Definition df-dip 28157

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 28156	. 2 class ·𝑖OLD
2		vu	. . 3 setvar 𝑢
3		cnv 28040	. . 3 class NrmCVec
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1519	. . . . 5 class 𝑢
7		cba 28042	. . . . 5 class BaseSet
8	6, 7	cfv 6217	. . . 4 class (BaseSet‘𝑢)
9		c1 10373	. . . . . . 7 class 1
10		c4 11531	. . . . . . 7 class 4
11		cfz 12731	. . . . . . 7 class ...
12	9, 10, 11	co 7007	. . . . . 6 class (1...4)
13		ci 10374	. . . . . . . 8 class i
14		vk	. . . . . . . . 9 setvar 𝑘
15	14	cv 1519	. . . . . . . 8 class 𝑘
16		cexp 13267	. . . . . . . 8 class ↑
17	13, 15, 16	co 7007	. . . . . . 7 class (i↑𝑘)
18	4	cv 1519	. . . . . . . . . 10 class 𝑥
19	5	cv 1519	. . . . . . . . . . 11 class 𝑦
20		cns 28043	. . . . . . . . . . . 12 class ·𝑠OLD
21	6, 20	cfv 6217	. . . . . . . . . . 11 class ( ·𝑠OLD ‘𝑢)
22	17, 19, 21	co 7007	. . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)
23		cpv 28041	. . . . . . . . . . 11 class +𝑣
24	6, 23	cfv 6217	. . . . . . . . . 10 class ( +𝑣 ‘𝑢)
25	18, 22, 24	co 7007	. . . . . . . . 9 class (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))
26		cnmcv 28046	. . . . . . . . . 10 class normCV
27	6, 26	cfv 6217	. . . . . . . . 9 class (normCV‘𝑢)
28	25, 27	cfv 6217	. . . . . . . 8 class ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))
29		c2 11529	. . . . . . . 8 class 2
30	28, 29, 16	co 7007	. . . . . . 7 class (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)
31		cmul 10377	. . . . . . 7 class ·
32	17, 30, 31	co 7007	. . . . . 6 class ((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
33	12, 32, 14	csu 14864	. . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
34		cdiv 11134	. . . . 5 class /
35	33, 10, 34	co 7007	. . . 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 7009	. . 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 5035	. 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 1520	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  28158