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Theorem cphphl 25190
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 2726 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2726 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2726 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2726 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2726 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 25189 . . 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 1142 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp1d 1139 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1534  wcel 2099  cin 3946  wss 3947  cmpt 5236  cima 5685  cfv 6554  (class class class)co 7424  0cc0 11158  +∞cpnf 11295  [,)cico 13380  csqrt 15238  Basecbs 17213  s cress 17242  Scalarcsca 17269  ·𝑖cip 17271  fldccnfld 21343  PreHilcphl 21620  normcnm 24576  NrmModcnlm 24580  ℂPreHilccph 25185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-nul 5311
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-rab 3420  df-v 3464  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-br 5154  df-opab 5216  df-mpt 5237  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fv 6562  df-ov 7427  df-cph 25187
This theorem is referenced by:  cphlvec  25194  cphcjcl  25202  cphipcl  25210  cphnmf  25214  cphipcj  25218  cphorthcom  25220  cphip0l  25221  cphip0r  25222  cphipeq0  25223  cphdir  25224  cphdi  25225  cph2di  25226  cphsubdir  25227  cphsubdi  25228  cph2subdi  25229  cphass  25230  cphassr  25231  ipcau  25257  nmparlem  25258  ipcn  25265  cphsscph  25270  hlphl  25384  cmscsscms  25392  bncssbn  25393  pjthlem2  25457
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