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Theorem ipf 22212
Description: Mapping for the inner product operation. (Contributed by NM, 28-Jan-2008.) (Revised by Mario Carneiro, 16-Nov-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipcl.1  |-  X  =  ( BaseSet `  U )
ipcl.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
ipf  |-  ( U  e.  NrmCVec  ->  P : ( X  X.  X ) --> CC )

Proof of Theorem ipf
Dummy variables  k  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ipcl.1 . . . . . . 7  |-  X  =  ( BaseSet `  U )
2 eqid 2436 . . . . . . 7  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2436 . . . . . . 7  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2436 . . . . . . 7  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . . . . 7  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval 22199 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  y  e.  X )  ->  (
x P y )  =  ( sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  U ) `  (
x ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 ) )
71, 5dipcl 22211 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  y  e.  X )  ->  (
x P y )  e.  CC )
86, 7eqeltrrd 2511 . . . . 5  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  y  e.  X )  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  U ) `  ( x ( +v
`  U ) ( ( _i ^ k
) ( .s OLD `  U ) y ) ) ) ^ 2 ) )  /  4
)  e.  CC )
983expib 1156 . . . 4  |-  ( U  e.  NrmCVec  ->  ( ( x  e.  X  /\  y  e.  X )  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  U ) `  ( x ( +v
`  U ) ( ( _i ^ k
) ( .s OLD `  U ) y ) ) ) ^ 2 ) )  /  4
)  e.  CC ) )
109ralrimivv 2797 . . 3  |-  ( U  e.  NrmCVec  ->  A. x  e.  X  A. y  e.  X  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  U
) `  ( x
( +v `  U
) ( ( _i
^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 )  e.  CC )
11 eqid 2436 . . . 4  |-  ( x  e.  X ,  y  e.  X  |->  ( 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.  (
( ( normCV `  U
) `  ( x
( +v `  U
) ( ( _i
^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 ) )
1211fmpt2 6418 . . 3  |-  ( A. x  e.  X  A. y  e.  X  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  U ) `  ( x ( +v
`  U ) ( ( _i ^ k
) ( .s OLD `  U ) y ) ) ) ^ 2 ) )  /  4
)  e.  CC  <->  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  U ) `  (
x ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 ) ) : ( X  X.  X ) --> CC )
1310, 12sylib 189 . 2  |-  ( U  e.  NrmCVec  ->  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  U ) `  (
x ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 ) ) : ( X  X.  X ) --> CC )
141, 2, 3, 4, 5dipfval 22198 . . 3  |-  ( U  e.  NrmCVec  ->  P  =  ( x  e.  X , 
y  e.  X  |->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  U
) `  ( x
( +v `  U
) ( ( _i
^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 ) ) )
1514feq1d 5580 . 2  |-  ( U  e.  NrmCVec  ->  ( P :
( X  X.  X
) --> CC  <->  ( x  e.  X ,  y  e.  X  |->  ( sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  U ) `  (
x ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) y ) ) ) ^ 2 ) )  /  4 ) ) : ( X  X.  X ) --> CC ) )
1613, 15mpbird 224 1  |-  ( U  e.  NrmCVec  ->  P : ( X  X.  X ) --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 936    = wceq 1652    e. wcel 1725   A.wral 2705    X. cxp 4876   -->wf 5450   ` cfv 5454  (class class class)co 6081    e. cmpt2 6083   CCcc 8988   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:  sspi  22238  hlipf  22412  hhip  22679
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-inf2 7596  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  ax-pre-sup 9068
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-rmo 2713  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-int 4051  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-se 4542  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-isom 5463  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-1o 6724  df-oadd 6728  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-sup 7446  df-oi 7479  df-card 7826  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-n0 10222  df-z 10283  df-uz 10489  df-rp 10613  df-fz 11044  df-fzo 11136  df-seq 11324  df-exp 11383  df-hash 11619  df-cj 11904  df-re 11905  df-im 11906  df-sqr 12040  df-abs 12041  df-clim 12282  df-sum 12480  df-grpo 21779  df-ablo 21870  df-vc 22025  df-nv 22071  df-va 22074  df-ba 22075  df-sm 22076  df-0v 22077  df-nmcv 22079  df-dip 22197
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