Theorem fwddifnval 36663

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 36661	. . . 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 7420	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	adantr 485	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5		dmeq 5892	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	eleq2d 2848	. . . . . . . 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 3337	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
9	8	rabbidv 3422	. . . . 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 7419	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
11	10	adantr 485	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12		oveq1 7419	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘))
13	12	oveq2d 7428	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (-1↑(𝑛 − 𝑘)) = (-1↑(𝑁 − 𝑘)))
14		fveq1 6880	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
15	13, 14	oveqan12d 7431	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))
16	11, 15	oveq12d 7430	. . . . . . 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 15764	. . . . 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 5197	. . . 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 11187	. . . . 5 ⊢ ℂ ∈ V
25		elpm2r 8840	. . . . 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 7223	. . . 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 7564	. 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 7435	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
32	31	oveq2d 7428	. . . . 5 ⊢ (𝑥 = 𝑋 → ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))
33	32	oveq2d 7428	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
34	33	sumeq2sdv 15761	. . 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 6716	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
39	38	adantr 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
40	37, 39	eleqtrrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
41	40	ralrimiva 3156	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42		oveq1 7419	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
43	42	eleq1d 2847	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
44	43	ralbidv 3187	. . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
45	44	elrab 3649	. . 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 15746	. . 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 6997	1 ⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142  ∀wral 3078  {crab 3415  Vcvv 3454   ⊆ wss 3904   ↦ cmpt 5191  dom cdm 5660  ⟶wf 6532  ‘cfv 6536  (class class class)co 7412   ∈ cmpo 7414   ↑_pm cpm 8823  ℂcc 11104  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111   − cmin 11447  -cneg 11448  ℕ₀cn0 12510  ...cfz 13541  ↑cexp 14104  Ccbc 14345  Σcsu 15744   △ⁿ cfwddifn 36660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pow 5335  ax-pr 5403  ax-un 7734  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-nel 3064  df-ral 3079  df-rex 3089  df-reu 3369  df-rab 3416  df-v 3456  df-sbc 3744  df-csb 3853  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-rn 5671  df-res 5672  df-ima 5673  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7861  df-1st 7984  df-2nd 7985  df-frecs 8276  df-wrecs 8307  df-recs 8356  df-rdg 8395  df-er 8692  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11251  df-mnf 11252  df-xr 11253  df-ltxr 11254  df-le 11255  df-sub 11449  df-neg 11450  df-nn 12240  df-n0 12511  df-z 12598  df-uz 12869  df-fz 13542  df-seq 14045  df-sum 15745  df-fwddifn 36661

This theorem is used by:  fwddifn0  36664  fwddifnp1  36665