MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ipval Unicode version

Theorem ipval 21222
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
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 21221 . . . 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 5795 . . 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 5785 . . . . . . . . 9  |-  ( x  =  A  ->  (
x G ( ( _i ^ k ) S y ) )  =  ( A G ( ( _i ^
k ) S y ) ) )
98fveq2d 5448 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  ( x G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S y ) ) ) )
109oveq1d 5793 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1110oveq2d 5794 . . . . . 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 12128 . . . . 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 5793 . . . 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 5786 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( _i ^ k
) S y )  =  ( ( _i
^ k ) S B ) )
1514oveq2d 5794 . . . . . . . . 9  |-  ( y  =  B  ->  ( A G ( ( _i
^ k ) S y ) )  =  ( A G ( ( _i ^ k
) S B ) ) )
1615fveq2d 5448 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  ( A G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S B ) ) ) )
1716oveq1d 5793 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) )
1817oveq2d 5794 . . . . . 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 12128 . . . . 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 5793 . . . 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 2256 . . . 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 5803 . . . 4  |-  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 )  e. 
_V
2313, 20, 21, 22ovmpt2 5903 . . 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 2308 . 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 1152 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 6    /\ wa 360    /\ w3a 939    = wceq 1619    e. wcel 1621   ` cfv 4659  (class class class)co 5778    e. cmpt2 5780   1c1 8692   _ici 8693    x. cmul 8696    / cdiv 9377   2c2 9749   4c4 9751   ...cfz 10734   ^cexp 11056   sum_csu 12109   NrmCVeccnv 21086   +vcpv 21087   BaseSetcba 21088   .s OLDcns 21089   normCVcnmcv 21092   .i OLDcdip 21219
This theorem is referenced by:  ipval2  21226  dipcl  21234  ipf  21235  sspival  21260
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-er 6614  df-en 6818  df-dom 6819  df-sdom 6820  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-n 9701  df-n0 9919  df-z 9978  df-uz 10184  df-fz 10735  df-seq 10999  df-sum 12110  df-dip 21220
  Copyright terms: Public domain W3C validator