Theorem ipval 30761

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 30760	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
7	6	oveqd 7377	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝑃𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵))
8		fvoveq1 7383	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))))
9	8	oveq1d 7375	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2))
10	9	oveq2d 7376	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)))
11	10	sumeq2sdv 15630	. . . . 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 7375	. . . 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 7368	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((i↑𝑘)𝑆𝑦) = ((i↑𝑘)𝑆𝐵))
14	13	oveq2d 7376	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴𝐺((i↑𝑘)𝑆𝑦)) = (𝐴𝐺((i↑𝑘)𝑆𝐵)))
15	14	fveq2d 6839	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))))
16	15	oveq1d 7375	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2))
17	16	oveq2d 7376	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)))
18	17	sumeq2sdv 15630	. . . . 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 7375	. . . 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 2737	. . . 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 7393	. . . 4 ⊢ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4) ∈ V
22	12, 19, 20, 21	ovmpo 7520	. . 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 2792	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
24	23	3impb 1115	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 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ‘cfv 6493  (class class class)co 7360   ∈ cmpo 7362  1c1 11031  ici 11032   · cmul 11035   / cdiv 11798  2c2 12204  4c4 12206  ...cfz 13427  ↑cexp 13988  Σcsu 15613  NrmCVeccnv 30642   +_𝑣 cpv 30643  BaseSetcba 30644   ·_𝑠OLD cns 30645  norm_CVcnmcv 30648  ·_𝑖OLDcdip 30758

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-pred 6260  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-1st 7935  df-2nd 7936  df-frecs 8225  df-wrecs 8256  df-recs 8305  df-rdg 8343  df-seq 13929  df-sum 15614  df-dip 30759

This theorem is referenced by:  ipval2  30765  dipcl  30770  ipf  30771