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Theorem cphnlm 24534
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 2736 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2736 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2736 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 eqid 2736 . . . 4 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2736 . . . 4 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 24532 . . 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 1145 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))))
87simp2d 1143 1 (𝑊 ∈ ℂPreHil → 𝑊 ∈ NrmMod)
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1541  wcel 2106  cin 3909  wss 3910  cmpt 5188  cima 5636  cfv 6496  (class class class)co 7356  0cc0 11050  +∞cpnf 11185  [,)cico 13265  csqrt 15117  Basecbs 17082  s cress 17111  Scalarcsca 17135  ·𝑖cip 17137  fldccnfld 20794  PreHilcphl 21026  normcnm 23930  NrmModcnlm 23934  ℂPreHilccph 24528
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707  ax-nul 5263
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2944  df-rab 3408  df-v 3447  df-sbc 3740  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-mpt 5189  df-xp 5639  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-iota 6448  df-fv 6504  df-ov 7359  df-cph 24530
This theorem is referenced by:  cphngp  24535  cphlmod  24536  cphnvc  24538  cphnmvs  24552  ipcnlem2  24606  ipcnlem1  24607  csscld  24611  cphsscph  24613
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