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Theorem ipval 22048
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  =  ( 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
ipval  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 ) )
Distinct variable groups:    k, G    k, N    S, k    U, k    A, k    B, k    k, X
Allowed substitution hint:    P( k)

Proof of Theorem ipval
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dipfval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
2 dipfval.2 . . . . 5  |-  G  =  ( +v `  U
)
3 dipfval.4 . . . . 5  |-  S  =  ( .s OLD `  U
)
4 dipfval.6 . . . . 5  |-  N  =  ( normCV `  U )
5 dipfval.7 . . . . 5  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5dipfval 22047 . . . 4  |-  ( 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 ) ) )
76oveqd 6038 . . 3  |-  ( U  e.  NrmCVec  ->  ( A P B )  =  ( A ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( N `  ( x G ( ( _i ^ k
) S y ) ) ) ^ 2 ) )  /  4
) ) B ) )
8 oveq1 6028 . . . . . . . . 9  |-  ( x  =  A  ->  (
x G ( ( _i ^ k ) S y ) )  =  ( A G ( ( _i ^
k ) S y ) ) )
98fveq2d 5673 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  ( x G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S y ) ) ) )
109oveq1d 6036 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1110oveq2d 6037 . . . . . 6  |-  ( x  =  A  ->  (
( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  =  ( ( _i
^ k )  x.  ( ( N `  ( A G ( ( _i ^ k ) S y ) ) ) ^ 2 ) ) )
1211sumeq2sdv 12426 . . . . 5  |-  ( x  =  A  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( N `  ( x G ( ( _i ^ k
) S y ) ) ) ^ 2 ) )  =  sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
1312oveq1d 6036 . . . 4  |-  ( x  =  A  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )
14 oveq2 6029 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( _i ^ k
) S y )  =  ( ( _i
^ k ) S B ) )
1514oveq2d 6037 . . . . . . . . 9  |-  ( y  =  B  ->  ( A G ( ( _i
^ k ) S y ) )  =  ( A G ( ( _i ^ k
) S B ) ) )
1615fveq2d 5673 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  ( A G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S B ) ) ) )
1716oveq1d 6036 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) )
1817oveq2d 6037 . . . . . 6  |-  ( y  =  B  ->  (
( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  =  ( ( _i
^ k )  x.  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) ) )
1918sumeq2sdv 12426 . . . . 5  |-  ( y  =  B  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( N `  ( A G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  =  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) ) )
2019oveq1d 6036 . . . 4  |-  ( y  =  B  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 ) )
21 eqid 2388 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S 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
) )
22 ovex 6046 . . . 4  |-  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 )  e. 
_V
2313, 20, 21, 22ovmpt2 6149 . . 3  |-  ( ( A  e.  X  /\  B  e.  X )  ->  ( A ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) B )  =  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 ) )
247, 23sylan9eq 2440 . 2  |-  ( ( U  e.  NrmCVec  /\  ( A  e.  X  /\  B  e.  X )
)  ->  ( A P B )  =  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 ) )
25243impb 1149 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ` cfv 5395  (class class class)co 6021    e. cmpt2 6023   1c1 8925   _ici 8926    x. cmul 8929    / cdiv 9610   2c2 9982   4c4 9984   ...cfz 10976   ^cexp 11310   sum_csu 12407   NrmCVeccnv 21912   +vcpv 21913   BaseSetcba 21914   .s OLDcns 21915   normCVcnmcv 21918   .i OLDcdip 22045
This theorem is referenced by:  ipval2  22052  dipcl  22060  ipf  22061  sspival  22086
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-iun 4038  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-er 6842  df-en 7047  df-dom 7048  df-sdom 7049  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-nn 9934  df-n0 10155  df-z 10216  df-uz 10422  df-fz 10977  df-seq 11252  df-sum 12408  df-dip 22046
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