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Theorem cphsca 24324
Description: A subcomplex pre-Hilbert space is a vector space over a subfield of fld. (Contributed by Mario Carneiro, 8-Oct-2015.)
Hypotheses
Ref Expression
cphsca.f 𝐹 = (Scalar‘𝑊)
cphsca.k 𝐾 = (Base‘𝐹)
Assertion
Ref Expression
cphsca (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))

Proof of Theorem cphsca
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eqid 2739 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2739 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2739 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 cphsca.f . . . 4 𝐹 = (Scalar‘𝑊)
5 cphsca.k . . . 4 𝐾 = (Base‘𝐹)
61, 2, 3, 4, 5iscph 24315 . . 3 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖𝑊)𝑥)))))
76simp1bi 1143 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
87simp3d 1142 1 (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1085   = wceq 1541  wcel 2109  cin 3890  wss 3891  cmpt 5161  cima 5591  cfv 6430  (class class class)co 7268  0cc0 10855  +∞cpnf 10990  [,)cico 13063  csqrt 14925  Basecbs 16893  s cress 16922  Scalarcsca 16946  ·𝑖cip 16948  fldccnfld 20578  PreHilcphl 20810  normcnm 23713  NrmModcnlm 23717  ℂPreHilccph 24311
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-nul 5233
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-xp 5594  df-cnv 5596  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-iota 6388  df-fv 6438  df-ov 7271  df-cph 24313
This theorem is referenced by:  cphsubrg  24325  cphreccl  24326  cphcjcl  24328  cphqss  24333  cphclm  24334  ipcau  24383  cphsscph  24396  hlprlem  24512  ishl2  24515
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