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Theorem lshplss 38390
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 2727 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2727 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lshplss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
6 lshplss.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 38388 . . . 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 231 . 2 (𝜑 → (𝑈𝑆𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
109simp1d 1140 1 (𝜑𝑈𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wrex 3065  cun 3942  {csn 4624  cfv 6542  Basecbs 17171  LModclmod 20732  LSubSpclss 20804  LSpanclspn 20844  LSHypclsh 38384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-iota 6494  df-fun 6544  df-fv 6550  df-lshyp 38386
This theorem is referenced by:  lshpnel  38392  lshpnelb  38393  lshpne0  38395  lshpdisj  38396  lshpcmp  38397  lshpsmreu  38518  lshpkrlem1  38519  lshpkrlem5  38523  lshpkr  38526  dochshpncl  40794  dochshpsat  40864  lclkrlem2f  40922
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