Definition df-dip 31360

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 31359	. 2 class ·𝑖OLD
2		vu	. . 3 setvar 𝑢
3		cnv 31243	. . 3 class NrmCVec
4		vx	. . . 4 setvar 𝑥
5		vy	. . . 4 setvar 𝑦
6	2	cv 1569	. . . . 5 class 𝑢
7		cba 31245	. . . . 5 class BaseSet
8	6, 7	cfv 6566	. . . 4 class (BaseSet‘𝑢)
9		c1 11240	. . . . . . 7 class 1
10		c4 12439	. . . . . . 7 class 4
11		cfz 13679	. . . . . . 7 class ...
12	9, 10, 11	co 7451	. . . . . 6 class (1...4)
13		ci 11241	. . . . . . . 8 class i
14		vk	. . . . . . . . 9 setvar 𝑘
15	14	cv 1569	. . . . . . . 8 class 𝑘
16		cexp 14244	. . . . . . . 8 class ↑
17	13, 15, 16	co 7451	. . . . . . 7 class (i↑𝑘)
18	4	cv 1569	. . . . . . . . . 10 class 𝑥
19	5	cv 1569	. . . . . . . . . . 11 class 𝑦
20		cns 31246	. . . . . . . . . . . 12 class ·𝑠OLD
21	6, 20	cfv 6566	. . . . . . . . . . 11 class ( ·𝑠OLD ‘𝑢)
22	17, 19, 21	co 7451	. . . . . . . . . 10 class ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)
23		cpv 31244	. . . . . . . . . . 11 class +𝑣
24	6, 23	cfv 6566	. . . . . . . . . 10 class ( +𝑣 ‘𝑢)
25	18, 22, 24	co 7451	. . . . . . . . 9 class (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))
26		cnmcv 31249	. . . . . . . . . 10 class normCV
27	6, 26	cfv 6566	. . . . . . . . 9 class (normCV‘𝑢)
28	25, 27	cfv 6566	. . . . . . . 8 class ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))
29		c2 12437	. . . . . . . 8 class 2
30	28, 29, 16	co 7451	. . . . . . 7 class (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)
31		cmul 11244	. . . . . . 7 class ·
32	17, 30, 31	co 7451	. . . . . 6 class ((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
33	12, 32, 14	csu 15897	. . . . 5 class Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2))
34		cdiv 12013	. . . . 5 class /
35	33, 10, 34	co 7451	. . . 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 7453	. . 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 5192	. 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 1570	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 used by:  dipfval  31361