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Theorem cphlvec 21488
Description: A complex pre-Hilbert space is a left vector space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphlvec  |-  ( W  e.  CPreHil  ->  W  e.  LVec )

Proof of Theorem cphlvec
StepHypRef Expression
1 cphphl 21484 . 2  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
2 phllvec 18531 . 2  |-  ( W  e.  PreHil  ->  W  e.  LVec )
31, 2syl 16 1  |-  ( W  e.  CPreHil  ->  W  e.  LVec )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    e. wcel 1802   LVecclvec 17616   PreHilcphl 18526   CPreHilccph 21479
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-nul 4562
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-xp 4991  df-cnv 4993  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fv 5582  df-ov 6280  df-phl 18528  df-cph 21481
This theorem is referenced by:  cphnvc  21489  cphsubrg  21493  cphreccl  21494  cphqss  21501  hlprlem  21673  ishl2  21676
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