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Theorem phllvec 21625
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 2726 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2726 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2726 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2726 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2726 . . 3 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6 eqid 2726 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isphl 21624 . 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 1142 1 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  wral 3051  cmpt 5236  cfv 6554  (class class class)co 7424  Basecbs 17213  *𝑟cstv 17268  Scalarcsca 17269  ·𝑖cip 17271  0gc0g 17454  *-Ringcsr 20817   LMHom clmhm 20997  LVecclvec 21080  ringLModcrglmod 21150  PreHilcphl 21620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-iota 6506  df-fv 6562  df-ov 7427  df-phl 21622
This theorem is referenced by:  phllmod  21626  phlssphl  21655  obsne0  21723  obslbs  21728  cphlvec  25194  phclm  25251  ipcau2  25253  tcphcph  25256
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