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Theorem cphsca 25213
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 2737 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2737 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2737 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 cphsca.f . . . 4 𝐹 = (Scalar‘𝑊)
5 cphsca.k . . . 4 𝐾 = (Base‘𝐹)
61, 2, 3, 4, 5iscph 25204 . . 3 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖𝑊)𝑥)))))
76simp1bi 1146 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
87simp3d 1145 1 (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  cin 3950  wss 3951  cmpt 5225  cima 5688  cfv 6561  (class class class)co 7431  0cc0 11155  +∞cpnf 11292  [,)cico 13389  csqrt 15272  Basecbs 17247  s cress 17274  Scalarcsca 17300  ·𝑖cip 17302  fldccnfld 21364  PreHilcphl 21642  normcnm 24589  NrmModcnlm 24593  ℂPreHilccph 25200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fv 6569  df-ov 7434  df-cph 25202
This theorem is referenced by:  cphsubrg  25214  cphreccl  25215  cphcjcl  25217  cphqss  25222  cphclm  25223  ipcau  25272  cphsscph  25285  hlprlem  25401  ishl2  25404
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