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Definition df-dip 30720
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 30719 . 2 class ·𝑖OLD
2 vu . . 3 setvar 𝑢
3 cnv 30603 . . 3 class NrmCVec
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1539 . . . . 5 class 𝑢
7 cba 30605 . . . . 5 class BaseSet
86, 7cfv 6561 . . . 4 class (BaseSet‘𝑢)
9 c1 11156 . . . . . . 7 class 1
10 c4 12323 . . . . . . 7 class 4
11 cfz 13547 . . . . . . 7 class ...
129, 10, 11co 7431 . . . . . 6 class (1...4)
13 ci 11157 . . . . . . . 8 class i
14 vk . . . . . . . . 9 setvar 𝑘
1514cv 1539 . . . . . . . 8 class 𝑘
16 cexp 14102 . . . . . . . 8 class
1713, 15, 16co 7431 . . . . . . 7 class (i↑𝑘)
184cv 1539 . . . . . . . . . 10 class 𝑥
195cv 1539 . . . . . . . . . . 11 class 𝑦
20 cns 30606 . . . . . . . . . . . 12 class ·𝑠OLD
216, 20cfv 6561 . . . . . . . . . . 11 class ( ·𝑠OLD𝑢)
2217, 19, 21co 7431 . . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD𝑢)𝑦)
23 cpv 30604 . . . . . . . . . . 11 class +𝑣
246, 23cfv 6561 . . . . . . . . . 10 class ( +𝑣𝑢)
2518, 22, 24co 7431 . . . . . . . . 9 class (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))
26 cnmcv 30609 . . . . . . . . . 10 class normCV
276, 26cfv 6561 . . . . . . . . 9 class (normCV𝑢)
2825, 27cfv 6561 . . . . . . . 8 class ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))
29 c2 12321 . . . . . . . 8 class 2
3028, 29, 16co 7431 . . . . . . 7 class (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)
31 cmul 11160 . . . . . . 7 class ·
3217, 30, 31co 7431 . . . . . 6 class ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
3312, 32, 14csu 15722 . . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
34 cdiv 11920 . . . . 5 class /
3533, 10, 34co 7431 . . . 4 class 𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)
364, 5, 8, 8, 35cmpo 7433 . . 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 5225 . 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 1540 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  30721
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