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Theorem cphphl 22879
 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 2621 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2621 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2621 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2621 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 22878 . . 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 1074 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp1d 1071 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ PreHil)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1036   = wceq 1480   ∈ wcel 1987   ∩ cin 3554   ⊆ wss 3555   ↦ cmpt 4673   “ cima 5077  ‘cfv 5847  (class class class)co 6604  0cc0 9880  +∞cpnf 10015  [,)cico 12119  √csqrt 13907  Basecbs 15781   ↾s cress 15782  Scalarcsca 15865  ·𝑖cip 15867  ℂfldccnfld 19665  PreHilcphl 19888  normcnm 22291  NrmModcnlm 22295  ℂPreHilccph 22874 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-nul 4749 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fv 5855  df-ov 6607  df-cph 22876 This theorem is referenced by:  cphlvec  22883  cphcjcl  22891  cphipcl  22899  cphnmf  22903  cphipcj  22907  cphorthcom  22909  cphip0l  22910  cphip0r  22911  cphipeq0  22912  cphdir  22913  cphdi  22914  cph2di  22915  cphsubdir  22916  cphsubdi  22917  cph2subdi  22918  cphass  22919  cphassr  22920  ipcau  22945  nmparlem  22946  ipcn  22953  hlphl  23069  pjthlem2  23117
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