Theorem fwddifnval 33619

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 33617	. . . 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 7158	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	adantr 483	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5		dmeq 5766	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	eleq2d 2898	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
7	6	adantl 484	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
8	4, 7	raleqbidv 3401	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
9	8	rabbidv 3480	. . . . 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 7157	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
11	10	adantr 483	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12		oveq1 7157	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘))
13	12	oveq2d 7166	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (-1↑(𝑛 − 𝑘)) = (-1↑(𝑁 − 𝑘)))
14		fveq1 6663	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
15	13, 14	oveqan12d 7169	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))
16	11, 15	oveq12d 7168	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
17	16	adantr 483	. . . . . 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 15057	. . . . 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 5143	. . . 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 484	. . 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 10612	. . . . 5 ⊢ ℂ ∈ V
25		elpm2r 8418	. . . . 5 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
26	24, 24, 25	mpanl12 700	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
27	22, 23, 26	syl2anc 586	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
28	24	mptrabex 6982	. . . 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 7296	. 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 7173	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
32	31	oveq2d 7166	. . . . 5 ⊢ (𝑥 = 𝑋 → ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))
33	32	oveq2d 7166	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
34	33	sumeq2sdv 15055	. . 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 484	. 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 6517	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
39	38	adantr 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
40	37, 39	eleqtrrd 2916	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
41	40	ralrimiva 3182	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42		oveq1 7157	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
43	42	eleq1d 2897	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
44	43	ralbidv 3197	. . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
45	44	elrab 3679	. . 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 585	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
47		sumex 15038	. . 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 6769	1 ⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  {crab 3142  Vcvv 3494   ⊆ wss 3935   ↦ cmpt 5138  dom cdm 5549  ⟶wf 6345  ‘cfv 6349  (class class class)co 7150   ∈ cmpo 7152   ↑_pm cpm 8401  ℂcc 10529  0cc0 10531  1c1 10532   + caddc 10534   · cmul 10536   − cmin 10864  -cneg 10865  ℕ₀cn0 11891  ...cfz 12886  ↑cexp 13423  Ccbc 13656  Σcsu 15036   △n cfwddifn 33616

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5182  ax-sep 5195  ax-nul 5202  ax-pow 5258  ax-pr 5321  ax-un 7455  ax-cnex 10587  ax-resscn 10588  ax-1cn 10589  ax-icn 10590  ax-addcl 10591  ax-addrcl 10592  ax-mulcl 10593  ax-mulrcl 10594  ax-mulcom 10595  ax-addass 10596  ax-mulass 10597  ax-distr 10598  ax-i2m1 10599  ax-1ne0 10600  ax-1rid 10601  ax-rnegex 10602  ax-rrecex 10603  ax-cnre 10604  ax-pre-lttri 10605  ax-pre-lttrn 10606  ax-pre-ltadd 10607  ax-pre-mulgt0 10608

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-fal 1546  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-pss 3953  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4561  df-pr 4563  df-tp 4565  df-op 4567  df-uni 4832  df-iun 4913  df-br 5059  df-opab 5121  df-mpt 5139  df-tr 5165  df-id 5454  df-eprel 5459  df-po 5468  df-so 5469  df-fr 5508  df-we 5510  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-pred 6142  df-ord 6188  df-on 6189  df-lim 6190  df-suc 6191  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-f1 6354  df-fo 6355  df-f1o 6356  df-fv 6357  df-riota 7108  df-ov 7153  df-oprab 7154  df-mpo 7155  df-om 7575  df-1st 7683  df-2nd 7684  df-wrecs 7941  df-recs 8002  df-rdg 8040  df-er 8283  df-pm 8403  df-en 8504  df-dom 8505  df-sdom 8506  df-pnf 10671  df-mnf 10672  df-xr 10673  df-ltxr 10674  df-le 10675  df-sub 10866  df-neg 10867  df-nn 11633  df-n0 11892  df-z 11976  df-uz 12238  df-fz 12887  df-seq 13364  df-sum 15037  df-fwddifn 33617

This theorem is referenced by:  fwddifn0  33620  fwddifnp1  33621