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Theorem phllvec 20772
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 2821 . . 3 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2821 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
3 eqid 2821 . . 3 (·𝑖𝑊) = (·𝑖𝑊)
4 eqid 2821 . . 3 (0g𝑊) = (0g𝑊)
5 eqid 2821 . . 3 (*𝑟‘(Scalar‘𝑊)) = (*𝑟‘(Scalar‘𝑊))
6 eqid 2821 . . 3 (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊))
71, 2, 3, 4, 5, 6isphl 20771 . 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 1141 1 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1083   = wceq 1533  wcel 2110  wral 3138  cmpt 5145  cfv 6354  (class class class)co 7155  Basecbs 16482  *𝑟cstv 16566  Scalarcsca 16567  ·𝑖cip 16569  0gc0g 16712  *-Ringcsr 19614   LMHom clmhm 19790  LVecclvec 19873  ringLModcrglmod 19940  PreHilcphl 20767
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-nul 5209
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-iota 6313  df-fv 6362  df-ov 7158  df-phl 20769
This theorem is referenced by:  phllmod  20773  phlssphl  20802  obsne0  20868  obslbs  20873  cphlvec  23778  phclm  23834  ipcau2  23836  tcphcph  23839
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