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Theorem cphnlm 23191
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 2771 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2771 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2771 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2771 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2771 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 23189 . . 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 1071   = wceq 1631  wcel 2145  cin 3722  wss 3723  cmpt 4864  cima 5253  cfv 6030  (class class class)co 6796  0cc0 10142  +∞cpnf 10277  [,)cico 12382  csqrt 14181  Basecbs 16064  s cress 16065  Scalarcsca 16152  ·𝑖cip 16154  fldccnfld 19961  PreHilcphl 20186  normcnm 22601  NrmModcnlm 22605  ℂPreHilccph 23185
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fv 6038  df-ov 6799  df-cph 23187
This theorem is referenced by:  cphngp  23192  cphlmod  23193  cphnvc  23195  cphnmvs  23209  ipcnlem2  23262  ipcnlem1  23263  csscld  23267
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