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Theorem dipfval 22198
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  =  ( .s OLD `  U
)
dipfval.6  |-  N  =  ( normCV `  U )
dipfval.7  |-  P  =  ( .i OLD `  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  =  ( .i OLD `  U
)
2 fveq2 5728 . . . . 5  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  ( BaseSet `  U )
)
3 dipfval.1 . . . . 5  |-  X  =  ( BaseSet `  U )
42, 3syl6eqr 2486 . . . 4  |-  ( u  =  U  ->  ( BaseSet
`  u )  =  X )
5 fveq2 5728 . . . . . . . . . 10  |-  ( u  =  U  ->  ( normCV `  u )  =  (
normCV
`  U ) )
6 dipfval.6 . . . . . . . . . 10  |-  N  =  ( normCV `  U )
75, 6syl6eqr 2486 . . . . . . . . 9  |-  ( u  =  U  ->  ( normCV `  u )  =  N )
8 fveq2 5728 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( +v `  u )  =  ( +v `  U
) )
9 dipfval.2 . . . . . . . . . . 11  |-  G  =  ( +v `  U
)
108, 9syl6eqr 2486 . . . . . . . . . 10  |-  ( u  =  U  ->  ( +v `  u )  =  G )
11 eqidd 2437 . . . . . . . . . 10  |-  ( u  =  U  ->  x  =  x )
12 fveq2 5728 . . . . . . . . . . . 12  |-  ( u  =  U  ->  ( .s OLD `  u )  =  ( .s OLD `  U ) )
13 dipfval.4 . . . . . . . . . . . 12  |-  S  =  ( .s OLD `  U
)
1412, 13syl6eqr 2486 . . . . . . . . . . 11  |-  ( u  =  U  ->  ( .s OLD `  u )  =  S )
1514oveqd 6098 . . . . . . . . . 10  |-  ( u  =  U  ->  (
( _i ^ k
) ( .s OLD `  u ) y )  =  ( ( _i
^ k ) S y ) )
1610, 11, 15oveq123d 6102 . . . . . . . . 9  |-  ( u  =  U  ->  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) )  =  ( x G ( ( _i ^
k ) S y ) ) )
177, 16fveq12d 5734 . . . . . . . 8  |-  ( u  =  U  ->  (
( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) )  =  ( N `  ( x G ( ( _i
^ k ) S y ) ) ) )
1817oveq1d 6096 . . . . . . 7  |-  ( u  =  U  ->  (
( ( normCV `  u
) `  ( x
( +v `  u
) ( ( _i
^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 )  =  ( ( N `
 ( x G ( ( _i ^
k ) S y ) ) ) ^
2 ) )
1918oveq2d 6097 . . . . . 6  |-  ( u  =  U  ->  (
( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  =  ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) ) )
2019sumeq2sdv 12498 . . . . 5  |-  ( u  =  U  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  u ) `  (
x ( +v `  u ) ( ( _i ^ k ) ( .s OLD `  u
) y ) ) ) ^ 2 ) )  =  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( N `  ( x G ( ( _i ^ k
) S y ) ) ) ^ 2 ) ) )
2120oveq1d 6096 . . . 4  |-  ( u  =  U  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  u ) `  ( x ( +v
`  u ) ( ( _i ^ k
) ( .s OLD `  u ) y ) ) ) ^ 2 ) )  /  4
)  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( N `  ( x G ( ( _i
^ k ) S y ) ) ) ^ 2 ) )  /  4 ) )
224, 4, 21mpt2eq123dv 6136 . . 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 ) ( .s OLD `  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 22197 . . 3  |-  .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
) ) )
24 fvex 5742 . . . . 5  |-  ( BaseSet `  U )  e.  _V
253, 24eqeltri 2506 . . . 4  |-  X  e. 
_V
2625, 25mpt2ex 6425 . . 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 5806 . 2  |-  ( U  e.  NrmCVec  ->  ( .i OLD `  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 2480 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 set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   _Vcvv 2956   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   1c1 8991   _ici 8992    x. cmul 8995    / cdiv 9677   2c2 10049   4c4 10051   ...cfz 11043   ^cexp 11382   sum_csu 12479   NrmCVeccnv 22063   +vcpv 22064   BaseSetcba 22065   .s OLDcns 22066   normCVcnmcv 22069   .i OLDcdip 22196
This theorem is referenced by:  ipval  22199  ipf  22212  dipcn  22219
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-sum 12480  df-dip 22197
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