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Theorem lshplss 38363
Description: A hyperplane is a subspace. (Contributed by NM, 3-Jul-2014.)
Hypotheses
Ref Expression
lshplss.s 𝑆 = (LSubSpβ€˜π‘Š)
lshplss.h 𝐻 = (LSHypβ€˜π‘Š)
lshplss.w (πœ‘ β†’ π‘Š ∈ LMod)
lshplss.u (πœ‘ β†’ π‘ˆ ∈ 𝐻)
Assertion
Ref Expression
lshplss (πœ‘ β†’ π‘ˆ ∈ 𝑆)

Proof of Theorem lshplss
Dummy variable 𝑣 is distinct from all other variables.
StepHypRef Expression
1 lshplss.u . . 3 (πœ‘ β†’ π‘ˆ ∈ 𝐻)
2 lshplss.w . . . 4 (πœ‘ β†’ π‘Š ∈ LMod)
3 eqid 2726 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
4 eqid 2726 . . . . 5 (LSpanβ€˜π‘Š) = (LSpanβ€˜π‘Š)
5 lshplss.s . . . . 5 𝑆 = (LSubSpβ€˜π‘Š)
6 lshplss.h . . . . 5 𝐻 = (LSHypβ€˜π‘Š)
73, 4, 5, 6islshp 38361 . . . 4 (π‘Š ∈ LMod β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  (Baseβ€˜π‘Š) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘Š)((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = (Baseβ€˜π‘Š))))
82, 7syl 17 . . 3 (πœ‘ β†’ (π‘ˆ ∈ 𝐻 ↔ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  (Baseβ€˜π‘Š) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘Š)((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = (Baseβ€˜π‘Š))))
91, 8mpbid 231 . 2 (πœ‘ β†’ (π‘ˆ ∈ 𝑆 ∧ π‘ˆ β‰  (Baseβ€˜π‘Š) ∧ βˆƒπ‘£ ∈ (Baseβ€˜π‘Š)((LSpanβ€˜π‘Š)β€˜(π‘ˆ βˆͺ {𝑣})) = (Baseβ€˜π‘Š)))
109simp1d 1139 1 (πœ‘ β†’ π‘ˆ ∈ 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064   βˆͺ cun 3941  {csn 4623  β€˜cfv 6536  Basecbs 17150  LModclmod 20703  LSubSpclss 20775  LSpanclspn 20815  LSHypclsh 38357
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-lshyp 38359
This theorem is referenced by:  lshpnel  38365  lshpnelb  38366  lshpne0  38368  lshpdisj  38369  lshpcmp  38370  lshpsmreu  38491  lshpkrlem1  38492  lshpkrlem5  38496  lshpkr  38499  dochshpncl  40767  dochshpsat  40837  lclkrlem2f  40895
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