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Definition df-dip 27684
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 27683 . 2 class ·𝑖OLD
2 vu . . 3 setvar 𝑢
3 cnv 27567 . . 3 class NrmCVec
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1522 . . . . 5 class 𝑢
7 cba 27569 . . . . 5 class BaseSet
86, 7cfv 5926 . . . 4 class (BaseSet‘𝑢)
9 c1 9975 . . . . . . 7 class 1
10 c4 11110 . . . . . . 7 class 4
11 cfz 12364 . . . . . . 7 class ...
129, 10, 11co 6690 . . . . . 6 class (1...4)
13 ci 9976 . . . . . . . 8 class i
14 vk . . . . . . . . 9 setvar 𝑘
1514cv 1522 . . . . . . . 8 class 𝑘
16 cexp 12900 . . . . . . . 8 class
1713, 15, 16co 6690 . . . . . . 7 class (i↑𝑘)
184cv 1522 . . . . . . . . . 10 class 𝑥
195cv 1522 . . . . . . . . . . 11 class 𝑦
20 cns 27570 . . . . . . . . . . . 12 class ·𝑠OLD
216, 20cfv 5926 . . . . . . . . . . 11 class ( ·𝑠OLD𝑢)
2217, 19, 21co 6690 . . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD𝑢)𝑦)
23 cpv 27568 . . . . . . . . . . 11 class +𝑣
246, 23cfv 5926 . . . . . . . . . 10 class ( +𝑣𝑢)
2518, 22, 24co 6690 . . . . . . . . 9 class (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))
26 cnmcv 27573 . . . . . . . . . 10 class normCV
276, 26cfv 5926 . . . . . . . . 9 class (normCV𝑢)
2825, 27cfv 5926 . . . . . . . 8 class ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))
29 c2 11108 . . . . . . . 8 class 2
3028, 29, 16co 6690 . . . . . . 7 class (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)
31 cmul 9979 . . . . . . 7 class ·
3217, 30, 31co 6690 . . . . . 6 class ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
3312, 32, 14csu 14460 . . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
34 cdiv 10722 . . . . 5 class /
3533, 10, 34co 6690 . . . 4 class 𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)
364, 5, 8, 8, 35cmpt2 6692 . . 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 4762 . 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 1523 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  27685
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