MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dipfval Structured version   Unicode version

Theorem dipfval 25739
Description: The inner product function on a normed complex vector space. The definition is meaningful for vector spaces that are also inner product spaces, i.e. satisfy the parallelogram law. (Contributed by NM, 10-Apr-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
dipfval  |-  ( 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 ) ) )
Distinct variable groups:    x, k,
y, G    k, N, x, y    S, k, x, y    U, k, x, y   
k, X, x, y
Allowed substitution hints:    P( x, y, k)

Proof of Theorem dipfval
Dummy variable  u is distinct from all other variables.
StepHypRef Expression
1 dipfval.7 . 2  |-  P  =  ( .iOLD `  U )
2 fveq2 5872 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 dipfval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2516 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
5 fveq2 5872 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
6 dipfval.6 . . . . . . . . . 10  |-  N  =  ( normCV `  U )
75, 6syl6eqr 2516 . . . . . . . . 9  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
8 fveq2 5872 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
9 dipfval.2 . . . . . . . . . . 11  |-  G  =  ( +v `  U
)
108, 9syl6eqr 2516 . . . . . . . . . 10  |-  ( u  =  U  ->  ( +v `  u )  =  G )
11 eqidd 2458 . . . . . . . . . 10  |-  ( u  =  U  ->  x  =  x )
12 fveq2 5872 . . . . . . . . . . . 12  |-  ( u  =  U  ->  ( .sOLD `  u )  =  ( .sOLD `  U ) )
13 dipfval.4 . . . . . . . . . . . 12  |-  S  =  ( .sOLD `  U )
1412, 13syl6eqr 2516 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( .sOLD `  u )  =  S )
1514oveqd 6313 . . . . . . . . . 10  |-  ( u  =  U  ->  (
( _i ^ k
) ( .sOLD `  u ) y )  =  ( ( _i
^ k ) S y ) )
1610, 11, 15oveq123d 6317 . . . . . . . . 9  |-  ( u  =  U  ->  (
x ( +v `  u ) ( ( _i ^ k ) ( .sOLD `  u ) y ) )  =  ( x G ( ( _i
^ k ) S y ) ) )
177, 16fveq12d 5878 . . . . . . . 8  |-  ( u  =  U  ->  (
( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) )  =  ( N `  ( x G ( ( _i
^ k ) S y ) ) ) )
1817oveq1d 6311 . . . . . . 7  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .sOLD `  u
) y ) ) ) ^ 2 )  =  ( ( N `
 ( x G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1918oveq2d 6312 . . . . . 6  |-  ( u  =  U  ->  (
( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  =  ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
2019sumeq2sdv 13538 . . . . 5  |-  ( u  =  U  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  =  sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
2120oveq1d 6311 . . . 4  |-  ( u  =  U  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .sOLD `  u ) y ) ) ) ^ 2 ) )  /  4
)  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )
224, 4, 21mpt2eq123dv 6358 . . 3  |-  ( u  =  U  ->  (
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
) )  =  ( x  e.  X , 
y  e.  X  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
23 df-dip 25738 . . 3  |-  .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
) ) )
24 fvex 5882 . . . . 5  |-  ( BaseSet `  U )  e.  _V
253, 24eqeltri 2541 . . . 4  |-  X  e. 
_V
2625, 25mpt2ex 6876 . . 3  |-  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )  e.  _V
2722, 23, 26fvmpt 5956 . 2  |-  ( U  e.  NrmCVec  ->  ( .iOLD `  U )  =  ( x  e.  X , 
y  e.  X  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( N `  (
x G ( ( _i ^ k ) S y ) ) ) ^ 2 ) )  /  4 ) ) )
281, 27syl5eq 2510 1  |-  ( 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 ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1395    e. wcel 1819   _Vcvv 3109   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1c1 9510   _ici 9511    x. cmul 9514    / cdiv 10227   2c2 10606   4c4 10608   ...cfz 11697   ^cexp 12169   sum_csu 13520   NrmCVeccnv 25604   +vcpv 25605   BaseSetcba 25606   .sOLDcns 25607   normCVcnmcv 25610   .iOLDcdip 25737
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-fal 1401  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111  df-sum 13521  df-dip 25738
This theorem is referenced by:  ipval  25740  ipf  25753  dipcn  25760
  Copyright terms: Public domain W3C validator