Theorem ipval 31311

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 31310	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
7	6	oveqd 7438	. . 3 ⊢ (𝑈 ∈ NrmCVec → (𝐴𝑃𝐵) = (𝐴(𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))𝐵))
8		fvoveq1 7444	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))))
9	8	oveq1d 7436	. . . . . . 7 ⊢ (𝑥 = 𝐴 → ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2))
10	9	oveq2d 7437	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)))
11	10	sumeq2sdv 15876	. . . . 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 7436	. . . 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 7429	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((i↑𝑘)𝑆𝑦) = ((i↑𝑘)𝑆𝐵))
14	13	oveq2d 7437	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (𝐴𝐺((i↑𝑘)𝑆𝑦)) = (𝐴𝐺((i↑𝑘)𝑆𝐵)))
15	14	fveq2d 6890	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦))) = (𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵))))
16	15	oveq1d 7436	. . . . . . 7 ⊢ (𝑦 = 𝐵 → ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2) = ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2))
17	16	oveq2d 7437	. . . . . 6 ⊢ (𝑦 = 𝐵 → ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)))
18	17	sumeq2sdv 15876	. . . . 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 7436	. . . 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 2760	. . . 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 7454	. . . 4 ⊢ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝐴𝐺((i↑𝑘)𝑆𝐵)))↑2)) / 4) ∈ V
22	12, 19, 20, 21	ovmpo 7581	. . 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 2815	. 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))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ‘cfv 6539  (class class class)co 7421   ∈ cmpo 7423  1c1 11205  ici 11206   · cmul 11209   / cdiv 11977  2c2 12401  4c4 12403  ...cfz 13643  ↑cexp 14208  Σcsu 15859  NrmCVeccnv 31192   +_𝑣 cpv 31193  BaseSetcba 31194   ·_𝑠OLD cns 31195  norm_CVcnmcv 31198  ·_𝑖OLDcdip 31308

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7752

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-pred 6304  df-iota 6494  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-ov 7424  df-oprab 7425  df-mpo 7426  df-1st 8002  df-2nd 8003  df-frecs 8303  df-wrecs 8334  df-recs 8383  df-rdg 8422  df-seq 14149  df-sum 15860  df-dip 31309

This theorem is used by:  ipval2  31315  dipcl  31320  ipf  31321