Theorem vtsval 30715

Description: Value of the Vinogradov trigonometric sums (Contributed by Thierry Arnoux, 1-Dec-2021.)

Hypotheses

Ref	Expression
vtsval.n	|- ( ph -> N e. NN0 )
vtsval.x	|- ( ph -> X e. CC )
vtsval.l	|- ( ph -> L : NN --> CC )

Assertion

Ref	Expression
vtsval	|- ( ph -> ( ( Lvts N ) ` X ) = sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) )

Distinct variable groups:   L, a   N, a   X, a

Allowed substitution hint:   ph( a)

Proof of Theorem vtsval

Dummy variables l n x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vtsval.l	. . . 4 |- ( ph -> L : NN --> CC )
2		cnex 10017	. . . . 5 |- CC e. _V
3		nnex 11026	. . . . 5 |- NN e. _V
4	2, 3	elmap 7886	. . . 4 |- ( L e. ( CC ^m NN ) <-> L : NN --> CC )
5	1, 4	sylibr 224	. . 3 |- ( ph -> L e. ( CC ^m NN ) )
6		vtsval.n	. . 3 |- ( ph -> N e. NN0 )
7		fveq1 6190	. . . . . . 7 |- ( l = L -> ( l ` a ) = ( L ` a ) )
8	7	oveq1d 6665	. . . . . 6 |- ( l = L -> ( ( l ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) = ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) )
9	8	sumeq2sdv 14435	. . . . 5 |- ( l = L -> sum_ a e. ( 1 ... n ) ( ( l ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) = sum_ a e. ( 1 ... n ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) )
10	9	mpteq2dv 4745	. . . 4 |- ( l = L -> ( x e. CC |-> sum_ a e. ( 1 ... n ) ( ( l ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) = ( x e. CC |-> sum_ a e. ( 1 ... n ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) )
11		oveq2 6658	. . . . . 6 |- ( n = N -> ( 1 ... n ) = ( 1 ... N ) )
12	11	sumeq1d 14431	. . . . 5 |- ( n = N -> sum_ a e. ( 1 ... n ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) = sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) )
13	12	mpteq2dv 4745	. . . 4 |- ( n = N -> ( x e. CC |-> sum_ a e. ( 1 ... n ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) = ( x e. CC |-> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) )
14		df-vts 30714	. . . 4 |- vts = ( l e. ( CC ^m NN ) , n e. NN0 |-> ( x e. CC |-> sum_ a e. ( 1 ... n ) ( ( l ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) )
15	2	mptex 6486	. . . 4 |- ( x e. CC |-> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) e. _V
16	10, 13, 14, 15	ovmpt2 6796	. . 3 |- ( ( L e. ( CC ^m NN ) /\ N e. NN0 ) -> ( Lvts N ) = ( x e. CC |-> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) )
17	5, 6, 16	syl2anc 693	. 2 |- ( ph -> ( Lvts N ) = ( x e. CC |-> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) ) )
18		oveq2 6658	. . . . . . 7 |- ( x = X -> ( a x. x ) = ( a x. X ) )
19	18	oveq2d 6666	. . . . . 6 |- ( x = X -> ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) = ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) )
20	19	fveq2d 6195	. . . . 5 |- ( x = X -> ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) = ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) )
21	20	oveq2d 6666	. . . 4 |- ( x = X -> ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) = ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) )
22	21	sumeq2sdv 14435	. . 3 |- ( x = X -> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) = sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) )
23	22	adantl 482	. 2 |- ( ( ph /\ x = X ) -> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. x ) ) ) ) = sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) )
24		vtsval.x	. 2 |- ( ph -> X e. CC )
25		sumex 14418	. . 3 |- sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) e. _V
26	25	a1i 11	. 2 |- ( ph -> sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) e. _V )
27	17, 23, 24, 26	fvmptd 6288	1 |- ( ph -> ( ( Lvts N ) ` X ) = sum_ a e. ( 1 ... N ) ( ( L ` a ) x. ( exp ` ( ( _i x. ( 2 x. pi ) ) x. ( a x. X ) ) ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   _Vcvv 3200   |-> cmpt 4729   -->wf 5884   ` cfv 5888  (class class class)co 6650   ^m cmap 7857   CCcc 9934   1c1 9937   _ici 9938   x. cmul 9941   NNcn 11020   2c2 11070   NN0cn0 11292   ...cfz 12326   sum_csu 14416   expce 14792   picpi 14797  vtscvts 30713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-seq 12802  df-sum 14417  df-vts 30714

This theorem is referenced by:  vtscl  30716  vtsprod  30717