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Theorem cphnlm 23693
Description: A subcomplex pre-Hilbert space is a normed module. (Contributed by Mario Carneiro, 7-Oct-2015.)
Assertion
Ref Expression
cphnlm (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)

Proof of Theorem cphnlm
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2825 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2825 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2825 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2825 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2825 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 23691 . . 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 1139 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp2d 1137 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1081   = wceq 1530  wcel 2107  cin 3938  wss 3939  cmpt 5142  cima 5556  cfv 6351  (class class class)co 7151  0cc0 10529  +∞cpnf 10664  [,)cico 12733  csqrt 14585  Basecbs 16475  s cress 16476  Scalarcsca 16560  ·𝑖cip 16562  fldccnfld 20463  PreHilcphl 20686  normcnm 23103  NrmModcnlm 23107  ℂPreHilccph 23687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-nul 5206
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-xp 5559  df-cnv 5561  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fv 6359  df-ov 7154  df-cph 23689
This theorem is referenced by:  cphngp  23694  cphlmod  23695  cphnvc  23697  cphnmvs  23711  ipcnlem2  23764  ipcnlem1  23765  csscld  23769  cphsscph  23771
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