Theorem dipfval 31127

Description: The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-2007.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)

Hypotheses

Ref	Expression
dipfval.1	⊢ 𝑋 = (BaseSet‘𝑈)
dipfval.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
dipfval.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
dipfval.6	⊢ 𝑁 = (normCV‘𝑈)
dipfval.7	⊢ 𝑃 = (·𝑖OLD‘𝑈)

Assertion

Ref	Expression
dipfval	⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))

Distinct variable groups:   𝑥,𝑘,𝑦,𝐺   𝑘,𝑁,𝑥,𝑦   𝑆,𝑘,𝑥,𝑦   𝑈,𝑘,𝑥,𝑦   𝑘,𝑋,𝑥,𝑦

Allowed substitution hints:   𝑃(𝑥, 𝑦, 𝑘)

Proof of Theorem dipfval

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dipfval.7	. 2 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
2		fveq2 6885	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		dipfval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2818	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
6		dipfval.6	. . . . . . . . . 10 ⊢ 𝑁 = (normCV‘𝑈)
7	5, 6	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝑁)
8		fveq2 6885	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
9		dipfval.2	. . . . . . . . . . 11 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
10	8, 9	eqtr4di 2818	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
11		eqidd 2766	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → 𝑥 = 𝑥)
12		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = ( ·𝑠OLD ‘𝑈))
13		dipfval.4	. . . . . . . . . . . 12 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
14	12, 13	eqtr4di 2818	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑆)
15	14	oveqd 7436	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦) = ((i↑𝑘)𝑆𝑦))
16	10, 11, 15	oveq123d 7440	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)) = (𝑥𝐺((i↑𝑘)𝑆𝑦)))
17	7, 16	fveq12d 6892	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))) = (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))))
18	17	oveq1d 7434	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2) = ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2))
19	18	oveq2d 7435	. . . . . 6 ⊢ (𝑢 = 𝑈 → ((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)))
20	19	sumeq2sdv 15780	. . . . 5 ⊢ (𝑢 = 𝑈 → Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) = Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)))
21	20	oveq1d 7434	. . . 4 ⊢ (𝑢 = 𝑈 → (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))
22	4, 4, 21	mpoeq123dv 7494	. . 3 ⊢ (𝑢 = 𝑈 → (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2)) / 4)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
23		df-dip 31126	. . 3 ⊢ ·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)))
24	3	fvexi 6899	. . . 4 ⊢ 𝑋 ∈ V
25	24, 24	mpoex 8082	. . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) ∈ V
26	22, 23, 25	fvmpt 6993	. 2 ⊢ (𝑈 ∈ NrmCVec → (·𝑖OLD‘𝑈) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
27	1, 26	eqtrid 2812	1 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7419   ∈ cmpo 7421  1c1 11118  ici 11119   · cmul 11122   / cdiv 11888  2c2 12312  4c4 12314  ...cfz 13553  ↑cexp 14117  Σcsu 15763  NrmCVeccnv 31009   +_𝑣 cpv 31010  BaseSetcba 31011   ·_𝑠OLD cns 31012  norm_CVcnmcv 31015  ·_𝑖OLDcdip 31125

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-seq 14058  df-sum 15764  df-dip 31126

This theorem is used by:  ipval  31128  ipf  31138  dipcn  31145