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Definition df-dip 26940
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 26939 . 2 class ·𝑖OLD
2 vu . . 3 setvar 𝑢
3 cnv 26823 . . 3 class NrmCVec
4 vx . . . 4 setvar 𝑥
5 vy . . . 4 setvar 𝑦
62cv 1474 . . . . 5 class 𝑢
7 cba 26825 . . . . 5 class BaseSet
86, 7cfv 5804 . . . 4 class (BaseSet‘𝑢)
9 c1 9816 . . . . . . 7 class 1
10 c4 10949 . . . . . . 7 class 4
11 cfz 12197 . . . . . . 7 class ...
129, 10, 11co 6549 . . . . . 6 class (1...4)
13 ci 9817 . . . . . . . 8 class i
14 vk . . . . . . . . 9 setvar 𝑘
1514cv 1474 . . . . . . . 8 class 𝑘
16 cexp 12722 . . . . . . . 8 class
1713, 15, 16co 6549 . . . . . . 7 class (i↑𝑘)
184cv 1474 . . . . . . . . . 10 class 𝑥
195cv 1474 . . . . . . . . . . 11 class 𝑦
20 cns 26826 . . . . . . . . . . . 12 class ·𝑠OLD
216, 20cfv 5804 . . . . . . . . . . 11 class ( ·𝑠OLD𝑢)
2217, 19, 21co 6549 . . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD𝑢)𝑦)
23 cpv 26824 . . . . . . . . . . 11 class +𝑣
246, 23cfv 5804 . . . . . . . . . 10 class ( +𝑣𝑢)
2518, 22, 24co 6549 . . . . . . . . 9 class (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))
26 cnmcv 26829 . . . . . . . . . 10 class normCV
276, 26cfv 5804 . . . . . . . . 9 class (normCV𝑢)
2825, 27cfv 5804 . . . . . . . 8 class ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))
29 c2 10947 . . . . . . . 8 class 2
3028, 29, 16co 6549 . . . . . . 7 class (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)
31 cmul 9820 . . . . . . 7 class ·
3217, 30, 31co 6549 . . . . . 6 class ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
3312, 32, 14csu 14264 . . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2))
34 cdiv 10563 . . . . 5 class /
3533, 10, 34co 6549 . . . 4 class 𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)
364, 5, 8, 8, 35cmpt2 6551 . . 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 4643 . 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 1475 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  26941
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