Theorem fwddifnval 32270

Description: The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.)

Hypotheses

Ref	Expression
fwddifnval.1	|- ( ph -> N e. NN0 )
fwddifnval.2	|- ( ph -> A C_ CC )
fwddifnval.3	|- ( ph -> F : A --> CC )
fwddifnval.4	|- ( ph -> X e. CC )
fwddifnval.5	|- ( ( ph /\ k e. ( 0 ... N ) ) -> ( X + k ) e. A )

Assertion

Ref	Expression
fwddifnval	|- ( ph -> ( ( N _/_\^nn F ) ` X ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) )

Distinct variable groups:   k, N   A, k   k, X   k, F   ph, k

Proof of Theorem fwddifnval

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

Step	Hyp	Ref	Expression
1		df-fwddifn 32268	. . . 4 |- _/_\^nn = ( n e. NN0 , f e. ( CC ^pm CC ) |-> ( x e. { y e. CC | A. k e. ( 0 ... n ) ( y + k ) e. dom f } |-> sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) ) )
2	1	a1i 11	. . 3 |- ( ph -> _/_\^nn = ( n e. NN0 , f e. ( CC ^pm CC ) |-> ( x e. { y e. CC | A. k e. ( 0 ... n ) ( y + k ) e. dom f } |-> sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) ) ) )
3		oveq2 6658	. . . . . . . 8 |- ( n = N -> ( 0 ... n ) = ( 0 ... N ) )
4	3	adantr 481	. . . . . . 7 |- ( ( n = N /\ f = F ) -> ( 0 ... n ) = ( 0 ... N ) )
5		dmeq 5324	. . . . . . . . 9 |- ( f = F -> dom f = dom F )
6	5	eleq2d 2687	. . . . . . . 8 |- ( f = F -> ( ( y + k ) e. dom f <-> ( y + k ) e. dom F ) )
7	6	adantl 482	. . . . . . 7 |- ( ( n = N /\ f = F ) -> ( ( y + k ) e. dom f <-> ( y + k ) e. dom F ) )
8	4, 7	raleqbidv 3152	. . . . . 6 |- ( ( n = N /\ f = F ) -> ( A. k e. ( 0 ... n ) ( y + k ) e. dom f <-> A. k e. ( 0 ... N ) ( y + k ) e. dom F ) )
9	8	rabbidv 3189	. . . . 5 |- ( ( n = N /\ f = F ) -> { y e. CC | A. k e. ( 0 ... n ) ( y + k ) e. dom f } = { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } )
10		oveq1 6657	. . . . . . . . 9 |- ( n = N -> ( n _C k ) = ( N _C k ) )
11	10	adantr 481	. . . . . . . 8 |- ( ( n = N /\ f = F ) -> ( n _C k ) = ( N _C k ) )
12		oveq1 6657	. . . . . . . . . 10 |- ( n = N -> ( n - k ) = ( N - k ) )
13	12	oveq2d 6666	. . . . . . . . 9 |- ( n = N -> ( -u 1 ^ ( n - k ) ) = ( -u 1 ^ ( N - k ) ) )
14		fveq1 6190	. . . . . . . . 9 |- ( f = F -> ( f ` ( x + k ) ) = ( F ` ( x + k ) ) )
15	13, 14	oveqan12d 6669	. . . . . . . 8 |- ( ( n = N /\ f = F ) -> ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) = ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) )
16	11, 15	oveq12d 6668	. . . . . . 7 |- ( ( n = N /\ f = F ) -> ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) = ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) )
17	16	adantr 481	. . . . . 6 |- ( ( ( n = N /\ f = F ) /\ k e. ( 0 ... n ) ) -> ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) = ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) )
18	4, 17	sumeq12dv 14437	. . . . 5 |- ( ( n = N /\ f = F ) -> sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) )
19	9, 18	mpteq12dv 4733	. . . 4 |- ( ( n = N /\ f = F ) -> ( x e. { y e. CC | A. k e. ( 0 ... n ) ( y + k ) e. dom f } |-> sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) ) = ( x e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } |-> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) ) )
20	19	adantl 482	. . 3 |- ( ( ph /\ ( n = N /\ f = F ) ) -> ( x e. { y e. CC | A. k e. ( 0 ... n ) ( y + k ) e. dom f } |-> sum_ k e. ( 0 ... n ) ( ( n _C k ) x. ( ( -u 1 ^ ( n - k ) ) x. ( f ` ( x + k ) ) ) ) ) = ( x e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } |-> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) ) )
21		fwddifnval.1	. . 3 |- ( ph -> N e. NN0 )
