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Definition df-dip 26733
Description: Define a function that maps a normed complex vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is (1st𝑤), the scalar product is (2nd𝑤), and the norm is 𝑛. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
df-dip ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
Distinct variable group:   𝑢,𝑘,𝑥,𝑦

Detailed syntax breakdown of Definition df-dip
StepHypRef Expression
1 cdip 26732 . 2 class ·𝑖OLD
2 vu . . 3 setvar 𝑢
3 cnv 26616 . . 3 class NrmCVec
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1473 . . . . 5 class 𝑢
7 cba 26618 . . . . 5 class BaseSet
86, 7cfv 5789 . . . 4 class (BaseSet‘𝑢)
9 c1 9793 . . . . . . 7 class 1
10 c4 10921 . . . . . . 7 class 4
11 cfz 12154 . . . . . . 7 class ...
129, 10, 11co 6526 . . . . . 6 class (1...4)
13 ci 9794 . . . . . . . 8 class i
14 vk . . . . . . . . 9 setvar 𝑘
1514cv 1473 . . . . . . . 8 class 𝑘
16 cexp 12679 . . . . . . . 8 class
1713, 15, 16co 6526 . . . . . . 7 class (i↑𝑘)
184cv 1473 . . . . . . . . . 10 class 𝑥
195cv 1473 . . . . . . . . . . 11 class 𝑦
20 cns 26619 . . . . . . . . . . . 12 class ·𝑠OLD
216, 20cfv 5789 . . . . . . . . . . 11 class ( ·𝑠OLD𝑢)
2217, 19, 21co 6526 . . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD𝑢)𝑦)
23 cpv 26617 . . . . . . . . . . 11 class +𝑣
246, 23cfv 5789 . . . . . . . . . 10 class ( +𝑣𝑢)
2518, 22, 24co 6526 . . . . . . . . 9 class (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))
26 cnmcv 26622 . . . . . . . . . 10 class normCV
276, 26cfv 5789 . . . . . . . . 9 class (normCV𝑢)
2825, 27cfv 5789 . . . . . . . 8 class ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))
29 c2 10919 . . . . . . . 8 class 2
3028, 29, 16co 6526 . . . . . . 7 class (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)
31 cmul 9797 . . . . . . 7 class ·
3217, 30, 31co 6526 . . . . . 6 class ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
3312, 32, 14csu 14212 . . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
34 cdiv 10535 . . . . 5 class /
3533, 10, 34co 6526 . . . 4 class 𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)
364, 5, 8, 8, 35cmpt2 6528 . . 3 class (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4))
372, 3, 36cmpt 4637 . 2 class (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
381, 37wceq 1474 1 wff ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
Colors of variables: wff setvar class
This definition is referenced by:  dipfval  26734
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