Definition df-dip 28480

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 (1^st ‘𝑤), the scalar product is (2^nd ‘𝑤), 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) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)))

Distinct variable group:   𝑢,𝑘,𝑥,𝑦

Detailed syntax breakdown of Definition df-dip

Step	Hyp	Ref	Expression
1		cdip 28479	. 2 class ·𝑖OLD
2		vu	. . 3 setvar 𝑢
3		cnv 28363	. . 3 class NrmCVec
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1536	. . . . 5 class 𝑢
7		cba 28365	. . . . 5 class BaseSet
8	6, 7	cfv 6357	. . . 4 class (BaseSet‘𝑢)
9		c1 10540	. . . . . . 7 class 1
10		c4 11697	. . . . . . 7 class 4
11		cfz 12895	. . . . . . 7 class ...
12	9, 10, 11	co 7158	. . . . . 6 class (1...4)
13		ci 10541	. . . . . . . 8 class i
14		vk	. . . . . . . . 9 setvar 𝑘
15	14	cv 1536	. . . . . . . 8 class 𝑘
16		cexp 13432	. . . . . . . 8 class ↑
17	13, 15, 16	co 7158	. . . . . . 7 class (i↑𝑘)
18	4	cv 1536	. . . . . . . . . 10 class 𝑥
19	5	cv 1536	. . . . . . . . . . 11 class 𝑦
20		cns 28366	. . . . . . . . . . . 12 class ·𝑠OLD
21	6, 20	cfv 6357	. . . . . . . . . . 11 class ( ·𝑠OLD ‘𝑢)
22	17, 19, 21	co 7158	. . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)
23		cpv 28364	. . . . . . . . . . 11 class +𝑣
24	6, 23	cfv 6357	. . . . . . . . . 10 class ( +𝑣 ‘𝑢)
25	18, 22, 24	co 7158	. . . . . . . . 9 class (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))
26		cnmcv 28369	. . . . . . . . . 10 class normCV
27	6, 26	cfv 6357	. . . . . . . . 9 class (normCV‘𝑢)
28	25, 27	cfv 6357	. . . . . . . 8 class ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))
29		c2 11695	. . . . . . . 8 class 2
30	28, 29, 16	co 7158	. . . . . . 7 class (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)
31		cmul 10544	. . . . . . 7 class ·
32	17, 30, 31	co 7158	. . . . . 6 class ((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
33	12, 32, 14	csu 15044	. . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
34		cdiv 11299	. . . . 5 class /
35	33, 10, 34	co 7158	. . . 4 class (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)
36	4, 5, 8, 8, 35	cmpo 7160	. . 3 class (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4))
37	2, 3, 36	cmpt 5148	. 2 class (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)))
38	1, 37	wceq 1537	1 wff ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)))

This definition is referenced by:  dipfval  28481