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Theorem cphnlm 22780
 Description: A complex 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 2610 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2610 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2610 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2610 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2610 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 22778 . . 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 1069 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp2d 1067 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977   ∩ cin 3539   ⊆ wss 3540   ↦ cmpt 4643   “ cima 5041  ‘cfv 5804  (class class class)co 6549  0cc0 9815  +∞cpnf 9950  [,)cico 12048  √csqrt 13821  Basecbs 15695   ↾s cress 15696  Scalarcsca 15771  ·𝑖cip 15773  ℂfldccnfld 19567  PreHilcphl 19788  normcnm 22191  NrmModcnlm 22195  ℂPreHilccph 22774 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-nul 4717 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fv 5812  df-ov 6552  df-cph 22776 This theorem is referenced by:  cphngp  22781  cphlmod  22782  cphnvc  22784  cphnmvs  22798  ipcnlem2  22851  ipcnlem1  22852  csscld  22856
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