Theorem dipcl 24231

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 = ( .iOLD ` 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.

Step	Hyp	Ref	Expression
1		ipcl.1	. . 3 |- X = ( BaseSet ` U )
2		eqid 2450	. . 3 |- ( +v ` U ) = ( +v ` U )
3		eqid 2450	. . 3 |- ( .sOLD ` U ) = ( .sOLD ` U )
4		eqid 2450	. . 3 |- ( normCV ` U ) = ( normCV ` U )
5		ipcl.7	. . 3 |- P = ( .iOLD ` U )
6	1, 2, 3, 4, 5	ipval 24219	. 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 ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) )
7		fzfid 11882	. . . 4 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( 1 ... 4 ) e. Fin )
8		ax-icn 9428	. . . . . . 7 |- _i e. CC
9		elfznn 11565	. . . . . . . 8 |- ( k e. ( 1 ... 4 ) -> k e. NN )
10	9	nnnn0d 10723	. . . . . . 7 |- ( k e. ( 1 ... 4 ) -> k e. NN0 )
11		expcl 11970	. . . . . . 7 |- ( ( _i e. CC /\ k e. NN0 ) -> ( _i ^ k ) e. CC )
12	8, 10, 11	sylancr 663	. . . . . 6 |- ( k e. ( 1 ... 4 ) -> ( _i ^ k ) e. CC )
13	12	adantl 466	. . . . 5 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ k e. ( 1 ... 4 ) ) -> ( _i ^ k ) e. CC )
14	1, 2, 3, 4, 5	ipval2lem4 24222	. . . . . 6 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ ( _i ^ k ) e. CC ) -> ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) e. CC )
15	12, 14	sylan2 474	. . . . 5 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ k e. ( 1 ... 4 ) ) -> ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) e. CC )
16	13, 15	mulcld 9493	. . . 4 |- ( ( ( U e. NrmCVec /\ A e. X /\ B e. X ) /\ k e. ( 1 ... 4 ) ) -> ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC )
17	7, 16	fsumcl 13298	. . 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 ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC )
18		4cn 10486	. . . 4 |- 4 e. CC
19		4ne0 10505	. . . 4 |- 4 =/= 0
20		divcl 10087	. . . 4 |- ( ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` 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 ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) e. CC )
21	18, 19, 20	mp3an23 1307	. . 3 |- ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) e. CC -> ( sum_ k e. ( 1 ... 4 ) ( ( _i ^ k ) x. ( ( ( normCV ` U ) ` ( A ( +v ` U ) ( ( _i ^ k ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) e. CC )
22	17, 21	syl 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 ) ( .sOLD ` U ) B ) ) ) ^ 2 ) ) / 4 ) e. CC )
23	6, 22	eqeltrd 2536	1 |- ( ( U e. NrmCVec /\ A e. X /\ B e. X ) -> ( A P B ) e. CC )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1370   e. wcel 1757   =/= wne 2641   ` cfv 5502  (class class class)co 6176   CCcc 9367   0cc0 9369   1c1 9370   _ici 9371   x. cmul 9374   / cdiv 10080   2c2 10458   4c4 10460   NN0cn0 10666   ...cfz 11524   ^cexp 11952   sum_csu 13251   NrmCVeccnv 24083   +vcpv 24084   BaseSetcba 24085   .sOLDcns 24086   norm_CVcnmcv 24089   .iOLDcdip 24216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1709  ax-7 1729  ax-8 1759  ax-9 1761  ax-10 1776  ax-11 1781  ax-12 1793  ax-13 1944  ax-ext 2429  ax-rep 4487  ax-sep 4497  ax-nul 4505  ax-pow 4554  ax-pr 4615  ax-un 6458  ax-inf2 7934  ax-cnex 9425  ax-resscn 9426  ax-1cn 9427  ax-icn 9428  ax-addcl 9429  ax-addrcl 9430  ax-mulcl 9431  ax-mulrcl 9432  ax-mulcom 9433  ax-addass 9434  ax-mulass 9435  ax-distr 9436  ax-i2m1 9437  ax-1ne0 9438  ax-1rid 9439  ax-rnegex 9440  ax-rrecex 9441  ax-cnre 9442  ax-pre-lttri 9443  ax-pre-lttrn 9444  ax-pre-ltadd 9445  ax-pre-mulgt0 9446  ax-pre-sup 9447

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-fal 1376  df-ex 1588  df-nf 1591  df-sb 1702  df-eu 2263  df-mo 2264  df-clab 2436  df-cleq 2442  df-clel 2445  df-nfc 2598  df-ne 2643  df-nel 2644  df-ral 2797  df-rex 2798  df-reu 2799  df-rmo 2800  df-rab 2801  df-v 3056  df-sbc 3271  df-csb 3373  df-dif 3415  df-un 3417  df-in 3419  df-ss 3426  df-pss 3428  df-nul 3722  df-if 3876  df-pw 3946  df-sn 3962  df-pr 3964  df-tp 3966  df-op 3968  df-uni 4176  df-int 4213  df-iun 4257  df-br 4377  df-opab 4435  df-mpt 4436  df-tr 4470  df-eprel 4716  df-id 4720  df-po 4725  df-so 4726  df-fr 4763  df-se 4764  df-we 4765  df-ord 4806  df-on 4807  df-lim 4808  df-suc 4809  df-xp 4930  df-rel 4931  df-cnv 4932  df-co 4933  df-dm 4934  df-rn 4935  df-res 4936  df-ima 4937  df-iota 5465  df-fun 5504  df-fn 5505  df-f 5506  df-f1 5507  df-fo 5508  df-f1o 5509  df-fv 5510  df-isom 5511  df-riota 6137  df-ov 6179  df-oprab 6180  df-mpt2 6181  df-om 6563  df-1st 6663  df-2nd 6664  df-recs 6918  df-rdg 6952  df-1o 7006  df-oadd 7010  df-er 7187  df-en 7397  df-dom 7398  df-sdom 7399  df-fin 7400  df-sup 7778  df-oi 7811  df-card 8196  df-pnf 9507  df-mnf 9508  df-xr 9509  df-ltxr 9510  df-le 9511  df-sub 9684  df-neg 9685  df-div 10081  df-nn 10410  df-2 10467  df-3 10468  df-4 10469  df-n0 10667  df-z 10734  df-uz 10949  df-rp 11079  df-fz 11525  df-fzo 11636  df-seq 11894  df-exp 11953  df-hash 12191  df-cj 12676  df-re 12677  df-im 12678  df-sqr 12812  df-abs 12813  df-clim 13054  df-sum 13252  df-grpo 23799  df-ablo 23890  df-vc 24045  df-nv 24091  df-va 24094  df-ba 24095  df-sm 24096  df-0v 24097  df-nmcv 24099  df-dip 24217

This theorem is referenced by:  ipf  24232  ipipcj  24234  ip1ilem  24347  ip2i  24349  ipasslem1  24352  ipasslem2  24353  ipasslem4  24355  ipasslem5  24356  ipasslem7  24357  ipasslem8  24358  ipasslem9  24359  ipasslem10  24360  ipasslem11  24361  dipdi  24364  ip2dii  24365  dipassr  24367  dipsubdir  24369  dipsubdi  24370  pythi  24371  siilem1  24372  siilem2  24373  siii  24374  ipblnfi  24377  ip2eqi  24378  htthlem  24440