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Definition df-dip 26173
Description: Define a function that maps a complex normed vector space to its inner product operation in case its norm satisfies the parallelogram identity (otherwise the operation is still defined, but not meaningful). Based on Exercise 4(a) of [ReedSimon] p. 63 and Theorem 6.44 of [Ponnusamy] p. 361. Vector addition is  ( 1st `  w
), the scalar product is  ( 2nd `  w
), and the norm is  n. (Contributed by NM, 10-Apr-2007.) (New usage is discouraged.)
Assertion
Ref Expression
df-dip  |-  .iOLD  =  ( u  e.  NrmCVec 
|->  ( x  e.  (
BaseSet `  u ) ,  y  e.  ( BaseSet `  u )  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  /  4
) ) )
Distinct variable group:    u, k, x, y

Detailed syntax breakdown of Definition df-dip
StepHypRef Expression
1 cdip 26172 . 2  class  .iOLD
2 vu . . 3  setvar  u
3 cnv 26039 . . 3  class  NrmCVec
4 vx . . . 4  setvar  x
5 vy . . . 4  setvar  y
62cv 1436 . . . . 5  class  u
7 cba 26041 . . . . 5  class  BaseSet
86, 7cfv 5601 . . . 4  class  ( BaseSet `  u )
9 c1 9539 . . . . . . 7  class  1
10 c4 10661 . . . . . . 7  class  4
11 cfz 11782 . . . . . . 7  class  ...
129, 10, 11co 6305 . . . . . 6  class  ( 1 ... 4 )
13 ci 9540 . . . . . . . 8  class  _i
14 vk . . . . . . . . 9  setvar  k
1514cv 1436 . . . . . . . 8  class  k
16 cexp 12269 . . . . . . . 8  class  ^
1713, 15, 16co 6305 . . . . . . 7  class  ( _i
^ k )
184cv 1436 . . . . . . . . . 10  class  x
195cv 1436 . . . . . . . . . . 11  class  y
20 cns 26042 . . . . . . . . . . . 12  class  .sOLD
216, 20cfv 5601 . . . . . . . . . . 11  class  ( .sOLD `  u )
2217, 19, 21co 6305 . . . . . . . . . 10  class  ( ( _i ^ k ) ( .sOLD `  u ) y )
23 cpv 26040 . . . . . . . . . . 11  class  +v
246, 23cfv 5601 . . . . . . . . . 10  class  ( +v
`  u )
2518, 22, 24co 6305 . . . . . . . . 9  class  ( x ( +v `  u
) ( ( _i
^ k ) ( .sOLD `  u
) y ) )
26 cnmcv 26045 . . . . . . . . . 10  class  normCV
276, 26cfv 5601 . . . . . . . . 9  class  ( normCV `  u )
2825, 27cfv 5601 . . . . . . . 8  class  ( (
normCV
`  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) )
29 c2 10659 . . . . . . . 8  class  2
3028, 29, 16co 6305 . . . . . . 7  class  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 )
31 cmul 9543 . . . . . . 7  class  x.
3217, 30, 31co 6305 . . . . . 6  class  ( ( _i ^ k )  x.  ( ( (
normCV
`  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )
3312, 32, 14csu 13730 . . . . 5  class  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .sOLD `  u ) y ) ) ) ^ 2 ) )
34 cdiv 10268 . . . . 5  class  /
3533, 10, 34co 6305 . . . 4  class  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  /  4
)
364, 5, 8, 8, 35cmpt2 6307 . . 3  class  ( x  e.  ( BaseSet `  u
) ,  y  e.  ( BaseSet `  u )  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .sOLD `  u
) y ) ) ) ^ 2 ) )  /  4 ) )
372, 3, 36cmpt 4484 . 2  class  ( u  e.  NrmCVec  |->  ( x  e.  ( BaseSet `  u ) ,  y  e.  ( BaseSet
`  u )  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .sOLD `  u
) y ) ) ) ^ 2 ) )  /  4 ) ) )
381, 37wceq 1437 1  wff  .iOLD  =  ( u  e.  NrmCVec 
|->  ( x  e.  (
BaseSet `  u ) ,  y  e.  ( BaseSet `  u )  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  /  4
) ) )
Colors of variables: wff setvar class
This definition is referenced by:  dipfval  26174
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