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Theorem cphlvec 23193
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 23189 . 2 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
2 phllvec 20190 . 2 (𝑊 ∈ PreHil → 𝑊 ∈ LVec)
31, 2syl 17 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ LVec)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2145  LVecclvec 19314  PreHilcphl 20185  ℂPreHilccph 23184
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4923
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-xp 5255  df-cnv 5257  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fv 6039  df-ov 6795  df-phl 20187  df-cph 23186
This theorem is referenced by:  cphnvc  23194  cphsubrg  23198  cphreccl  23199  cphqss  23206  hlprlem  23381  ishl2  23384
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