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Theorem cphsca 23791
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 2824 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2824 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2824 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 cphsca.f . . . 4 𝐹 = (Scalar‘𝑊)
5 cphsca.k . . . 4 𝐾 = (Base‘𝐹)
61, 2, 3, 4, 5iscph 23782 . . 3 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖𝑊)𝑥)))))
76simp1bi 1142 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
87simp3d 1141 1 (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2115  cin 3918  wss 3919  cmpt 5132  cima 5545  cfv 6343  (class class class)co 7149  0cc0 10535  +∞cpnf 10670  [,)cico 12737  csqrt 14592  Basecbs 16483  s cress 16484  Scalarcsca 16568  ·𝑖cip 16570  fldccnfld 20098  PreHilcphl 20320  normcnm 23190  NrmModcnlm 23194  ℂPreHilccph 23778
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5053  df-opab 5115  df-mpt 5133  df-xp 5548  df-cnv 5550  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fv 6351  df-ov 7152  df-cph 23780
This theorem is referenced by:  cphsubrg  23792  cphreccl  23793  cphcjcl  23795  cphqss  23800  cphclm  23801  ipcau  23849  cphsscph  23862  hlprlem  23978  ishl2  23981
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