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Theorem dipfval 21291
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 5541 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 dipfval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2346 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
5 fveq2 5541 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
6 dipfval.6 . . . . . . . . . 10  |-  N  =  ( normCV `  U )
75, 6syl6eqr 2346 . . . . . . . . 9  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
8 fveq2 5541 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
9 dipfval.2 . . . . . . . . . . 11  |-  G  =  ( +v `  U
)
108, 9syl6eqr 2346 . . . . . . . . . 10  |-  ( u  =  U  ->  ( +v `  u )  =  G )
11 eqidd 2297 . . . . . . . . . 10  |-  ( u  =  U  ->  x  =  x )
12 fveq2 5541 . . . . . . . . . . . 12  |-  ( u  =  U  ->  ( .s OLD `  u )  =  ( .s OLD `  U ) )
13 dipfval.4 . . . . . . . . . . . 12  |-  S  =  ( .s OLD `  U
)
1412, 13syl6eqr 2346 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( .s OLD `  u )  =  S )
1514oveqd 5891 . . . . . . . . . 10  |-  ( u  =  U  ->  (
( _i ^ k
) ( .s OLD `  u ) y )  =  ( ( _i
^ k ) S y ) )
1610, 11, 15oveq123d 5895 . . . . . . . . 9  |-  ( u  =  U  ->  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) )  =  ( x G ( ( _i ^
k ) S y ) ) )
177, 16fveq12d 5547 . . . . . . . 8  |-  ( u  =  U  ->  (
( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) )  =  ( N `  ( x G ( ( _i
^ k ) S y ) ) ) )
1817oveq1d 5889 . . . . . . 7  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 )  =  ( ( N `
 ( x G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1918oveq2d 5890 . . . . . 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 12193 . . . . 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 5889 . . . 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 5926 . . 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 21290 . . 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 5555 . . . . 5  |-  ( BaseSet `  U )  e.  _V
253, 24eqeltri 2366 . . . 4  |-  X  e. 
_V
2625, 25mpt2ex 6214 . . 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 5618 . 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 2340 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 1632    e. wcel 1696   _Vcvv 2801   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   1c1 8754   _ici 8755    x. cmul 8758    / cdiv 9439   2c2 9811   4c4 9813   ...cfz 10798   ^cexp 11120   sum_csu 12174   NrmCVeccnv 21156   +vcpv 21157   BaseSetcba 21158   .s OLDcns 21159   normCVcnmcv 21162   .i OLDcdip 21289
This theorem is referenced by:  ipval  21292  ipf  21305  dipcn  21312
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-sum 12175  df-dip 21290
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