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Theorem ipval 24243
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 24242 . . . 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 6210 . . 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 6200 . . . . . . . . 9  |-  ( x  =  A  ->  (
x G ( ( _i ^ k ) S y ) )  =  ( A G ( ( _i ^
k ) S y ) ) )
98fveq2d 5796 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  ( x G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S y ) ) ) )
109oveq1d 6208 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1110oveq2d 6209 . . . . . 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 13292 . . . . 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 6208 . . . 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 6201 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( _i ^ k
) S y )  =  ( ( _i
^ k ) S B ) )
1514oveq2d 6209 . . . . . . . . 9  |-  ( y  =  B  ->  ( A G ( ( _i
^ k ) S y ) )  =  ( A G ( ( _i ^ k
) S B ) ) )
1615fveq2d 5796 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  ( A G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S B ) ) ) )
1716oveq1d 6208 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) )
1817oveq2d 6209 . . . . . 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 13292 . . . . 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 6208 . . . 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 2451 . . . 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 6218 . . . 4  |-  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 )  e. 
_V
2313, 20, 21, 22ovmpt2 6329 . . 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 2512 . 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 1184 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 369    /\ w3a 965    = wceq 1370    e. wcel 1758   ` cfv 5519  (class class class)co 6193    |-> cmpt2 6195   1c1 9387   _ici 9388    x. cmul 9391    / cdiv 10097   2c2 10475   4c4 10477   ...cfz 11547   ^cexp 11975   sum_csu 13274   NrmCVeccnv 24107   +vcpv 24108   BaseSetcba 24109   .sOLDcns 24110   normCVcnmcv 24113   .iOLDcdip 24240
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4571  ax-pr 4632  ax-un 6475  ax-cnex 9442  ax-resscn 9443  ax-1cn 9444  ax-icn 9445  ax-addcl 9446  ax-addrcl 9447  ax-mulcl 9448  ax-mulrcl 9449  ax-mulcom 9450  ax-addass 9451  ax-mulass 9452  ax-distr 9453  ax-i2m1 9454  ax-1ne0 9455  ax-1rid 9456  ax-rnegex 9457  ax-rrecex 9458  ax-cnre 9459  ax-pre-lttri 9460  ax-pre-lttrn 9461  ax-pre-ltadd 9462  ax-pre-mulgt0 9463
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-nel 2647  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-csb 3390  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-pss 3445  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-tp 3983  df-op 3985  df-uni 4193  df-iun 4274  df-br 4394  df-opab 4452  df-mpt 4453  df-tr 4487  df-eprel 4733  df-id 4737  df-po 4742  df-so 4743  df-fr 4780  df-we 4782  df-ord 4823  df-on 4824  df-lim 4825  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-riota 6154  df-ov 6196  df-oprab 6197  df-mpt2 6198  df-om 6580  df-1st 6680  df-2nd 6681  df-recs 6935  df-rdg 6969  df-er 7204  df-en 7414  df-dom 7415  df-sdom 7416  df-pnf 9524  df-mnf 9525  df-xr 9526  df-ltxr 9527  df-le 9528  df-sub 9701  df-neg 9702  df-nn 10427  df-n0 10684  df-z 10751  df-uz 10966  df-fz 11548  df-seq 11917  df-sum 13275  df-dip 24241
This theorem is referenced by:  ipval2  24247  dipcl  24255  ipf  24256  sspival  24281
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