Theorem ipval 30731

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 30730	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
7	6	oveqd 7447	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝑃𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵))
8		fvoveq1 7453	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))))
9	8	oveq1d 7445	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2))
10	9	oveq2d 7446	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)))
11	10	sumeq2sdv 15735	. . . . 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 7445	. . . 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 7438	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((i↑𝑘)𝑆𝑦) = ((i↑𝑘)𝑆𝐵))
14	13	oveq2d 7446	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴𝐺((i↑𝑘)𝑆𝑦)) = (𝐴𝐺((i↑𝑘)𝑆𝐵)))
15	14	fveq2d 6910	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))))
16	15	oveq1d 7445	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2))
17	16	oveq2d 7446	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)))
18	17	sumeq2sdv 15735	. . . . 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 7445	. . . 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 2734	. . . 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 7463	. . . 4 ⊢ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4) ∈ V
22	12, 19, 20, 21	ovmpo 7592	. . 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 2794	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐵 ∈ 𝑋)) → (𝐴𝑃𝐵) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4))
24	23	3impb 1114	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 1086   = wceq 1536   ∈ wcel 2105  ‘cfv 6562  (class class class)co 7430   ∈ cmpo 7432  1c1 11153  ici 11154   · cmul 11157   / cdiv 11917  2c2 12318  4c4 12320  ...cfz 13543  ↑cexp 14098  Σcsu 15718  NrmCVeccnv 30612   +_𝑣 cpv 30613  BaseSetcba 30614   ·_𝑠OLD cns 30615  norm_CVcnmcv 30618  ·_𝑖OLDcdip 30728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-pred 6322  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-seq 14039  df-sum 15719  df-dip 30729

This theorem is referenced by:  ipval2  30735  dipcl  30740  ipf  30741