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Theorem cphphl 22779
 Description: A complex 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 2610 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2610 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2610 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2610 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 22778 . . 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 1069 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp1d 1066 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643   “ cima 5041  ‘cfv 5804  (class class class)co 6549  0cc0 9815  +∞cpnf 9950  [,)cico 12048  √csqrt 13821  Basecbs 15695   ↾s cress 15696  Scalarcsca 15771  ·𝑖cip 15773  ℂfldccnfld 19567  PreHilcphl 19788  normcnm 22191  NrmModcnlm 22195  ℂPreHilccph 22774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-cph 22776 This theorem is referenced by:  cphlvec  22783  cphcjcl  22791  cphipcl  22799  cphnmf  22803  cphipcj  22807  cphorthcom  22809  cphip0l  22810  cphip0r  22811  cphipeq0  22812  cphdir  22813  cphdi  22814  cph2di  22815  cphsubdir  22816  cphsubdi  22817  cph2subdi  22818  cphass  22819  cphassr  22820  ipcau  22845  nmparlem  22846  ipcn  22853  hlphl  22969  pjthlem2  23017
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