Theorem fwddifnval 36521

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

Hypotheses

Ref	Expression
fwddifnval.1	⊢ (𝜑 → 𝑁 ∈ ℕ0)
fwddifnval.2	⊢ (𝜑 → 𝐴 ⊆ ℂ)
fwddifnval.3	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
fwddifnval.4	⊢ (𝜑 → 𝑋 ∈ ℂ)
fwddifnval.5	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)

Assertion

Ref	Expression
fwddifnval	⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))

Distinct variable groups:   𝑘,𝑁   𝐴,𝑘   𝑘,𝑋   𝑘,𝐹   𝜑,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 36519	. . . 4 ⊢ △n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
2	1	a1i 11	. . 3 ⊢ (𝜑 → △n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))))))
3		oveq2 7408	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	adantr 485	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5		dmeq 5883	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	eleq2d 2851	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
7	6	adantl 486	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
8	4, 7	raleqbidv 3339	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
9	8	rabbidv 3424	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} = {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
10		oveq1 7407	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12		oveq1 7407	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘))
13	12	oveq2d 7416	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (-1↑(𝑛 − 𝑘)) = (-1↑(𝑁 − 𝑘)))
14		fveq1 6870	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
15	13, 14	oveqan12d 7419	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))
16	11, 15	oveq12d 7418	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
17	16	adantr 485	. . . . . 6 ⊢ (((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
18	4, 17	sumeq12dv 15745	. . . . 5 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
19	9, 18	mpteq12dv 5191	. . . 4 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
20	19	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑛 = 𝑁 ∧ 𝑓 = 𝐹)) → (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
21		fwddifnval.1	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
22		fwddifnval.3	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
23		fwddifnval.2	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
24		cnex 11169	. . . . 5 ⊢ ℂ ∈ V
25		elpm2r 8830	. . . . 5 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
26	24, 24, 25	mpanl12 714	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
27	22, 23, 26	syl2anc 595	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
28	24	mptrabex 7213	. . . 4 ⊢ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))) ∈ V
29	28	a1i 11	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))) ∈ V)
30	2, 20, 21, 27, 29	ovmpod 7552	. 2 ⊢ (𝜑 → (𝑁 △n 𝐹) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
31		fvoveq1 7423	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
32	31	oveq2d 7416	. . . . 5 ⊢ (𝑥 = 𝑋 → ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))
33	32	oveq2d 7416	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
34	33	sumeq2sdv 15742	. . 3 ⊢ (𝑥 = 𝑋 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
35	34	adantl 486	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
36		fwddifnval.4	. . 3 ⊢ (𝜑 → 𝑋 ∈ ℂ)
37		fwddifnval.5	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)
38	22	fdmd 6706	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
39	38	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
40	37, 39	eleqtrrd 2868	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
41	40	ralrimiva 3157	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42		oveq1 7407	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
43	42	eleq1d 2850	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
44	43	ralbidv 3188	. . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
45	44	elrab 3653	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↔ (𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
46	36, 41, 45	sylanbrc 594	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
47		sumex 15727	. . 3 ⊢ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V
48	47	a1i 11	. 2 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V)
49	30, 35, 46, 48	fvmptd 6987	1 ⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079  {crab 3417  Vcvv 3457   ⊆ wss 3907   ↦ cmpt 5185  dom cdm 5651  ⟶wf 6521  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402   ↑_pm cpm 8813  ℂcc 11086  0cc0 11088  1c1 11089   + caddc 11091   · cmul 11093   − cmin 11429  -cneg 11430  ℕ₀cn0 12492  ...cfz 13523  ↑cexp 14085  Ccbc 14326  Σcsu 15725   △ⁿ cfwddifn 36518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5231  ax-sep 5250  ax-nul 5260  ax-pow 5326  ax-pr 5394  ax-un 7722  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5105  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6291  df-ord 6352  df-on 6353  df-lim 6354  df-suc 6355  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-er 8682  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-nn 12222  df-n0 12493  df-z 12580  df-uz 12851  df-fz 13524  df-seq 14026  df-sum 15726  df-fwddifn 36519

This theorem is referenced by:  fwddifn0  36522  fwddifnp1  36523