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Theorem cphphl 25205
Description: A subcomplex pre-Hilbert space is a pre-Hilbert space. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphphl (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)

Proof of Theorem cphphl
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2737 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2737 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2737 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2737 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 25204 . . 3 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖𝑊)𝑥)))))
76simp1bi 1146 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp1d 1143 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951  cmpt 5225  cima 5688  cfv 6561  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,)cico 13389  csqrt 15272  Basecbs 17247  s cress 17274  Scalarcsca 17300  ·𝑖cip 17302  fldccnfld 21364  PreHilcphl 21642  normcnm 24589  NrmModcnlm 24593  ℂPreHilccph 25200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-cph 25202
This theorem is referenced by:  cphlvec  25209  cphcjcl  25217  cphipcl  25225  cphnmf  25229  cphipcj  25233  cphorthcom  25235  cphip0l  25236  cphip0r  25237  cphipeq0  25238  cphdir  25239  cphdi  25240  cph2di  25241  cphsubdir  25242  cphsubdi  25243  cph2subdi  25244  cphass  25245  cphassr  25246  ipcau  25272  nmparlem  25273  ipcn  25280  cphsscph  25285  hlphl  25399  cmscsscms  25407  bncssbn  25408  pjthlem2  25472
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