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Theorem ipval 30450
Description: Value of the inner product. The definition is meaningful for normed complex vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law, although for convenience we define it for any normed complex vector space. The vector (group) addition operation is 𝐺, the scalar product is 𝑆, the norm is 𝑁, and the set of vectors is 𝑋. Equation 6.45 of [Ponnusamy] p. 361. (Contributed by NM, 31-Jan-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
ipval ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝑃𝐡) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4))
Distinct variable groups:   π‘˜,𝐺   π‘˜,𝑁   𝑆,π‘˜   π‘ˆ,π‘˜   𝐴,π‘˜   𝐡,π‘˜   π‘˜,𝑋
Allowed substitution hint:   𝑃(π‘˜)
Proof of Theorem ipval
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dipfval.1 . . . . 5 𝑋 = (BaseSetβ€˜π‘ˆ)
2 dipfval.2 . . . . 5 𝐺 = ( +𝑣 β€˜π‘ˆ)
3 dipfval.4 . . . . 5 𝑆 = ( ·𝑠OLD β€˜π‘ˆ)
4 dipfval.6 . . . . 5 𝑁 = (normCVβ€˜π‘ˆ)
5 dipfval.7 . . . . 5 𝑃 = (·𝑖OLDβ€˜π‘ˆ)
61, 2, 3, 4, 5dipfval 30449 . . . 4 (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)))
76oveqd 7419 . . 3 (π‘ˆ ∈ NrmCVec β†’ (𝐴𝑃𝐡) = (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4))𝐡))
8 fvoveq1 7425 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦))) = (π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦))))
98oveq1d 7417 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2) = ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2))
109oveq2d 7418 . . . . . 6 (π‘₯ = 𝐴 β†’ ((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) = ((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)))
1110sumeq2sdv 15652 . . . . 5 (π‘₯ = 𝐴 β†’ Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) = Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)))
1211oveq1d 7417 . . . 4 (π‘₯ = 𝐴 β†’ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4))
13 oveq2 7410 . . . . . . . . . 10 (𝑦 = 𝐡 β†’ ((iβ†‘π‘˜)𝑆𝑦) = ((iβ†‘π‘˜)𝑆𝐡))
1413oveq2d 7418 . . . . . . . . 9 (𝑦 = 𝐡 β†’ (𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)) = (𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))
1514fveq2d 6886 . . . . . . . 8 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦))) = (π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡))))
1615oveq1d 7417 . . . . . . 7 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2) = ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2))
1716oveq2d 7418 . . . . . 6 (𝑦 = 𝐡 β†’ ((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) = ((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)))
1817sumeq2sdv 15652 . . . . 5 (𝑦 = 𝐡 β†’ Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) = Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)))
1918oveq1d 7417 . . . 4 (𝑦 = 𝐡 β†’ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4))
20 eqid 2724 . . . 4 (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4)) = (π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4))
21 ovex 7435 . . . 4 (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4) ∈ V
2212, 19, 20, 21ovmpo 7561 . . 3 ((𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴(π‘₯ ∈ 𝑋, 𝑦 ∈ 𝑋 ↦ (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2)) / 4))𝐡) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4))
237, 22sylan9eq 2784 . 2 ((π‘ˆ ∈ NrmCVec ∧ (𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋)) β†’ (𝐴𝑃𝐡) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4))
24233impb 1112 1 ((π‘ˆ ∈ NrmCVec ∧ 𝐴 ∈ 𝑋 ∧ 𝐡 ∈ 𝑋) β†’ (𝐴𝑃𝐡) = (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  β€˜cfv 6534  (class class class)co 7402   ∈ cmpo 7404  1c1 11108  ici 11109   Β· cmul 11112   / cdiv 11870  2c2 12266  4c4 12268  ...cfz 13485  β†‘cexp 14028  Ξ£csu 15634  NrmCVeccnv 30331   +𝑣 cpv 30332  BaseSetcba 30333   ·𝑠OLD cns 30334  normCVcnmcv 30337  Β·π‘–OLDcdip 30447
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3960  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-tr 5257  df-id 5565  df-eprel 5571  df-po 5579  df-so 5580  df-fr 5622  df-we 5624  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-pred 6291  df-ord 6358  df-on 6359  df-lim 6360  df-suc 6361  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-riota 7358  df-ov 7405  df-oprab 7406  df-mpo 7407  df-om 7850  df-1st 7969  df-2nd 7970  df-frecs 8262  df-wrecs 8293  df-recs 8367  df-rdg 8406  df-er 8700  df-en 8937  df-dom 8938  df-sdom 8939  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-n0 12472  df-z 12558  df-uz 12822  df-fz 13486  df-seq 13968  df-sum 15635  df-dip 30448
This theorem is referenced by:  ipval2  30454  dipcl  30459  ipf  30460
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