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Definition df-dip 27865
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 27864 . 2 class ·𝑖OLD
2 vu . . 3 setvar 𝑢
3 cnv 27748 . . 3 class NrmCVec
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1631 . . . . 5 class 𝑢
7 cba 27750 . . . . 5 class BaseSet
86, 7cfv 6049 . . . 4 class (BaseSet‘𝑢)
9 c1 10129 . . . . . . 7 class 1
10 c4 11264 . . . . . . 7 class 4
11 cfz 12519 . . . . . . 7 class ...
129, 10, 11co 6813 . . . . . 6 class (1...4)
13 ci 10130 . . . . . . . 8 class i
14 vk . . . . . . . . 9 setvar 𝑘
1514cv 1631 . . . . . . . 8 class 𝑘
16 cexp 13054 . . . . . . . 8 class
1713, 15, 16co 6813 . . . . . . 7 class (i↑𝑘)
184cv 1631 . . . . . . . . . 10 class 𝑥
195cv 1631 . . . . . . . . . . 11 class 𝑦
20 cns 27751 . . . . . . . . . . . 12 class ·𝑠OLD
216, 20cfv 6049 . . . . . . . . . . 11 class ( ·𝑠OLD𝑢)
2217, 19, 21co 6813 . . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD𝑢)𝑦)
23 cpv 27749 . . . . . . . . . . 11 class +𝑣
246, 23cfv 6049 . . . . . . . . . 10 class ( +𝑣𝑢)
2518, 22, 24co 6813 . . . . . . . . 9 class (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))
26 cnmcv 27754 . . . . . . . . . 10 class normCV
276, 26cfv 6049 . . . . . . . . 9 class (normCV𝑢)
2825, 27cfv 6049 . . . . . . . 8 class ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))
29 c2 11262 . . . . . . . 8 class 2
3028, 29, 16co 6813 . . . . . . 7 class (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)
31 cmul 10133 . . . . . . 7 class ·
3217, 30, 31co 6813 . . . . . 6 class ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
3312, 32, 14csu 14615 . . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
34 cdiv 10876 . . . . 5 class /
3533, 10, 34co 6813 . . . 4 class 𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)
364, 5, 8, 8, 35cmpt2 6815 . . 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 4881 . 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 1632 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  27866
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