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Theorem cphlvec 22878
Description: A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphlvec (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)

Proof of Theorem cphlvec
StepHypRef Expression
1 cphphl 22874 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2 phllvec 19888 . 2 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1992  LVecclvec 19016  PreHilcphl 19883  ℂPreHilccph 22869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-nul 4754
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fv 5858  df-ov 6608  df-phl 19885  df-cph 22871
This theorem is referenced by:  cphnvc  22879  cphsubrg  22883  cphreccl  22884  cphqss  22891  hlprlem  23066  ishl2  23069
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