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Theorem dipcl 21288
Description: An inner product is a complex number. (Contributed by NM, 1-Feb-2007.) (Revised by Mario Carneiro, 5-May-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
ipcl.1  |-  X  =  ( BaseSet `  U )
ipcl.7  |-  P  =  ( .i OLD `  U
)
Assertion
Ref Expression
dipcl  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )

Proof of Theorem dipcl
Dummy variable  k is distinct from all other variables.
StepHypRef Expression
1 ipcl.1 . . 3  |-  X  =  ( BaseSet `  U )
2 eqid 2283 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2283 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2283 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . 3  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval 21276 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  =  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( ( _i
^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  /  4 ) )
7 fzfid 11035 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
1 ... 4 )  e. 
Fin )
8 ax-icn 8796 . . . . . . 7  |-  _i  e.  CC
9 elfznn 10819 . . . . . . . 8  |-  ( k  e.  ( 1 ... 4 )  ->  k  e.  NN )
109nnnn0d 10018 . . . . . . 7  |-  ( k  e.  ( 1 ... 4 )  ->  k  e.  NN0 )
11 expcl 11121 . . . . . . 7  |-  ( ( _i  e.  CC  /\  k  e.  NN0 )  -> 
( _i ^ k
)  e.  CC )
128, 10, 11sylancr 644 . . . . . 6  |-  ( k  e.  ( 1 ... 4 )  ->  (
_i ^ k )  e.  CC )
1312adantl 452 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  (
1 ... 4 ) )  ->  ( _i ^
k )  e.  CC )
141, 2, 3, 4, 5ipval2lem4 21279 . . . . . 6  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  ( _i ^ k
)  e.  CC )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1512, 14sylan2 460 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  (
1 ... 4 ) )  ->  ( ( (
normCV
`  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 )  e.  CC )
1613, 15mulcld 8855 . . . 4  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  (
1 ... 4 ) )  ->  ( ( _i
^ k )  x.  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )
177, 16fsumcl 12206 . . 3  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  sum_ k  e.  ( 1 ... 4
) ( ( _i
^ k )  x.  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC )
18 4cn 9820 . . . 4  |-  4  e.  CC
19 4re 9819 . . . . 5  |-  4  e.  RR
20 4pos 9832 . . . . 5  |-  0  <  4
2119, 20gt0ne0ii 9309 . . . 4  |-  4  =/=  0
22 divcl 9430 . . . 4  |-  ( (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( ( _i
^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC  /\  4  e.  CC  /\  4  =/=  0 )  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  /  4 )  e.  CC )
2318, 21, 22mp3an23 1269 . . 3  |-  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  e.  CC  ->  (
sum_ k  e.  ( 1 ... 4 ) ( ( _i ^
k )  x.  (
( ( normCV `  U
) `  ( A
( +v `  U
) ( ( _i
^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  /  4 )  e.  CC )
2417, 23syl 15 . 2  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( sum_ k  e.  ( 1 ... 4 ) ( ( _i ^ k
)  x.  ( ( ( normCV `  U ) `  ( A ( +v `  U ) ( ( _i ^ k ) ( .s OLD `  U
) B ) ) ) ^ 2 ) )  /  4 )  e.  CC )
256, 24eqeltrd 2357 1  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  ( A P B )  e.  CC )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738   _ici 8739    x. cmul 8742    / cdiv 9423   2c2 9795   4c4 9797   NN0cn0 9965   ...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:  ipf  21289  ipipcj  21291  ip1ilem  21404  ip2i  21406  ipasslem1  21409  ipasslem2  21410  ipasslem4  21412  ipasslem5  21413  ipasslem7  21414  ipasslem8  21415  ipasslem9  21416  ipasslem10  21417  ipasslem11  21418  dipdi  21421  ip2dii  21422  dipassr  21424  dipsubdir  21426  dipsubdi  21427  pythi  21428  siilem1  21429  siilem2  21430  siii  21431  ipblnfi  21434  ip2eqi  21435  htthlem  21497
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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