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Theorem dipfval 28485
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 = ( +𝑣𝑈)
dipfval.4 𝑆 = ( ·𝑠OLD𝑈)
dipfval.6 𝑁 = (normCV𝑈)
dipfval.7 𝑃 = (·𝑖OLD𝑈)
Assertion
Ref Expression
dipfval (𝑈 ∈ NrmCVec → 𝑃 = (𝑥𝑋, 𝑦𝑋 ↦ (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.
StepHypRef Expression
1 dipfval.7 . 2 𝑃 = (·𝑖OLD𝑈)
2 fveq2 6645 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 dipfval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2851 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5 fveq2 6645 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
6 dipfval.6 . . . . . . . . . 10 𝑁 = (normCV𝑈)
75, 6eqtr4di 2851 . . . . . . . . 9 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
8 fveq2 6645 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
9 dipfval.2 . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
108, 9eqtr4di 2851 . . . . . . . . . 10 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
11 eqidd 2799 . . . . . . . . . 10 (𝑢 = 𝑈𝑥 = 𝑥)
12 fveq2 6645 . . . . . . . . . . . 12 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
13 dipfval.4 . . . . . . . . . . . 12 𝑆 = ( ·𝑠OLD𝑈)
1412, 13eqtr4di 2851 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
1514oveqd 7152 . . . . . . . . . 10 (𝑢 = 𝑈 → ((i↑𝑘)( ·𝑠OLD𝑢)𝑦) = ((i↑𝑘)𝑆𝑦))
1610, 11, 15oveq123d 7156 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)) = (𝑥𝐺((i↑𝑘)𝑆𝑦)))
177, 16fveq12d 6652 . . . . . . . 8 (𝑢 = 𝑈 → ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))) = (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))))
1817oveq1d 7150 . . . . . . 7 (𝑢 = 𝑈 → (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2) = ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2))
1918oveq2d 7151 . . . . . 6 (𝑢 = 𝑈 → ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)))
2019sumeq2sdv 15053 . . . . 5 (𝑢 = 𝑈 → Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) = Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)))
2120oveq1d 7150 . . . 4 (𝑢 = 𝑈 → (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4) = (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4))
224, 4, 21mpoeq123dv 7208 . . 3 (𝑢 = 𝑈 → (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)) = (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
23 df-dip 28484 . . 3 ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
243fvexi 6659 . . . 4 𝑋 ∈ V
2524, 24mpoex 7760 . . 3 (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) ∈ V
2622, 23, 25fvmpt 6745 . 2 (𝑈 ∈ NrmCVec → (·𝑖OLD𝑈) = (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
271, 26syl5eq 2845 1 (𝑈 ∈ NrmCVec → 𝑃 = (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  wcel 2111  cfv 6324  (class class class)co 7135  cmpo 7137  1c1 10527  ici 10528   · cmul 10531   / cdiv 11286  2c2 11680  4c4 11682  ...cfz 12885  cexp 13425  Σcsu 15034  NrmCVeccnv 28367   +𝑣 cpv 28368  BaseSetcba 28369   ·𝑠OLD cns 28370  normCVcnmcv 28373  ·𝑖OLDcdip 28483
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12886  df-seq 13365  df-sum 15035  df-dip 28484
This theorem is referenced by:  ipval  28486  ipf  28496  dipcn  28503
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