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Theorem dipcl 22164
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 2404 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2404 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2404 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . 3  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval 22152 . 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 11267 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
1 ... 4 )  e. 
Fin )
8 ax-icn 9005 . . . . . . 7  |-  _i  e.  CC
9 elfznn 11036 . . . . . . . 8  |-  ( k  e.  ( 1 ... 4 )  ->  k  e.  NN )
109nnnn0d 10230 . . . . . . 7  |-  ( k  e.  ( 1 ... 4 )  ->  k  e.  NN0 )
11 expcl 11354 . . . . . . 7  |-  ( ( _i  e.  CC  /\  k  e.  NN0 )  -> 
( _i ^ k
)  e.  CC )
128, 10, 11sylancr 645 . . . . . 6  |-  ( k  e.  ( 1 ... 4 )  ->  (
_i ^ k )  e.  CC )
1312adantl 453 . . . . 5  |-  ( ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  /\  k  e.  (
1 ... 4 ) )  ->  ( _i ^
k )  e.  CC )
141, 2, 3, 4, 5ipval2lem4 22155 . . . . . 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 461 . . . . 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 9064 . . . 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 12482 . . 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 10030 . . . 4  |-  4  e.  CC
19 4re 10029 . . . . 5  |-  4  e.  RR
20 4pos 10042 . . . . 5  |-  0  <  4
2119, 20gt0ne0ii 9519 . . . 4  |-  4  =/=  0
22 divcl 9640 . . . 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 1271 . . 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 16 . 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 2478 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567   ` cfv 5413  (class class class)co 6040   CCcc 8944   0cc0 8946   1c1 8947   _ici 8948    x. cmul 8951    / cdiv 9633   2c2 10005   4c4 10007   NN0cn0 10177   ...cfz 10999   ^cexp 11337   sum_csu 12434   NrmCVeccnv 22016   +vcpv 22017   BaseSetcba 22018   .s OLDcns 22019   normCVcnmcv 22022   .i OLDcdip 22149
This theorem is referenced by:  ipf  22165  ipipcj  22167  ip1ilem  22280  ip2i  22282  ipasslem1  22285  ipasslem2  22286  ipasslem4  22288  ipasslem5  22289  ipasslem7  22290  ipasslem8  22291  ipasslem9  22292  ipasslem10  22293  ipasslem11  22294  dipdi  22297  ip2dii  22298  dipassr  22300  dipsubdir  22302  dipsubdi  22303  pythi  22304  siilem1  22305  siilem2  22306  siii  22307  ipblnfi  22310  ip2eqi  22311  htthlem  22373
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-4 10016  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-fz 11000  df-fzo 11091  df-seq 11279  df-exp 11338  df-hash 11574  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-clim 12237  df-sum 12435  df-grpo 21732  df-ablo 21823  df-vc 21978  df-nv 22024  df-va 22027  df-ba 22028  df-sm 22029  df-0v 22030  df-nmcv 22032  df-dip 22150
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