Theorem fwddifnval 36473

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 36471	. . . 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 7398	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
4	3	adantr 484	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5		dmeq 5875	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
6	5	eleq2d 2847	. . . . . . . 8 ⊢ (𝑓 = 𝐹 → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
7	6	adantl 485	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
8	4, 7	raleqbidv 3335	. . . . . 6 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
9	8	rabbidv 3420	. . . . 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 7397	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
11	10	adantr 484	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12		oveq1 7397	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (𝑛 − 𝑘) = (𝑁 − 𝑘))
13	12	oveq2d 7406	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (-1↑(𝑛 − 𝑘)) = (-1↑(𝑁 − 𝑘)))
14		fveq1 6860	. . . . . . . . 9 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
15	13, 14	oveqan12d 7409	. . . . . . . 8 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))))
16	11, 15	oveq12d 7408	. . . . . . 7 ⊢ ((𝑛 = 𝑁 ∧ 𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛 − 𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
17	16	adantr 484	. . . . . 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 15723	. . . . 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 5184	. . . 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 485	. . 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 11147	. . . . 5 ⊢ ℂ ∈ V
25		elpm2r 8819	. . . . 5 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
26	24, 24, 25	mpanl12 712	. . . 4 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
27	22, 23, 26	syl2anc 593	. . 3 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
28	24	mptrabex 7203	. . . 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 7542	. 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 7413	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
32	31	oveq2d 7406	. . . . 5 ⊢ (𝑥 = 𝑋 → ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘))))
33	32	oveq2d 7406	. . . 4 ⊢ (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
34	33	sumeq2sdv 15720	. . 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 485	. 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 6696	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
39	38	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
40	37, 39	eleqtrrd 2864	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
41	40	ralrimiva 3153	. . 3 ⊢ (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42		oveq1 7397	. . . . . 6 ⊢ (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
43	42	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
44	43	ralbidv 3184	. . . 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 592	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
47		sumex 15705	. . 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 6977	1 ⊢ (𝜑 → ((𝑁 △n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁 − 𝑘)) · (𝐹‘(𝑋 + 𝑘)))))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  {crab 3413  Vcvv 3453   ⊆ wss 3902   ↦ cmpt 5178  dom cdm 5643  ⟶wf 6511  ‘cfv 6515  (class class class)co 7390   ∈ cmpo 7392   ↑_pm cpm 8802  ℂcc 11064  0cc0 11066  1c1 11067   + caddc 11069   · cmul 11071   − cmin 11407  -cneg 11408  ℕ₀cn0 12474  ...cfz 13505  ↑cexp 14067  Ccbc 14308  Σcsu 15703   △ⁿ cfwddifn 36470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-rep 5224  ax-sep 5243  ax-nul 5253  ax-pow 5319  ax-pr 5387  ax-un 7712  ax-cnex 11122  ax-resscn 11123  ax-1cn 11124  ax-icn 11125  ax-addcl 11126  ax-addrcl 11127  ax-mulcl 11128  ax-mulrcl 11129  ax-mulcom 11130  ax-addass 11131  ax-mulass 11132  ax-distr 11133  ax-i2m1 11134  ax-1ne0 11135  ax-1rid 11136  ax-rnegex 11137  ax-rrecex 11138  ax-cnre 11139  ax-pre-lttri 11140  ax-pre-lttrn 11141  ax-pre-ltadd 11142  ax-pre-mulgt0 11143

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1098  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3061  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4948  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5538  df-eprel 5543  df-po 5551  df-so 5552  df-fr 5596  df-we 5598  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-pred 6282  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-iota 6471  df-fun 6517  df-fn 6518  df-f 6519  df-f1 6520  df-fo 6521  df-f1o 6522  df-fv 6523  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7841  df-1st 7964  df-2nd 7965  df-frecs 8255  df-wrecs 8286  df-recs 8335  df-rdg 8374  df-er 8671  df-pm 8804  df-en 8921  df-dom 8922  df-sdom 8923  df-pnf 11211  df-mnf 11212  df-xr 11213  df-ltxr 11214  df-le 11215  df-sub 11409  df-neg 11410  df-nn 12204  df-n0 12475  df-z 12562  df-uz 12833  df-fz 13506  df-seq 14008  df-sum 15704  df-fwddifn 36471

This theorem is referenced by:  fwddifn0  36474  fwddifnp1  36475