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Theorem dipfval 24102
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  =  ( .sOLD `  U )
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .iOLD `  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  =  ( .iOLD `  U )
2 fveq2 5696 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 dipfval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2493 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
5 fveq2 5696 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
6 dipfval.6 . . . . . . . . . 10  |-  N  =  ( normCV `  U )
75, 6syl6eqr 2493 . . . . . . . . 9  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
8 fveq2 5696 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
9 dipfval.2 . . . . . . . . . . 11  |-  G  =  ( +v `  U
)
108, 9syl6eqr 2493 . . . . . . . . . 10  |-  ( u  =  U  ->  ( +v `  u )  =  G )
11 eqidd 2444 . . . . . . . . . 10  |-  ( u  =  U  ->  x  =  x )
12 fveq2 5696 . . . . . . . . . . . 12  |-  ( u  =  U  ->  ( .sOLD `  u )  =  ( .sOLD `  U ) )
13 dipfval.4 . . . . . . . . . . . 12  |-  S  =  ( .sOLD `  U )
1412, 13syl6eqr 2493 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( .sOLD `  u )  =  S )
1514oveqd 6113 . . . . . . . . . 10  |-  ( u  =  U  ->  (
( _i ^ k
) ( .sOLD `  u ) y )  =  ( ( _i
^ k ) S y ) )
1610, 11, 15oveq123d 6117 . . . . . . . . 9  |-  ( u  =  U  ->  (
x ( +v `  u ) ( ( _i ^ k ) ( .sOLD `  u ) y ) )  =  ( x G ( ( _i
^ k ) S y ) ) )
177, 16fveq12d 5702 . . . . . . . 8  |-  ( u  =  U  ->  (
( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) )  =  ( N `  ( x G ( ( _i
^ k ) S y ) ) ) )
1817oveq1d 6111 . . . . . . 7  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .sOLD `  u
) y ) ) ) ^ 2 )  =  ( ( N `
 ( x G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1918oveq2d 6112 . . . . . 6  |-  ( u  =  U  ->  (
( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  =  ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
2019sumeq2sdv 13186 . . . . 5  |-  ( u  =  U  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  =  sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
2120oveq1d 6111 . . . 4  |-  ( u  =  U  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  /  4
)  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )
224, 4, 21mpt2eq123dv 6153 . . 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 ) ( .sOLD `  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 24101 . . 3  |-  .iOLD  =  ( 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
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  /  4
) ) )
24 fvex 5706 . . . . 5  |-  ( BaseSet `  U )  e.  _V
253, 24eqeltri 2513 . . . 4  |-  X  e. 
_V
2625, 25mpt2ex 6655 . . 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 5779 . 2  |-  ( U  e.  NrmCVec  ->  ( .iOLD `  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 2487 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 setvar class
Syntax hints:    -> wi 4    = wceq 1369    e. wcel 1756   _Vcvv 2977   ` cfv 5423  (class class class)co 6096    e. cmpt2 6098   1c1 9288   _ici 9289    x. cmul 9292    / cdiv 9998   2c2 10376   4c4 10378   ...cfz 11442   ^cexp 11870   sum_csu 13168   NrmCVeccnv 23967   +vcpv 23968   BaseSetcba 23969   .sOLDcns 23970   normCVcnmcv 23973   .iOLDcdip 24100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-fal 1375  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-1st 6582  df-2nd 6583  df-recs 6837  df-rdg 6871  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-nn 10328  df-n0 10585  df-z 10652  df-uz 10867  df-fz 11443  df-seq 11812  df-sum 13169  df-dip 24101
This theorem is referenced by:  ipval  24103  ipf  24116  dipcn  24123
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