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Theorem dipfval 21275
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  =  ( +v `  U
)
dipfval.4  |-  S  =  ( .s OLD `  U
)
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dipfval  |-  ( U  e.  NrmCVec  ->  P  =  ( x  e.  X , 
y  e.  X  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 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  =  ( .i OLD `  U
)
2 fveq2 5525 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 dipfval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2333 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
5 fveq2 5525 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
6 dipfval.6 . . . . . . . . . 10  |-  N  =  ( normCV `  U )
75, 6syl6eqr 2333 . . . . . . . . 9  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
8 fveq2 5525 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
9 dipfval.2 . . . . . . . . . . 11  |-  G  =  ( +v `  U
)
108, 9syl6eqr 2333 . . . . . . . . . 10  |-  ( u  =  U  ->  ( +v `  u )  =  G )
11 eqidd 2284 . . . . . . . . . 10  |-  ( u  =  U  ->  x  =  x )
12 fveq2 5525 . . . . . . . . . . . 12  |-  ( u  =  U  ->  ( .s OLD `  u )  =  ( .s OLD `  U ) )
13 dipfval.4 . . . . . . . . . . . 12  |-  S  =  ( .s OLD `  U
)
1412, 13syl6eqr 2333 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( .s OLD `  u )  =  S )
1514oveqd 5875 . . . . . . . . . 10  |-  ( u  =  U  ->  (
( _i ^ k
) ( .s OLD `  u ) y )  =  ( ( _i
^ k ) S y ) )
1610, 11, 15oveq123d 5879 . . . . . . . . 9  |-  ( u  =  U  ->  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) )  =  ( x G ( ( _i ^
k ) S y ) ) )
177, 16fveq12d 5531 . . . . . . . 8  |-  ( u  =  U  ->  (
( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) )  =  ( N `  ( x G ( ( _i
^ k ) S y ) ) ) )
1817oveq1d 5873 . . . . . . 7  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 )  =  ( ( N `
 ( x G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1918oveq2d 5874 . . . . . 6  |-  ( u  =  U  ->  (
( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  =  ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
2019sumeq2sdv 12177 . . . . 5  |-  ( u  =  U  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )  =  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( N `  ( x G ( ( _i ^ k
) S y ) ) ) ^ 2 ) ) )
2120oveq1d 5873 . . . 4  |-  ( u  =  U  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4
)  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )
224, 4, 21mpt2eq123dv 5910 . . 3  |-  ( u  =  U  ->  (
x  e.  ( BaseSet `  u ) ,  y  e.  ( BaseSet `  u
)  |->  ( sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )  /  4 ) )  =  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
23 df-dip 21274 . . 3  |-  .i OLD  =  ( u  e.  NrmCVec 
|->  ( x  e.  (
BaseSet `  u ) ,  y  e.  ( BaseSet `  u )  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4
) ) )
24 fvex 5539 . . . . 5  |-  ( BaseSet `  U )  e.  _V
253, 24eqeltri 2353 . . . 4  |-  X  e. 
_V
2625, 25mpt2ex 6198 . . 3  |-  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )  e.  _V
2722, 23, 26fvmpt 5602 . 2  |-  ( U  e.  NrmCVec  ->  ( .i OLD `  U )  =  ( x  e.  X , 
y  e.  X  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
281, 27syl5eq 2327 1  |-  ( U  e.  NrmCVec  ->  P  =  ( x  e.  X , 
y  e.  X  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   _Vcvv 2788   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   1c1 8738   _ici 8739    x. cmul 8742    / cdiv 9423   2c2 9795   4c4 9797   ...cfz 10782   ^cexp 11104   sum_csu 12158   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s OLDcns 21143   normCVcnmcv 21146   .i OLDcdip 21273
This theorem is referenced by:  ipval  21276  ipf  21289  dipcn  21296
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-sum 12159  df-dip 21274
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