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

Theorem cphlvec 23472
 Description: A subcomplex 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 23468 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2 phllvec 20465 . 2 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∈ wcel 2048  LVecclvec 19586  PreHilcphl 20460  ℂPreHilccph 23463 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-ext 2745  ax-nul 5061 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-xp 5406  df-cnv 5408  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fv 6190  df-ov 6973  df-phl 20462  df-cph 23465 This theorem is referenced by:  cphnvc  23473  cphsubrg  23477  cphreccl  23478  cphqss  23485  hlprlem  23663  ishl2  23666
 Copyright terms: Public domain W3C validator