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Theorem cphnmfval 23907
 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 2758 . . 3 (Scalar‘𝑊) = (Scalar‘𝑊)
5 eqid 2758 . . 3 (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊))
61, 2, 3, 4, 5iscph 23885 . 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 1144 1 (𝑊 ∈ ℂPreHil → 𝑁 = (𝑥𝑉 ↦ (√‘(𝑥 , 𝑥))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ∩ cin 3859   ⊆ wss 3860   ↦ cmpt 5116   “ cima 5531  ‘cfv 6340  (class class class)co 7156  0cc0 10588  +∞cpnf 10723  [,)cico 12794  √csqrt 14653  Basecbs 16555   ↾s cress 16556  Scalarcsca 16640  ·𝑖cip 16642  ℂfldccnfld 20180  PreHilcphl 20403  normcnm 23292  NrmModcnlm 23296  ℂPreHilccph 23881 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-nul 5180 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5037  df-opab 5099  df-mpt 5117  df-xp 5534  df-cnv 5536  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6299  df-fv 6348  df-ov 7159  df-cph 23883 This theorem is referenced by:  cphnm  23908  cphnmf  23910  cphtcphnm  23944  cphsscph  23965
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