MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipval Structured version   Visualization version   GIF version

Theorem ipval 30512
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 30511 . . . 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 7437 . . 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 7443 . . . . . . . 8 (π‘₯ = 𝐴 β†’ (π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦))) = (π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦))))
98oveq1d 7435 . . . . . . 7 (π‘₯ = 𝐴 β†’ ((π‘β€˜(π‘₯𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2) = ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2))
109oveq2d 7436 . . . . . 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 15682 . . . . 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 7435 . . . 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 7428 . . . . . . . . . 10 (𝑦 = 𝐡 β†’ ((iβ†‘π‘˜)𝑆𝑦) = ((iβ†‘π‘˜)𝑆𝐡))
1413oveq2d 7436 . . . . . . . . 9 (𝑦 = 𝐡 β†’ (𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)) = (𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))
1514fveq2d 6901 . . . . . . . 8 (𝑦 = 𝐡 β†’ (π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦))) = (π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡))))
1615oveq1d 7435 . . . . . . 7 (𝑦 = 𝐡 β†’ ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝑦)))↑2) = ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2))
1716oveq2d 7436 . . . . . 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 15682 . . . . 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 7435 . . . 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 2728 . . . 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 7453 . . . 4 (Ξ£π‘˜ ∈ (1...4)((iβ†‘π‘˜) Β· ((π‘β€˜(𝐴𝐺((iβ†‘π‘˜)𝑆𝐡)))↑2)) / 4) ∈ V
2212, 19, 20, 21ovmpo 7581 . . 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 2788 . 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 1113 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 1085   = wceq 1534   ∈ wcel 2099  β€˜cfv 6548  (class class class)co 7420   ∈ cmpo 7422  1c1 11139  ici 11140   Β· cmul 11143   / cdiv 11901  2c2 12297  4c4 12299  ...cfz 13516  β†‘cexp 14058  Ξ£csu 15664  NrmCVeccnv 30393   +𝑣 cpv 30394  BaseSetcba 30395   ·𝑠OLD cns 30396  normCVcnmcv 30399  Β·π‘–OLDcdip 30509
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740  ax-cnex 11194  ax-resscn 11195  ax-1cn 11196  ax-icn 11197  ax-addcl 11198  ax-addrcl 11199  ax-mulcl 11200  ax-mulrcl 11201  ax-mulcom 11202  ax-addass 11203  ax-mulass 11204  ax-distr 11205  ax-i2m1 11206  ax-1ne0 11207  ax-1rid 11208  ax-rnegex 11209  ax-rrecex 11210  ax-cnre 11211  ax-pre-lttri 11212  ax-pre-lttrn 11213  ax-pre-ltadd 11214  ax-pre-mulgt0 11215
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-nel 3044  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-ord 6372  df-on 6373  df-lim 6374  df-suc 6375  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-riota 7376  df-ov 7423  df-oprab 7424  df-mpo 7425  df-om 7871  df-1st 7993  df-2nd 7994  df-frecs 8286  df-wrecs 8317  df-recs 8391  df-rdg 8430  df-er 8724  df-en 8964  df-dom 8965  df-sdom 8966  df-pnf 11280  df-mnf 11281  df-xr 11282  df-ltxr 11283  df-le 11284  df-sub 11476  df-neg 11477  df-nn 12243  df-n0 12503  df-z 12589  df-uz 12853  df-fz 13517  df-seq 13999  df-sum 15665  df-dip 30510
This theorem is referenced by:  ipval2  30516  dipcl  30521  ipf  30522
  Copyright terms: Public domain W3C validator