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Theorem ipval 25317
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 25316 . . . 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 6301 . . 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 6291 . . . . . . . . 9  |-  ( x  =  A  ->  (
x G ( ( _i ^ k ) S y ) )  =  ( A G ( ( _i ^
k ) S y ) ) )
98fveq2d 5870 . . . . . . . 8  |-  ( x  =  A  ->  ( N `  ( x G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S y ) ) ) )
109oveq1d 6299 . . . . . . 7  |-  ( x  =  A  ->  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 )  =  ( ( N `
 ( A G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1110oveq2d 6300 . . . . . 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 13489 . . . . 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 6299 . . . 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 6292 . . . . . . . . . 10  |-  ( y  =  B  ->  (
( _i ^ k
) S y )  =  ( ( _i
^ k ) S B ) )
1514oveq2d 6300 . . . . . . . . 9  |-  ( y  =  B  ->  ( A G ( ( _i
^ k ) S y ) )  =  ( A G ( ( _i ^ k
) S B ) ) )
1615fveq2d 5870 . . . . . . . 8  |-  ( y  =  B  ->  ( N `  ( A G ( ( _i
^ k ) S y ) ) )  =  ( N `  ( A G ( ( _i ^ k ) S B ) ) ) )
1716oveq1d 6299 . . . . . . 7  |-  ( y  =  B  ->  (
( N `  ( A G ( ( _i
^ k ) S y ) ) ) ^ 2 )  =  ( ( N `  ( A G ( ( _i ^ k ) S B ) ) ) ^ 2 ) )
1817oveq2d 6300 . . . . . 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 13489 . . . . 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 6299 . . . 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 2467 . . . 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 6309 . . . 4  |-  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( A G ( ( _i
^ k ) S B ) ) ) ^ 2 ) )  /  4 )  e. 
_V
2313, 20, 21, 22ovmpt2 6422 . . 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 2528 . 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 1192 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 973    = wceq 1379    e. wcel 1767   ` cfv 5588  (class class class)co 6284    |-> cmpt2 6286   1c1 9493   _ici 9494    x. cmul 9497    / cdiv 10206   2c2 10585   4c4 10587   ...cfz 11672   ^cexp 12134   sum_csu 13471   NrmCVeccnv 25181   +vcpv 25182   BaseSetcba 25183   .sOLDcns 25184   normCVcnmcv 25187   .iOLDcdip 25314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568  ax-pre-mulgt0 9569
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6245  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-om 6685  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-sub 9807  df-neg 9808  df-nn 10537  df-n0 10796  df-z 10865  df-uz 11083  df-fz 11673  df-seq 12076  df-sum 13472  df-dip 25315
This theorem is referenced by:  ipval2  25321  dipcl  25329  ipf  25330  sspival  25355
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