Theorem dipfval 31304

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 2814	. . . 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 2814	. . . . . . . . 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 2814	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
11		eqidd 2762	. . . . . . . . . 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 2814	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑆)
15	14	oveqd 7437	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦) = ((i↑𝑘)𝑆𝑦))
16	10, 11, 15	oveq123d 7441	. . . . . . . . 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 7435	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2) = ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2))
19	18	oveq2d 7436	. . . . . 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 15870	. . . . 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 7435	. . . 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 7495	. . 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 31303	. . 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 8092	. . 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 2808	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 2145  ‘cfv 6538  (class class class)co 7420   ∈ cmpo 7422  1c1 11201  ici 11202   · cmul 11205   / cdiv 11973  2c2 12397  4c4 12399  ...cfz 13639  ↑cexp 14204  Σcsu 15853  NrmCVeccnv 31186   +_𝑣 cpv 31187  BaseSetcba 31188   ·_𝑠OLD cns 31189  norm_CVcnmcv 31192  ·_𝑖OLDcdip 31302

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  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 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-frecs 8299  df-wrecs 8330  df-recs 8379  df-rdg 8418  df-seq 14145  df-sum 15854  df-dip 31303

This theorem is used by:  ipval  31305  ipf  31315  dipcn  31322