Definition df-dip 30733

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 30732	. 2 class ·𝑖OLD
2		vu	. . 3 setvar 𝑢
3		cnv 30616	. . 3 class NrmCVec
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1536	. . . . 5 class 𝑢
7		cba 30618	. . . . 5 class BaseSet
8	6, 7	cfv 6573	. . . 4 class (BaseSet‘𝑢)
9		c1 11185	. . . . . . 7 class 1
10		c4 12350	. . . . . . 7 class 4
11		cfz 13567	. . . . . . 7 class ...
12	9, 10, 11	co 7448	. . . . . 6 class (1...4)
13		ci 11186	. . . . . . . 8 class i
14		vk	. . . . . . . . 9 setvar 𝑘
15	14	cv 1536	. . . . . . . 8 class 𝑘
16		cexp 14112	. . . . . . . 8 class ↑
17	13, 15, 16	co 7448	. . . . . . 7 class (i↑𝑘)
18	4	cv 1536	. . . . . . . . . 10 class 𝑥
19	5	cv 1536	. . . . . . . . . . 11 class 𝑦
20		cns 30619	. . . . . . . . . . . 12 class ·𝑠OLD
21	6, 20	cfv 6573	. . . . . . . . . . 11 class ( ·𝑠OLD ‘𝑢)
22	17, 19, 21	co 7448	. . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)
23		cpv 30617	. . . . . . . . . . 11 class +𝑣
24	6, 23	cfv 6573	. . . . . . . . . 10 class ( +𝑣 ‘𝑢)
25	18, 22, 24	co 7448	. . . . . . . . 9 class (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))
26		cnmcv 30622	. . . . . . . . . 10 class normCV
27	6, 26	cfv 6573	. . . . . . . . 9 class (normCV‘𝑢)
28	25, 27	cfv 6573	. . . . . . . 8 class ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))
29		c2 12348	. . . . . . . 8 class 2
30	28, 29, 16	co 7448	. . . . . . 7 class (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)
31		cmul 11189	. . . . . . 7 class ·
32	17, 30, 31	co 7448	. . . . . 6 class ((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
33	12, 32, 14	csu 15734	. . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
34		cdiv 11947	. . . . 5 class /
35	33, 10, 34	co 7448	. . . 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 7450	. . 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 5249	. 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  30734