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Definition df-dip 21220
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 21219 . 2  class  .i OLD
2 vu . . 3  set  u
3 cnv 21086 . . 3  class  NrmCVec
4 vx . . . 4  set  x
5 vy . . . 4  set  y
62cv 1618 . . . . 5  class  u
7 cba 21088 . . . . 5  class  BaseSet
86, 7cfv 4659 . . . 4  class  ( BaseSet `  u )
9 c1 8692 . . . . . . 7  class  1
10 c4 9751 . . . . . . 7  class  4
11 cfz 10734 . . . . . . 7  class  ...
129, 10, 11co 5778 . . . . . 6  class  ( 1 ... 4 )
13 ci 8693 . . . . . . . 8  class  _i
14 vk . . . . . . . . 9  set  k
1514cv 1618 . . . . . . . 8  class  k
16 cexp 11056 . . . . . . . 8  class  ^
1713, 15, 16co 5778 . . . . . . 7  class  ( _i
^ k )
184cv 1618 . . . . . . . . . 10  class  x
195cv 1618 . . . . . . . . . . 11  class  y
20 cns 21089 . . . . . . . . . . . 12  class  .s OLD
216, 20cfv 4659 . . . . . . . . . . 11  class  ( .s
OLD `  u )
2217, 19, 21co 5778 . . . . . . . . . 10  class  ( ( _i ^ k ) ( .s OLD `  u
) y )
23 cpv 21087 . . . . . . . . . . 11  class  +v
246, 23cfv 4659 . . . . . . . . . 10  class  ( +v
`  u )
2518, 22, 24co 5778 . . . . . . . . 9  class  ( x ( +v `  u
) ( ( _i
^ k ) ( .s OLD `  u
) y ) )
26 cnmcv 21092 . . . . . . . . . 10  class  normCV
276, 26cfv 4659 . . . . . . . . 9  class  ( normCV `  u )
2825, 27cfv 4659 . . . . . . . 8  class  ( (
normCV
`  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) )
29 c2 9749 . . . . . . . 8  class  2
3028, 29, 16co 5778 . . . . . . 7  class  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 )
31 cmul 8696 . . . . . . 7  class  x.
3217, 30, 31co 5778 . . . . . 6  class  ( ( _i ^ k )  x.  ( ( (
normCV
`  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )
3312, 32, 14csu 12109 . . . . 5  class  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )
34 cdiv 9377 . . . . 5  class  /
3533, 10, 34co 5778 . . . 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 5780 . . 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 4037 . 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  21221
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