MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  phllvec Structured version   Visualization version   GIF version

Theorem phllvec 20623
Description: A pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
phllvec (𝑊 ∈ PreHil → 𝑊 ∈ LVec)

Proof of Theorem phllvec
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2739 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2739 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2739 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2739 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2739 . . 3 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6 eqid 2739 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isphl 20622 . 2 (𝑊 ∈ PreHil ↔ (𝑊 ∈ LVec ∧ (Scalar‘𝑊) ∈ *-Ring ∧ ∀𝑥 ∈ (Base‘𝑊)((𝑦 ∈ (Base‘𝑊) ↦ (𝑦(·𝑖𝑊)𝑥)) ∈ (𝑊 LMHom (ringLMod‘(Scalar‘𝑊))) ∧ ((𝑥(·𝑖𝑊)𝑥) = (0g‘(Scalar‘𝑊)) → 𝑥 = (0g𝑊)) ∧ ∀𝑦 ∈ (Base‘𝑊)((*𝑟‘(Scalar‘𝑊))‘(𝑥(·𝑖𝑊)𝑦)) = (𝑦(·𝑖𝑊)𝑥))))
87simp1bi 1147 1 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1089   = wceq 1543  wcel 2112  wral 3064  cmpt 5151  cfv 6400  (class class class)co 7234  Basecbs 16792  *𝑟cstv 16836  Scalarcsca 16837  ·𝑖cip 16839  0gc0g 16976  *-Ringcsr 19912   LMHom clmhm 20088  LVecclvec 20171  ringLModcrglmod 20238  PreHilcphl 20618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2710  ax-nul 5215
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4836  df-br 5070  df-opab 5132  df-mpt 5152  df-iota 6358  df-fv 6408  df-ov 7237  df-phl 20620
This theorem is referenced by:  phllmod  20624  phlssphl  20653  obsne0  20719  obslbs  20724  cphlvec  24103  phclm  24160  ipcau2  24162  tcphcph  24165
  Copyright terms: Public domain W3C validator