Theorem ipval 31036

Description: Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, the norm is 𝑁, and the set of vectors is 𝑋. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-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
ipval	⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))

Distinct variable groups:   𝑘,𝐺   𝑘,𝑁   𝑆,𝑘   𝑈,𝑘   𝐴,𝑘   𝐵,𝑘   𝑘,𝑋

Allowed substitution hint:   𝑃(𝑘)

Proof of Theorem ipval

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dipfval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
2		dipfval.2	. . . . 5 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3		dipfval.4	. . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
4		dipfval.6	. . . . 5 ⊢ 𝑁 = (normCV‘𝑈)
5		dipfval.7	. . . . 5 ⊢ 𝑃 = (·𝑖OLD‘𝑈)
6	1, 2, 3, 4, 5	dipfval 31035	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
7	6	oveqd 7429	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝑃𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵))
8		fvoveq1 7435	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))))
9	8	oveq1d 7427	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2))
10	9	oveq2d 7428	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)))
11	10	sumeq2sdv 15756	. . . . 5 ⊢ (𝑥 = 𝐴 → Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)))
12	11	oveq1d 7427	. . . 4 ⊢ (𝑥 = 𝐴 → (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))
13		oveq2 7420	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((i↑𝑘)𝑆𝑦) = ((i↑𝑘)𝑆𝐵))
14	13	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴𝐺((i↑𝑘)𝑆𝑦)) = (𝐴𝐺((i↑𝑘)𝑆𝐵)))
15	14	fveq2d 6887	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))))
16	15	oveq1d 7427	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2))
17	16	oveq2d 7428	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)))
18	17	sumeq2sdv 15756	. . . . 5 ⊢ (𝑦 = 𝐵 → Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)))
19	18	oveq1d 7427	. . . 4 ⊢ (𝑦 = 𝐵 → (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
20		eqid 2763	. . . 4 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))
21		ovex 7445	. . . 4 ⊢ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4) ∈ V
22	12, 19, 20, 21	ovmpo 7572	. . 3 ⊢ ((𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
23	7, 22	sylan9eq 2818	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
24	23	3impb 1132	1 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ‘cfv 6538  (class class class)co 7412   ∈ cmpo 7414  1c1 11102  ici 11103   · cmul 11106   / cdiv 11872  2c2 12296  4c4 12298  ...cfz 13536  ↑cexp 14099  Σcsu 15739  NrmCVeccnv 30917   +_𝑣 cpv 30918  BaseSetcba 30919   ·_𝑠OLD cns 30920  norm_CVcnmcv 30923  ·_𝑖OLDcdip 31033

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  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 6304  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-seq 14040  df-sum 15740  df-dip 31034

This theorem is referenced by:  ipval2  31040  dipcl  31045  ipf  31046