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Theorem dipfval 27887
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 6353 . . . . 5 (𝑢 = 𝑈 → (BaseSet‘𝑢) = (BaseSet‘𝑈))
3 dipfval.1 . . . . 5 𝑋 = (BaseSet‘𝑈)
42, 3syl6eqr 2812 . . . 4 (𝑢 = 𝑈 → (BaseSet‘𝑢) = 𝑋)
5 fveq2 6353 . . . . . . . . . 10 (𝑢 = 𝑈 → (normCV𝑢) = (normCV𝑈))
6 dipfval.6 . . . . . . . . . 10 𝑁 = (normCV𝑈)
75, 6syl6eqr 2812 . . . . . . . . 9 (𝑢 = 𝑈 → (normCV𝑢) = 𝑁)
8 fveq2 6353 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( +𝑣𝑢) = ( +𝑣𝑈))
9 dipfval.2 . . . . . . . . . . 11 𝐺 = ( +𝑣𝑈)
108, 9syl6eqr 2812 . . . . . . . . . 10 (𝑢 = 𝑈 → ( +𝑣𝑢) = 𝐺)
11 eqidd 2761 . . . . . . . . . 10 (𝑢 = 𝑈𝑥 = 𝑥)
12 fveq2 6353 . . . . . . . . . . . 12 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = ( ·𝑠OLD𝑈))
13 dipfval.4 . . . . . . . . . . . 12 𝑆 = ( ·𝑠OLD𝑈)
1412, 13syl6eqr 2812 . . . . . . . . . . 11 (𝑢 = 𝑈 → ( ·𝑠OLD𝑢) = 𝑆)
1514oveqd 6831 . . . . . . . . . 10 (𝑢 = 𝑈 → ((i↑𝑘)( ·𝑠OLD𝑢)𝑦) = ((i↑𝑘)𝑆𝑦))
1610, 11, 15oveq123d 6835 . . . . . . . . 9 (𝑢 = 𝑈 → (𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)) = (𝑥𝐺((i↑𝑘)𝑆𝑦)))
177, 16fveq12d 6359 . . . . . . . 8 (𝑢 = 𝑈 → ((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦))) = (𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦))))
1817oveq1d 6829 . . . . . . 7 (𝑢 = 𝑈 → (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2) = ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2))
1918oveq2d 6830 . . . . . 6 (𝑢 = 𝑈 → ((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) = ((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)))
2019sumeq2sdv 14654 . . . . 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 6829 . . . 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, 21mpt2eq123dv 6883 . . 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 27886 . . 3 ·𝑖OLD = (𝑢 ∈ NrmCVec ↦ (𝑥 ∈ (BaseSet‘𝑢), 𝑦 ∈ (BaseSet‘𝑢) ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · (((normCV𝑢)‘(𝑥( +𝑣𝑢)((i↑𝑘)( ·𝑠OLD𝑢)𝑦)))↑2)) / 4)))
24 fvex 6363 . . . . 5 (BaseSet‘𝑈) ∈ V
253, 24eqeltri 2835 . . . 4 𝑋 ∈ V
2625, 25mpt2ex 7416 . . 3 (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)) ∈ V
2722, 23, 26fvmpt 6445 . 2 (𝑈 ∈ NrmCVec → (·𝑖OLD𝑈) = (𝑥𝑋, 𝑦𝑋 ↦ (Σ𝑘 ∈ (1...4)((i↑𝑘) · ((𝑁‘(𝑥𝐺((i↑𝑘)𝑆𝑦)))↑2)) / 4)))
281, 27syl5eq 2806 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 1632  wcel 2139  Vcvv 3340  cfv 6049  (class class class)co 6814  cmpt2 6816  1c1 10149  ici 10150   · cmul 10153   / cdiv 10896  2c2 11282  4c4 11284  ...cfz 12539  cexp 13074  Σcsu 14635  NrmCVeccnv 27769   +𝑣 cpv 27770  BaseSetcba 27771   ·𝑠OLD cns 27772  normCVcnmcv 27775  ·𝑖OLDcdip 27885
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-fal 1638  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-z 11590  df-uz 11900  df-fz 12540  df-seq 13016  df-sum 14636  df-dip 27886
This theorem is referenced by:  ipval  27888  ipf  27898  dipcn  27905
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