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Theorem cphsca 22973
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 2621 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2621 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2621 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 cphsca.f . . . 4 𝐹 = (Scalar‘𝑊)
5 cphsca.k . . . 4 𝐾 = (Base‘𝐹)
61, 2, 3, 4, 5iscph 22964 . . 3 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖𝑊)𝑥)))))
76simp1bi 1075 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
87simp3d 1074 1 (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1037   = wceq 1482  wcel 1989  cin 3571  wss 3572  cmpt 4727  cima 5115  cfv 5886  (class class class)co 6647  0cc0 9933  +∞cpnf 10068  [,)cico 12174  csqrt 13967  Basecbs 15851  s cress 15852  Scalarcsca 15938  ·𝑖cip 15940  fldccnfld 19740  PreHilcphl 19963  normcnm 22375  NrmModcnlm 22379  ℂPreHilccph 22960
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-nul 4787
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fv 5894  df-ov 6650  df-cph 22962
This theorem is referenced by:  cphsubrg  22974  cphreccl  22975  cphcjcl  22977  cphqss  22982  cphclm  22983  ipcau  23031  hlprlem  23157  ishl2  23160
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