MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cphlvec Structured version   Unicode version

Theorem cphlvec 21350
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 21346 . 2  |-  ( W  e.  CPreHil  ->  W  e.  PreHil )
2 phllvec 18424 . 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 1762   LVecclvec 17524   PreHilcphl 18419   CPreHilccph 21341
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-nul 4569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-xp 4998  df-cnv 5000  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fv 5587  df-ov 6278  df-phl 18421  df-cph 21343
This theorem is referenced by:  cphnvc  21351  cphsubrg  21355  cphreccl  21356  cphqss  21363  hlprlem  21535  ishl2  21538
  Copyright terms: Public domain W3C validator