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Theorem ipval 25814
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  =  ( .sOLD `  U )
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .iOLD `  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  =  ( .sOLD `  U )
4 dipfval.6 . . . . 5  |-  N  =  ( normCV `  U )
5 dipfval.7 . . . . 5  |-  P  =  ( .iOLD `  U )
61, 2, 3, 4, 5dipfval 25813 . . . 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 6287 . . 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 6277 . . . . . . . . 9  |-  ( x  =  A  ->  (
x G ( ( _i ^ k ) S y ) )  =  ( A G ( ( _i ^
k ) S y ) ) )
98fveq2d 5852 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  ( x G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S y ) ) ) )
109oveq1d 6285 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1110oveq2d 6286 . . . . . 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 13611 . . . . 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 6285 . . . 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 6278 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( _i ^ k
) S y )  =  ( ( _i
^ k ) S B ) )
1514oveq2d 6286 . . . . . . . . 9  |-  ( y  =  B  ->  ( A G ( ( _i
^ k ) S y ) )  =  ( A G ( ( _i ^ k
) S B ) ) )
1615fveq2d 5852 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  ( A G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S B ) ) ) )
1716oveq1d 6285 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) )
1817oveq2d 6286 . . . . . 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 13611 . . . . 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 6285 . . . 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 2454 . . . 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 6298 . . . 4  |-  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 )  e. 
_V
2313, 20, 21, 22ovmpt2 6411 . . 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 2515 . 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 1190 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 setvar class
Syntax hints:    -> wi 4    /\ wa 367    /\ w3a 971    = wceq 1398    e. wcel 1823   ` cfv 5570  (class class class)co 6270    |-> cmpt2 6272   1c1 9482   _ici 9483    x. cmul 9486    / cdiv 10202   2c2 10581   4c4 10583   ...cfz 11675   ^cexp 12151   sum_csu 13593   NrmCVeccnv 25678   +vcpv 25679   BaseSetcba 25680   .sOLDcns 25681   normCVcnmcv 25684   .iOLDcdip 25811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-seq 12093  df-sum 13594  df-dip 25812
This theorem is referenced by:  ipval2  25818  dipcl  25826  ipf  25827  sspival  25852
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