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Theorem cphnlm 22885
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 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 22883 . . 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)))))
87simp2d 1072 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1036   = wceq 1480  wcel 1987  cin 3555  wss 3556  cmpt 4675  cima 5079  cfv 5849  (class class class)co 6607  0cc0 9883  +∞cpnf 10018  [,)cico 12122  csqrt 13910  Basecbs 15784  s cress 15785  Scalarcsca 15868  ·𝑖cip 15870  fldccnfld 19668  PreHilcphl 19891  normcnm 22294  NrmModcnlm 22298  ℂPreHilccph 22879
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 4751
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 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-xp 5082  df-cnv 5084  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fv 5857  df-ov 6610  df-cph 22881
This theorem is referenced by:  cphngp  22886  cphlmod  22887  cphnvc  22889  cphnmvs  22903  ipcnlem2  22956  ipcnlem1  22957  csscld  22961
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