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Theorem phllvec 21665
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 2735 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2735 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2735 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2735 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2735 . . 3 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6 eqid 2735 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isphl 21664 . 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 1144 1 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  wral 3059  cmpt 5231  cfv 6563  (class class class)co 7431  Basecbs 17245  *𝑟cstv 17300  Scalarcsca 17301  ·𝑖cip 17303  0gc0g 17486  *-Ringcsr 20856   LMHom clmhm 21036  LVecclvec 21119  ringLModcrglmod 21189  PreHilcphl 21660
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-iota 6516  df-fv 6571  df-ov 7434  df-phl 21662
This theorem is referenced by:  phllmod  21666  phlssphl  21695  obsne0  21763  obslbs  21768  cphlvec  25223  phclm  25280  ipcau2  25282  tcphcph  25285
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