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Theorem dipfval 30721
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 6906 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 dipfval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3eqtr4di 2795 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5 fveq2 6906 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
6 dipfval.6 . . . . . . . . . 10 𝑁 = (normCV𝑈)
75, 6eqtr4di 2795 . . . . . . . . 9 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
8 fveq2 6906 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
9 dipfval.2 . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
108, 9eqtr4di 2795 . . . . . . . . . 10 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
11 eqidd 2738 . . . . . . . . . 10 (𝑢 = 𝑈𝑥 = 𝑥)
12 fveq2 6906 . . . . . . . . . . . 12 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
13 dipfval.4 . . . . . . . . . . . 12 𝑆 = ( ·𝑠OLD𝑈)
1412, 13eqtr4di 2795 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
1514oveqd 7448 . . . . . . . . . 10 (𝑢 = 𝑈 → ((i↑𝑘)( ·𝑠OLD𝑢)𝑦) = ((i↑𝑘)𝑆𝑦))
1610, 11, 15oveq123d 7452 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)) = (𝑥𝐺((i↑𝑘)𝑆𝑦)))
177, 16fveq12d 6913 . . . . . . . 8 (𝑢 = 𝑈 → ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))) = (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))))
1817oveq1d 7446 . . . . . . 7 (𝑢 = 𝑈 → (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2) = ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2))
1918oveq2d 7447 . . . . . 6 (𝑢 = 𝑈 → ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)))
2019sumeq2sdv 15739 . . . . 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 7446 . . . 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 7508 . . 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 30720 . . 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 6920 . . . 4 𝑋 ∈ V
2524, 24mpoex 8104 . . 3 (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) ∈ V
2622, 23, 25fvmpt 7016 . 2 (𝑈 ∈ NrmCVec → (·𝑖OLD𝑈) = (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
271, 26eqtrid 2789 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 1540  wcel 2108  cfv 6561  (class class class)co 7431  cmpo 7433  1c1 11156  ici 11157   · cmul 11160   / cdiv 11920  2c2 12321  4c4 12323  ...cfz 13547  cexp 14102  Σcsu 15722  NrmCVeccnv 30603   +𝑣 cpv 30604  BaseSetcba 30605   ·𝑠OLD cns 30606  normCVcnmcv 30609  ·𝑖OLDcdip 30719
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-seq 14043  df-sum 15723  df-dip 30720
This theorem is referenced by:  ipval  30722  ipf  30732  dipcn  30739
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