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Theorem cphphl 24688
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 2733 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2733 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2733 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2733 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2733 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 24687 . . 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 1088   = wceq 1542  wcel 2107  cin 3948  wss 3949  cmpt 5232  cima 5680  cfv 6544  (class class class)co 7409  0cc0 11110  +∞cpnf 11245  [,)cico 13326  csqrt 15180  Basecbs 17144  s cress 17173  Scalarcsca 17200  ·𝑖cip 17202  fldccnfld 20944  PreHilcphl 21177  normcnm 24085  NrmModcnlm 24089  ℂPreHilccph 24683
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fv 6552  df-ov 7412  df-cph 24685
This theorem is referenced by:  cphlvec  24692  cphcjcl  24700  cphipcl  24708  cphnmf  24712  cphipcj  24716  cphorthcom  24718  cphip0l  24719  cphip0r  24720  cphipeq0  24721  cphdir  24722  cphdi  24723  cph2di  24724  cphsubdir  24725  cphsubdi  24726  cph2subdi  24727  cphass  24728  cphassr  24729  ipcau  24755  nmparlem  24756  ipcn  24763  cphsscph  24768  hlphl  24882  cmscsscms  24890  bncssbn  24891  pjthlem2  24955
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