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Theorem dipcl 22211
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 2436 . . 3  |-  ( +v
`  U )  =  ( +v `  U
)
3 eqid 2436 . . 3  |-  ( .s
OLD `  U )  =  ( .s OLD `  U )
4 eqid 2436 . . 3  |-  ( normCV `  U )  =  (
normCV
`  U )
5 ipcl.7 . . 3  |-  P  =  ( .i OLD `  U
)
61, 2, 3, 4, 5ipval 22199 . 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 11312 . . . 4  |-  ( ( U  e.  NrmCVec  /\  A  e.  X  /\  B  e.  X )  ->  (
1 ... 4 )  e. 
Fin )
8 ax-icn 9049 . . . . . . 7  |-  _i  e.  CC
9 elfznn 11080 . . . . . . . 8  |-  ( k  e.  ( 1 ... 4 )  ->  k  e.  NN )
109nnnn0d 10274 . . . . . . 7  |-  ( k  e.  ( 1 ... 4 )  ->  k  e.  NN0 )
11 expcl 11399 . . . . . . 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 22202 . . . . . 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 9108 . . . 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 12527 . . 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 10074 . . . 4  |-  4  e.  CC
19 4re 10073 . . . . 5  |-  4  e.  RR
20 4pos 10086 . . . . 5  |-  0  <  4
2119, 20gt0ne0ii 9563 . . . 4  |-  4  =/=  0
22 divcl 9684 . . . 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 2510 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 1652    e. wcel 1725    =/= wne 2599   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991   _ici 8992    x. cmul 8995    / cdiv 9677   2c2 10049   4c4 10051   NN0cn0 10221   ...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:  ipf  22212  ipipcj  22214  ip1ilem  22327  ip2i  22329  ipasslem1  22332  ipasslem2  22333  ipasslem4  22335  ipasslem5  22336  ipasslem7  22337  ipasslem8  22338  ipasslem9  22339  ipasslem10  22340  ipasslem11  22341  dipdi  22344  ip2dii  22345  dipassr  22347  dipsubdir  22349  dipsubdi  22350  pythi  22351  siilem1  22352  siilem2  22353  siii  22354  ipblnfi  22357  ip2eqi  22358  htthlem  22420
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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