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Theorem cphphl 23776
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 2798 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2798 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2798 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2798 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2798 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 23775 . . 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 1538  wcel 2111  cin 3880  wss 3881  cmpt 5110  cima 5522  cfv 6324  (class class class)co 7135  0cc0 10526  +∞cpnf 10661  [,)cico 12728  csqrt 14584  Basecbs 16475  s cress 16476  Scalarcsca 16560  ·𝑖cip 16562  fldccnfld 20091  PreHilcphl 20313  normcnm 23183  NrmModcnlm 23187  ℂPreHilccph 23771
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-nul 5174
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fv 6332  df-ov 7138  df-cph 23773
This theorem is referenced by:  cphlvec  23780  cphcjcl  23788  cphipcl  23796  cphnmf  23800  cphipcj  23804  cphorthcom  23806  cphip0l  23807  cphip0r  23808  cphipeq0  23809  cphdir  23810  cphdi  23811  cph2di  23812  cphsubdir  23813  cphsubdi  23814  cph2subdi  23815  cphass  23816  cphassr  23817  ipcau  23842  nmparlem  23843  ipcn  23850  cphsscph  23855  hlphl  23969  cmscsscms  23977  bncssbn  23978  pjthlem2  24042
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