22		fwddifnval.3	. . . 4 |- ( ph -> F : A --> CC )
23		fwddifnval.2	. . . 4 |- ( ph -> A C_ CC )
24		cnex 10017	. . . . 5 |- CC e. _V
25		elpm2r 7875	. . . . 5 |- ( ( ( CC e. _V /\ CC e. _V ) /\ ( F : A --> CC /\ A C_ CC ) ) -> F e. ( CC ^pm CC ) )
26	24, 24, 25	mpanl12 718	. . . 4 |- ( ( F : A --> CC /\ A C_ CC ) -> F e. ( CC ^pm CC ) )
27	22, 23, 26	syl2anc 693	. . 3 |- ( ph -> F e. ( CC ^pm CC ) )
28	24	mptrabex 6488	. . . 4 |- ( x e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } |-> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) ) e. _V
29	28	a1i 11	. . 3 |- ( ph -> ( x e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } |-> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) ) e. _V )
30	2, 20, 21, 27, 29	ovmpt2d 6788	. 2 |- ( ph -> ( N _/_\^nn F ) = ( x e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } |-> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) ) )
31		oveq1 6657	. . . . . . 7 |- ( x = X -> ( x + k ) = ( X + k ) )
32	31	fveq2d 6195	. . . . . 6 |- ( x = X -> ( F ` ( x + k ) ) = ( F ` ( X + k ) ) )
33	32	oveq2d 6666	. . . . 5 |- ( x = X -> ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) = ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) )
34	33	oveq2d 6666	. . . 4 |- ( x = X -> ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) = ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) )
35	34	sumeq2sdv 14435	. . 3 |- ( x = X -> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) )
36	35	adantl 482	. 2 |- ( ( ph /\ x = X ) -> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( x + k ) ) ) ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) )
37		fwddifnval.4	. . 3 |- ( ph -> X e. CC )
38		fwddifnval.5	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( X + k ) e. A )
39		fdm 6051	. . . . . . 7 |- ( F : A --> CC -> dom F = A )
40	22, 39	syl 17	. . . . . 6 |- ( ph -> dom F = A )
41	40	adantr 481	. . . . 5 |- ( ( ph /\ k e. ( 0 ... N ) ) -> dom F = A )
42	38, 41	eleqtrrd 2704	. . . 4 |- ( ( ph /\ k e. ( 0 ... N ) ) -> ( X + k ) e. dom F )
43	42	ralrimiva 2966	. . 3 |- ( ph -> A. k e. ( 0 ... N ) ( X + k ) e. dom F )
44		oveq1 6657	. . . . . 6 |- ( y = X -> ( y + k ) = ( X + k ) )
45	44	eleq1d 2686	. . . . 5 |- ( y = X -> ( ( y + k ) e. dom F <-> ( X + k ) e. dom F ) )
46	45	ralbidv 2986	. . . 4 |- ( y = X -> ( A. k e. ( 0 ... N ) ( y + k ) e. dom F <-> A. k e. ( 0 ... N ) ( X + k ) e. dom F ) )
47	46	elrab 3363	. . 3 |- ( X e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } <-> ( X e. CC /\ A. k e. ( 0 ... N ) ( X + k ) e. dom F ) )
48	37, 43, 47	sylanbrc 698	. 2 |- ( ph -> X e. { y e. CC | A. k e. ( 0 ... N ) ( y + k ) e. dom F } )
49		sumex 14418	. . 3 |- sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) e. _V
50	49	a1i 11	. 2 |- ( ph -> sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) e. _V )
51	30, 36, 48, 50	fvmptd 6288	1 |- ( ph -> ( ( N _/_\^nn F ) ` X ) = sum_ k e. ( 0 ... N ) ( ( N _C k ) x. ( ( -u 1 ^ ( N - k ) ) x. ( F ` ( X + k ) ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   C_ wss 3574   |-> cmpt 4729   dom cdm 5114   -->wf 5884   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ^pm cpm 7858   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267   NN0cn0 11292   ...cfz 12326   ^cexp 12860   _C cbc 13089   sum_csu 14416   _/_\^nn cfwddifn 32267

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-pm 7860  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-fwddifn 32268

This theorem is referenced by:  fwddifn0  32271  fwddifnp1  32272