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Theorem ipf 21289
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 2283 . . . . . . 7  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2283 . . . . . . 7  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2283 . . . . . . 7  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . . . . . 7  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval 21276 . . . . . 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 21288 . . . . . 6  |-  ( ( U  e.  NrmCVec  /\  x  e.  X  /\  y  e.  X )  ->  (
x P y )  e.  CC )
86, 7eqeltrrd 2358 . . . . 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 1154 . . . 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 2634 . . 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 2283 . . . 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 6191 . . 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 188 . 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 21275 . . 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 5379 . 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 223 1  |-  ( U  e.  NrmCVec  ->  P : ( X  X.  X ) --> CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543    X. cxp 4687   -->wf 5251   ` cfv 5255  (class class class)co 5858    e. cmpt2 5860   CCcc 8735   1c1 8738   _ici 8739    x. cmul 8742    / cdiv 9423   2c2 9795   4c4 9797   ...cfz 10782   ^cexp 11104   sum_csu 12158   NrmCVeccnv 21140   +vcpv 21141   BaseSetcba 21142   .s OLDcns 21143   normCVcnmcv 21146   .i OLDcdip 21273
This theorem is referenced by:  sspi  21315  hlipf  21489  hhip  21756
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-sup 7194  df-oi 7225  df-card 7572  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-n0 9966  df-z 10025  df-uz 10231  df-rp 10355  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-clim 11962  df-sum 12159  df-grpo 20858  df-ablo 20949  df-vc 21102  df-nv 21148  df-va 21151  df-ba 21152  df-sm 21153  df-0v 21154  df-nmcv 21156  df-dip 21274
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