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Theorem lshplss 33085
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 2605 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
4 eqid 2605 . . . . 5 (LSpan‘𝑊) = (LSpan‘𝑊)
5 lshplss.s . . . . 5 𝑆 = (LSubSp‘𝑊)
6 lshplss.h . . . . 5 𝐻 = (LSHyp‘𝑊)
73, 4, 5, 6islshp 33083 . . . 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 220 . 2 (𝜑 → (𝑈𝑆𝑈 ≠ (Base‘𝑊) ∧ ∃𝑣 ∈ (Base‘𝑊)((LSpan‘𝑊)‘(𝑈 ∪ {𝑣})) = (Base‘𝑊)))
109simp1d 1065 1 (𝜑𝑈𝑆)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  w3a 1030   = wceq 1474  wcel 1975  wne 2775  wrex 2892  cun 3533  {csn 4120  cfv 5786  Basecbs 15637  LModclmod 18628  LSubSpclss 18695  LSpanclspn 18734  LSHypclsh 33079
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 2228  ax-ext 2585  ax-sep 4699  ax-nul 4708  ax-pr 4824
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 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-rab 2900  df-v 3170  df-sbc 3398  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-sn 4121  df-pr 4123  df-op 4127  df-uni 4363  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-iota 5750  df-fun 5788  df-fv 5794  df-lshyp 33081
This theorem is referenced by:  lshpnel  33087  lshpnelb  33088  lshpne0  33090  lshpdisj  33091  lshpcmp  33092  lshpsmreu  33213  lshpkrlem1  33214  lshpkrlem5  33218  lshpkr  33221  dochshpncl  35490  dochshpsat  35560  lclkrlem2f  35618
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