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Theorem cphphl 24519
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 2736 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2736 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2736 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2736 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2736 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 24518 . . 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 1145 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp1d 1142 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cin 3907  wss 3908  cmpt 5186  cima 5634  cfv 6493  (class class class)co 7353  0cc0 11047  +∞cpnf 11182  [,)cico 13258  csqrt 15110  Basecbs 17075  s cress 17104  Scalarcsca 17128  ·𝑖cip 17130  fldccnfld 20781  PreHilcphl 21013  normcnm 23916  NrmModcnlm 23920  ℂPreHilccph 24514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5261
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-rab 3406  df-v 3445  df-sbc 3738  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fv 6501  df-ov 7356  df-cph 24516
This theorem is referenced by:  cphlvec  24523  cphcjcl  24531  cphipcl  24539  cphnmf  24543  cphipcj  24547  cphorthcom  24549  cphip0l  24550  cphip0r  24551  cphipeq0  24552  cphdir  24553  cphdi  24554  cph2di  24555  cphsubdir  24556  cphsubdi  24557  cph2subdi  24558  cphass  24559  cphassr  24560  ipcau  24586  nmparlem  24587  ipcn  24594  cphsscph  24599  hlphl  24713  cmscsscms  24721  bncssbn  24722  pjthlem2  24786
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