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Theorem cphnlm 23379
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 2777 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2777 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2777 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2777 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2777 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 23377 . . 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 1136 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp2d 1134 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1601  wcel 2106  cin 3790  wss 3791  cmpt 4965  cima 5358  cfv 6135  (class class class)co 6922  0cc0 10272  +∞cpnf 10408  [,)cico 12489  csqrt 14380  Basecbs 16255  s cress 16256  Scalarcsca 16341  ·𝑖cip 16343  fldccnfld 20142  PreHilcphl 20367  normcnm 22789  NrmModcnlm 22793  ℂPreHilccph 23373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-nul 5025
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3399  df-sbc 3652  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-nul 4141  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4672  df-br 4887  df-opab 4949  df-mpt 4966  df-xp 5361  df-cnv 5363  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fv 6143  df-ov 6925  df-cph 23375
This theorem is referenced by:  cphngp  23380  cphlmod  23381  cphnvc  23383  cphnmvs  23397  ipcnlem2  23450  ipcnlem1  23451  csscld  23455
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