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Theorem lshplss 34087
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 2620 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2620 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lshplss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
6 lshplss.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 34085 . . . 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 222 . 2 (𝜑 → (𝑈𝑆𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
109simp1d 1071 1 (𝜑𝑈𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  w3a 1036   = wceq 1481  wcel 1988  wne 2791  wrex 2910  cun 3565  {csn 4168  cfv 5876  Basecbs 15838  LModclmod 18844  LSubSpclss 18913  LSpanclspn 18952  LSHypclsh 34081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-iota 5839  df-fun 5878  df-fv 5884  df-lshyp 34083
This theorem is referenced by:  lshpnel  34089  lshpnelb  34090  lshpne0  34092  lshpdisj  34093  lshpcmp  34094  lshpsmreu  34215  lshpkrlem1  34216  lshpkrlem5  34220  lshpkr  34223  dochshpncl  36492  dochshpsat  36562  lclkrlem2f  36620
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