Definition df-dip 30787

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 30786	. 2 class ·𝑖OLD
2		vu	. . 3 setvar 𝑢
3		cnv 30670	. . 3 class NrmCVec
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1541	. . . . 5 class 𝑢
7		cba 30672	. . . . 5 class BaseSet
8	6, 7	cfv 6492	. . . 4 class (BaseSet‘𝑢)
9		c1 11030	. . . . . . 7 class 1
10		c4 12229	. . . . . . 7 class 4
11		cfz 13452	. . . . . . 7 class ...
12	9, 10, 11	co 7360	. . . . . 6 class (1...4)
13		ci 11031	. . . . . . . 8 class i
14		vk	. . . . . . . . 9 setvar 𝑘
15	14	cv 1541	. . . . . . . 8 class 𝑘
16		cexp 14014	. . . . . . . 8 class ↑
17	13, 15, 16	co 7360	. . . . . . 7 class (i↑𝑘)
18	4	cv 1541	. . . . . . . . . 10 class 𝑥
19	5	cv 1541	. . . . . . . . . . 11 class 𝑦
20		cns 30673	. . . . . . . . . . . 12 class ·𝑠OLD
21	6, 20	cfv 6492	. . . . . . . . . . 11 class ( ·𝑠OLD ‘𝑢)
22	17, 19, 21	co 7360	. . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)
23		cpv 30671	. . . . . . . . . . 11 class +𝑣
24	6, 23	cfv 6492	. . . . . . . . . 10 class ( +𝑣 ‘𝑢)
25	18, 22, 24	co 7360	. . . . . . . . 9 class (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))
26		cnmcv 30676	. . . . . . . . . 10 class normCV
27	6, 26	cfv 6492	. . . . . . . . 9 class (normCV‘𝑢)
28	25, 27	cfv 6492	. . . . . . . 8 class ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))
29		c2 12227	. . . . . . . 8 class 2
30	28, 29, 16	co 7360	. . . . . . 7 class (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)
31		cmul 11034	. . . . . . 7 class ·
32	17, 30, 31	co 7360	. . . . . 6 class ((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
33	12, 32, 14	csu 15639	. . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
34		cdiv 11798	. . . . 5 class /
35	33, 10, 34	co 7360	. . . 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 7362	. . 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 5167	. 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 1542	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  30788