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Definition df-dip 21104
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  |-  .i OLD  =  ( 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
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4
) ) )
Distinct variable group:    u, k, x, y

Detailed syntax breakdown of Definition df-dip
StepHypRef Expression
1 cdip 21103 . 2  class  .i OLD
2 vu . . 3  set  u
3 cnv 20970 . . 3  class  NrmCVec
4 vx . . . 4  set  x
5 vy . . . 4  set  y
62cv 1618 . . . . 5  class  u
7 cba 20972 . . . . 5  class  BaseSet
86, 7cfv 4592 . . . 4  class  ( BaseSet `  u )
9 c1 8618 . . . . . . 7  class  1
10 c4 9677 . . . . . . 7  class  4
11 cfz 10660 . . . . . . 7  class  ...
129, 10, 11co 5710 . . . . . 6  class  ( 1 ... 4 )
13 ci 8619 . . . . . . . 8  class  _i
14 vk . . . . . . . . 9  set  k
1514cv 1618 . . . . . . . 8  class  k
16 cexp 10982 . . . . . . . 8  class  ^
1713, 15, 16co 5710 . . . . . . 7  class  ( _i
^ k )
184cv 1618 . . . . . . . . . 10  class  x
195cv 1618 . . . . . . . . . . 11  class  y
20 cns 20973 . . . . . . . . . . . 12  class  .s OLD
216, 20cfv 4592 . . . . . . . . . . 11  class  ( .s
OLD `  u )
2217, 19, 21co 5710 . . . . . . . . . 10  class  ( ( _i ^ k ) ( .s OLD `  u
) y )
23 cpv 20971 . . . . . . . . . . 11  class  +v
246, 23cfv 4592 . . . . . . . . . 10  class  ( +v
`  u )
2518, 22, 24co 5710 . . . . . . . . 9  class  ( x ( +v `  u
) ( ( _i
^ k ) ( .s OLD `  u
) y ) )
26 cnmcv 20976 . . . . . . . . . 10  class  normCV
276, 26cfv 4592 . . . . . . . . 9  class  ( normCV `  u )
2825, 27cfv 4592 . . . . . . . 8  class  ( (
normCV
`  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) )
29 c2 9675 . . . . . . . 8  class  2
3028, 29, 16co 5710 . . . . . . 7  class  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 )
31 cmul 8622 . . . . . . 7  class  x.
3217, 30, 31co 5710 . . . . . 6  class  ( ( _i ^ k )  x.  ( ( (
normCV
`  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )
3312, 32, 14csu 12035 . . . . 5  class  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )
34 cdiv 9303 . . . . 5  class  /
3533, 10, 34co 5710 . . . 4  class  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4
)
364, 5, 8, 8, 35cmpt2 5712 . . 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 ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )  /  4 ) )
372, 3, 36cmpt 3974 . 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 ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )  /  4 ) ) )
381, 37wceq 1619 1  wff  .i OLD  =  ( 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
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4
) ) )
Colors of variables: wff set class
This definition is referenced by:  dipfval  21105
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