Users' Mathboxes Mathbox for Norm Megill < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  lshplss Structured version   Visualization version   GIF version

Theorem lshplss 38485
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 2728 . . . . 5 (Baseβ€˜π‘Š) = (Baseβ€˜π‘Š)
4 eqid 2728 . . . . 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 38483 . . . 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 2937  βˆƒwrex 3067   βˆͺ cun 3947  {csn 4632  β€˜cfv 6553  Basecbs 17187  LModclmod 20750  LSubSpclss 20822  LSpanclspn 20862  LSHypclsh 38479
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 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-iota 6505  df-fun 6555  df-fv 6561  df-lshyp 38481
This theorem is referenced by:  lshpnel  38487  lshpnelb  38488  lshpne0  38490  lshpdisj  38491  lshpcmp  38492  lshpsmreu  38613  lshpkrlem1  38614  lshpkrlem5  38618  lshpkr  38621  dochshpncl  40889  dochshpsat  40959  lclkrlem2f  41017
  Copyright terms: Public domain W3C validator