝑈𝑆𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))))
82, 7syl 17 . . 3 (𝜑 → (𝑈𝐻 ↔ (𝑈𝑆𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊))))
91, 8mpbid 232 . 2 (𝜑 → (𝑈𝑆𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
109simp1d 1141 1 (𝜑𝑈𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  w3a 1086   = wceq 1537  wcel 2106  wne 2938  wrex 3068  cun 3961  {csn 4631  cfv 6563  Basecbs 17245  LModclmod 20875  LSubSpclss 20947  LSpanclspn 20987  LSHypclsh 38957
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-lshyp 38959
This theorem is referenced by:  lshpnel  38965  lshpnelb  38966  lshpne0  38968  lshpdisj  38969  lshpcmp  38970  lshpsmreu  39091  lshpkrlem1  39092  lshpkrlem5  39096  lshpkr  39099  dochshpncl  41367  dochshpsat  41437  lclkrlem2f  41495
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