Theorem dipfval 30790

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 6842	. . . . 5 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3		dipfval.1	. . . . 5 ⊢ 𝑋 = (BaseSet‘𝑈)
4	2, 3	eqtr4di 2790	. . . 4 ⊢ (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5		fveq2 6842	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = (normCV‘𝑈))
6		dipfval.6	. . . . . . . . . 10 ⊢ 𝑁 = (normCV‘𝑈)
7	5, 6	eqtr4di 2790	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (normCV‘𝑢) = 𝑁)
8		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = ( +𝑣 ‘𝑈))
9		dipfval.2	. . . . . . . . . . 11 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
10	8, 9	eqtr4di 2790	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ( +𝑣 ‘𝑢) = 𝐺)
11		eqidd 2738	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → 𝑥 = 𝑥)
12		fveq2 6842	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = ( ·𝑠OLD ‘𝑈))
13		dipfval.4	. . . . . . . . . . . 12 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
14	12, 13	eqtr4di 2790	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑈 → ( ·𝑠OLD ‘𝑢) = 𝑆)
15	14	oveqd 7385	. . . . . . . . . 10 ⊢ (𝑢 = 𝑈 → ((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦) = ((i↑𝑘)𝑆𝑦))
16	10, 11, 15	oveq123d 7389	. . . . . . . . 9 ⊢ (𝑢 = 𝑈 → (𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)) = (𝑥𝐺((i↑𝑘)𝑆𝑦)))
17	7, 16	fveq12d 6849	. . . . . . . 8 ⊢ (𝑢 = 𝑈 → ((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦))) = (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))))
18	17	oveq1d 7383	. . . . . . 7 ⊢ (𝑢 = 𝑈 → (((normCV‘𝑢)‘(𝑥( +𝑣 ‘𝑢)((i↑𝑘)( ·𝑠OLD ‘𝑢)𝑦)))↑2) = ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2))
19	18	oveq2d 7384	. . . . . 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 15638	. . . . 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 7383	. . . 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 7443	. . 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 30789	. . 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 6856	. . . 4 ⊢ 𝑋 ∈ V
25	24, 24	mpoex 8033	. . 3 ⊢ (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) ∈ V
26	22, 23, 25	fvmpt 6949	. 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 2784	1 ⊢ (𝑈 ∈ NrmCVec → 𝑃 = (𝑥 ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368   ∈ cmpo 7370  1c1 11039  ici 11040   · cmul 11043   / cdiv 11806  2c2 12212  4c4 12214  ...cfz 13435  ↑cexp 13996  Σcsu 15621  NrmCVeccnv 30672   +_𝑣 cpv 30673  BaseSetcba 30674   ·_𝑠OLD cns 30675  norm_CVcnmcv 30678  ·_𝑖OLDcdip 30788

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 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

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 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-pred 6267  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-frecs 8233  df-wrecs 8264  df-recs 8313  df-rdg 8351  df-seq 13937  df-sum 15622  df-dip 30789

This theorem is referenced by:  ipval  30791  ipf  30801  dipcn  30808