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Theorem cphnmfval 23212
Description: The value of the norm in a subcomplex pre-Hilbert space is the square root of the inner product of a vector with itself. (Contributed by Mario Carneiro, 7-Oct-2015.)
Hypotheses
Ref Expression
nmsq.v 𝑉 = (Base‘𝑊)
nmsq.h , = (·𝑖𝑊)
nmsq.n 𝑁 = (norm‘𝑊)
Assertion
Ref Expression
cphnmfval (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
Distinct variable groups:   𝑥, ,   𝑥,𝑉   𝑥,𝑊
Allowed substitution hint:   𝑁(𝑥)

Proof of Theorem cphnmfval
StepHypRef Expression
1 nmsq.v . . 3 𝑉 = (Base‘𝑊)
2 nmsq.h . . 3 , = (·𝑖𝑊)
3 nmsq.n . . 3 𝑁 = (norm‘𝑊)
4 eqid 2771 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2771 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 23190 . 2 (𝑊 ∈ ℂPreHil ↔ ((𝑊 ∈ PreHil ∧ 𝑊 ∈ NrmMod ∧ (Scalar‘𝑊) = (ℂflds (Base‘(Scalar‘𝑊)))) ∧ (√ “ ((Base‘(Scalar‘𝑊)) ∩ (0[,)+∞))) ⊆ (Base‘(Scalar‘𝑊)) ∧ 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥)))))
76simp3bi 1141 1 (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1071   = wceq 1631  wcel 2145  cin 3723  wss 3724  cmpt 4864  cima 5253  cfv 6032  (class class class)co 6794  0cc0 10139  +∞cpnf 10274  [,)cico 12383  csqrt 14182  Basecbs 16065  s cress 16066  Scalarcsca 16153  ·𝑖cip 16155  fldccnfld 19962  PreHilcphl 20187  normcnm 22602  NrmModcnlm 22606  ℂPreHilccph 23186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-nul 4924
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fv 6040  df-ov 6797  df-cph 23188
This theorem is referenced by:  cphnm  23213  cphnmf  23215  cphtchnm  23249
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