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Theorem cphsca 25227
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 2735 . . . 4 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2735 . . . 4 (·𝑖𝑊) = (·𝑖𝑊)
3 eqid 2735 . . . 4 (norm‘𝑊) = (norm‘𝑊)
4 cphsca.f . . . 4 𝐹 = (Scalar‘𝑊)
5 cphsca.k . . . 4 𝐾 = (Base‘𝐹)
61, 2, 3, 4, 5iscph 25218 . . 3 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)) ∧ (√ “ (𝐾 ∩ (0[,)+∞))) ⊆ 𝐾 ∧ (norm‘𝑊) = (𝑥 ∈ (Base‘𝑊) ↦ (√‘(𝑥(·𝑖𝑊)𝑥)))))
76simp1bi 1144 . 2 (𝑊 ∈ ℂPreHil → (𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ 𝐹 = (ℂflds 𝐾)))
87simp3d 1143 1 (𝑊 ∈ ℂPreHil → 𝐹 = (ℂflds 𝐾))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1086   = wceq 1537  wcel 2106  cin 3962  wss 3963  cmpt 5231  cima 5692  cfv 6563  (class class class)co 7431  0cc0 11153  +∞cpnf 11290  [,)cico 13386  csqrt 15269  Basecbs 17245  s cress 17274  Scalarcsca 17301  ·𝑖cip 17303  fldccnfld 21382  PreHilcphl 21660  normcnm 24605  NrmModcnlm 24609  ℂPreHilccph 25214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-nul 5312
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fv 6571  df-ov 7434  df-cph 25216
This theorem is referenced by:  cphsubrg  25228  cphreccl  25229  cphcjcl  25231  cphqss  25236  cphclm  25237  ipcau  25286  cphsscph  25299  hlprlem  25415  ishl2  25418
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