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Theorem dipfval 29473
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 = (BaseSetβ€˜π‘ˆ)
dipfval.2 𝐺 = ( +𝑣 β€˜π‘ˆ)
dipfval.4 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
dipfval.6 𝑁 = (normCVβ€˜π‘ˆ)
dipfval.7 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
Assertion
Ref Expression
dipfval (π‘ˆ ∈ NrmCVec β†’ 𝑃 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4)))
Distinct variable groups:   π‘₯,π‘˜,𝑦,𝐺   π‘˜,𝑁,π‘₯,𝑦   𝑆,π‘˜,π‘₯,𝑦   π‘ˆ,π‘˜,π‘₯,𝑦   π‘˜,𝑋,π‘₯,𝑦
Allowed substitution hints:   𝑃(π‘₯,𝑦,π‘˜)

Proof of Theorem dipfval
Dummy variable 𝑒 is distinct from all other variables.
StepHypRef Expression
1 dipfval.7 . 2 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
2 fveq2 6839 . . . . 5 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = (BaseSetβ€˜π‘ˆ))
3 dipfval.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
42, 3eqtr4di 2795 . . . 4 (𝑒 = π‘ˆ β†’ (BaseSetβ€˜π‘’) = 𝑋)
5 fveq2 6839 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = (normCVβ€˜π‘ˆ))
6 dipfval.6 . . . . . . . . . 10 𝑁 = (normCVβ€˜π‘ˆ)
75, 6eqtr4di 2795 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (normCVβ€˜π‘’) = 𝑁)
8 fveq2 6839 . . . . . . . . . . 11 (𝑒 = π‘ˆ β†’ ( +𝑣 β€˜π‘’) = ( +𝑣 β€˜π‘ˆ))
9 dipfval.2 . . . . . . . . . . 11 𝐺 = ( +𝑣 β€˜π‘ˆ)
108, 9eqtr4di 2795 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ ( +𝑣 β€˜π‘’) = 𝐺)
11 eqidd 2738 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ π‘₯ = π‘₯)
12 fveq2 6839 . . . . . . . . . . . 12 (𝑒 = π‘ˆ β†’ ( ·𝑠OLD β€˜π‘’) = ( ·𝑠OLD β€˜π‘ˆ))
13 dipfval.4 . . . . . . . . . . . 12 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
1412, 13eqtr4di 2795 . . . . . . . . . . 11 (𝑒 = π‘ˆ β†’ ( ·𝑠OLD β€˜π‘’) = 𝑆)
1514oveqd 7368 . . . . . . . . . 10 (𝑒 = π‘ˆ β†’ ((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦) = ((iβ†‘π‘˜)𝑆𝑦))
1610, 11, 15oveq123d 7372 . . . . . . . . 9 (𝑒 = π‘ˆ β†’ (π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)) = (π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))
177, 16fveq12d 6846 . . . . . . . 8 (𝑒 = π‘ˆ β†’ ((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦))) = (π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦))))
1817oveq1d 7366 . . . . . . 7 (𝑒 = π‘ˆ β†’ (((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)))↑2) = ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2))
1918oveq2d 7367 . . . . . 6 (𝑒 = π‘ˆ β†’ ((iβ†‘π‘˜) Β· (((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)))↑2)) = ((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)))
2019sumeq2sdv 15549 . . . . 5 (𝑒 = π‘ˆ β†’ Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· (((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)))↑2)) = Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)))
2120oveq1d 7366 . . . 4 (𝑒 = π‘ˆ β†’ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· (((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)))↑2)) / 4) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4))
224, 4, 21mpoeq123dv 7426 . . 3 (𝑒 = π‘ˆ β†’ (π‘₯ ∈ (BaseSetβ€˜π‘’), 𝑦 ∈ (BaseSetβ€˜π‘’) ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· (((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)))↑2)) / 4)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4)))
23 df-dip 29472 . . 3 ·𝑖OLD = (𝑒 ∈ NrmCVec ↦ (π‘₯ ∈ (BaseSetβ€˜π‘’), 𝑦 ∈ (BaseSetβ€˜π‘’) ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· (((normCVβ€˜π‘’)β€˜(π‘₯( +𝑣 β€˜π‘’)((iβ†‘π‘˜)( ·𝑠OLD β€˜π‘’)𝑦)))↑2)) / 4)))
243fvexi 6853 . . . 4 𝑋 ∈ V
2524, 24mpoex 8004 . . 3 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4)) ∈ V
2622, 23, 25fvmpt 6945 . 2 (π‘ˆ ∈ NrmCVec β†’ (·𝑖OLDβ€˜π‘ˆ) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4)))
271, 26eqtrid 2789 1 (π‘ˆ ∈ NrmCVec β†’ 𝑃 = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1541   ∈ wcel 2106  β€˜cfv 6493  (class class class)co 7351   ∈ cmpo 7353  1c1 11010  ici 11011   Β· cmul 11014   / cdiv 11770  2c2 12166  4c4 12168  ...cfz 13378  β†‘cexp 13921  Ξ£csu 15530  NrmCVeccnv 29355   +𝑣 cpv 29356  BaseSetcba 29357   ·𝑠OLD cns 29358  normCVcnmcv 29361  Β·π‘–OLDcdip 29471
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-rep 5240  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7664  ax-cnex 11065  ax-resscn 11066  ax-1cn 11067  ax-icn 11068  ax-addcl 11069  ax-addrcl 11070  ax-mulcl 11071  ax-mulrcl 11072  ax-mulcom 11073  ax-addass 11074  ax-mulass 11075  ax-distr 11076  ax-i2m1 11077  ax-1ne0 11078  ax-1rid 11079  ax-rnegex 11080  ax-rrecex 11081  ax-cnre 11082  ax-pre-lttri 11083  ax-pre-lttrn 11084  ax-pre-ltadd 11085  ax-pre-mulgt0 11086
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7307  df-ov 7354  df-oprab 7355  df-mpo 7356  df-om 7795  df-1st 7913  df-2nd 7914  df-frecs 8204  df-wrecs 8235  df-recs 8309  df-rdg 8348  df-er 8606  df-en 8842  df-dom 8843  df-sdom 8844  df-pnf 11149  df-mnf 11150  df-xr 11151  df-ltxr 11152  df-le 11153  df-sub 11345  df-neg 11346  df-nn 12112  df-n0 12372  df-z 12458  df-uz 12722  df-fz 13379  df-seq 13861  df-sum 15531  df-dip 29472
This theorem is referenced by:  ipval  29474  ipf  29484  dipcn  29491
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