Theorem fwddifnval 36668

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 36666	. . . 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 7424	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	adantr 486	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5		dmeq 5895	. . . . . . . . 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 487	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
8	4, 7	raleqbidv 3340	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
9	8	rabbidv 3425	. . . . 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 7423	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
11	10	adantr 486	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12		oveq1 7423	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘))
13	12	oveq2d 7432	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (-1↑(𝑛 − 𝑘)) = (-1↑(𝑁 − 𝑘)))
14		fveq1 6884	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
15	13, 14	oveqan12d 7435	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))
16	11, 15	oveq12d 7434	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
17	16	adantr 486	. . . . . 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 15775	. . . . 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 5200	. . . 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 487	. . 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 11192	. . . . 5 ⊢ ℂ ∈ V
25		elpm2r 8844	. . . . 5 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
26	24, 24, 25	mpanl12 715	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
27	22, 23, 26	syl2anc 596	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
28	24	mptrabex 7227	. . . 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 7568	. 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 7439	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
32	31	oveq2d 7432	. . . . 5 ⊢ (𝑥 = 𝑋 → ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))
33	32	oveq2d 7432	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
34	33	sumeq2sdv 15773	. . 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 487	. 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 6720	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
39	38	adantr 486	. . . . 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 3159	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42		oveq1 7423	. . . . . 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 3190	. . . 4 ⊢ (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
45	44	elrab 3652	. . 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 595	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
47		sumex 15758	. . 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 7001	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 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  {crab 3418  Vcvv 3457   ⊆ wss 3906   ↦ cmpt 5194  dom cdm 5663  ⟶wf 6536  ‘cfv 6540  (class class class)co 7416   ∈ cmpo 7418   ↑_pm cpm 8827  ℂcc 11109  0cc0 11111  1c1 11112   + caddc 11114   · cmul 11116   − cmin 11452  -cneg 11453  ℕ₀cn0 12515  ...cfz 13546  ↑cexp 14110  Ccbc 14351  Σcsu 15756   △ⁿ cfwddifn 36665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7738  ax-cnex 11167  ax-resscn 11168  ax-1cn 11169  ax-icn 11170  ax-addcl 11171  ax-addrcl 11172  ax-mulcl 11173  ax-mulrcl 11174  ax-mulcom 11175  ax-addass 11176  ax-mulass 11177  ax-distr 11178  ax-i2m1 11179  ax-1ne0 11180  ax-1rid 11181  ax-rnegex 11182  ax-rrecex 11183  ax-cnre 11184  ax-pre-lttri 11185  ax-pre-lttrn 11186  ax-pre-ltadd 11187  ax-pre-mulgt0 11188

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7373  df-ov 7419  df-oprab 7420  df-mpo 7421  df-om 7865  df-1st 7988  df-2nd 7989  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-er 8696  df-pm 8829  df-en 8946  df-dom 8947  df-sdom 8948  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11454  df-neg 11455  df-nn 12245  df-n0 12516  df-z 12603  df-uz 12874  df-fz 13547  df-seq 14051  df-sum 15757  df-fwddifn 36666

This theorem is used by:  fwddifn0  36669  fwddifnp1  36670