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Theorem fwddifnval 32245
Description: The value of the forward difference operator at a point. (Contributed by Scott Fenton, 28-May-2020.)
Hypotheses
Ref Expression
fwddifnval.1 (𝜑𝑁 ∈ ℕ0)
fwddifnval.2 (𝜑𝐴 ⊆ ℂ)
fwddifnval.3 (𝜑𝐹:𝐴⟶ℂ)
fwddifnval.4 (𝜑𝑋 ∈ ℂ)
fwddifnval.5 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)
Assertion
Ref Expression
fwddifnval (𝜑 → ((𝑁n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (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.
StepHypRef Expression
1 df-fwddifn 32243 . . . 4 n = (𝑛 ∈ ℕ0, 𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))))
21a1i 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↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))))))
3 oveq2 6643 . . . . . . . 8 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43adantr 481 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5 dmeq 5313 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
65eleq2d 2685 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
76adantl 482 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
84, 7raleqbidv 3147 . . . . . 6 ((𝑛 = 𝑁𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
98rabbidv 3184 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} = {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
10 oveq1 6642 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
1110adantr 481 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 6642 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312oveq2d 6651 . . . . . . . . 9 (𝑛 = 𝑁 → (-1↑(𝑛𝑘)) = (-1↑(𝑁𝑘)))
14 fveq1 6177 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
1513, 14oveqan12d 6654 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))
1611, 15oveq12d 6653 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
1716adantr 481 . . . . . 6 (((𝑛 = 𝑁𝑓 = 𝐹) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
184, 17sumeq12dv 14418 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
199, 18mpteq12dv 4724 . . . 4 ((𝑛 = 𝑁𝑓 = 𝐹) → (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
2019adantl 482 . . 3 ((𝜑 ∧ (𝑛 = 𝑁𝑓 = 𝐹)) → (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} ↦ Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))))) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
21 fwddifnval.1 . . 3 (𝜑𝑁 ∈ ℕ0)
22 fwddifnval.3 . . . 4 (𝜑𝐹:𝐴⟶ℂ)
23 fwddifnval.2 . . . 4 (𝜑𝐴 ⊆ ℂ)
24 cnex 10002 . . . . 5 ℂ ∈ V
25 elpm2r 7860 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2624, 24, 25mpanl12 717 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
2722, 23, 26syl2anc 692 . . 3 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
2824mptrabex 6473 . . . 4 (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))) ∈ V
2928a1i 11 . . 3 (𝜑 → (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))) ∈ V)
302, 20, 21, 27, 29ovmpt2d 6773 . 2 (𝜑 → (𝑁n 𝐹) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
31 oveq1 6642 . . . . . . 7 (𝑥 = 𝑋 → (𝑥 + 𝑘) = (𝑋 + 𝑘))
3231fveq2d 6182 . . . . . 6 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
3332oveq2d 6651 . . . . 5 (𝑥 = 𝑋 → ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘))))
3433oveq2d 6651 . . . 4 (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3534sumeq2sdv 14416 . . 3 (𝑥 = 𝑋 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3635adantl 482 . 2 ((𝜑𝑥 = 𝑋) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
37 fwddifnval.4 . . 3 (𝜑𝑋 ∈ ℂ)
38 fwddifnval.5 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)
39 fdm 6038 . . . . . . 7 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
4022, 39syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
4140adantr 481 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
4238, 41eleqtrrd 2702 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
4342ralrimiva 2963 . . 3 (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
44 oveq1 6642 . . . . . 6 (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
4544eleq1d 2684 . . . . 5 (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
4645ralbidv 2983 . . . 4 (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4746elrab 3357 . . 3 (𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↔ (𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4837, 43, 47sylanbrc 697 . 2 (𝜑𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
49 sumex 14399 . . 3 Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V
5049a1i 11 . 2 (𝜑 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V)
5130, 36, 48, 50fvmptd 6275 1 (𝜑 → ((𝑁n 𝐹)‘𝑋) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1481  wcel 1988  wral 2909  {crab 2913  Vcvv 3195  wss 3567  cmpt 4720  dom cdm 5104  wf 5872  cfv 5876  (class class class)co 6635  cmpt2 6637  pm cpm 7843  cc 9919  0cc0 9921  1c1 9922   + caddc 9924   · cmul 9926  cmin 10251  -cneg 10252  0cn0 11277  ...cfz 12311  cexp 12843  Ccbc 13072  Σcsu 14397  n cfwddifn 32242
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-rep 4762  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934  ax-cnex 9977  ax-resscn 9978  ax-1cn 9979  ax-icn 9980  ax-addcl 9981  ax-addrcl 9982  ax-mulcl 9983  ax-mulrcl 9984  ax-mulcom 9985  ax-addass 9986  ax-mulass 9987  ax-distr 9988  ax-i2m1 9989  ax-1ne0 9990  ax-1rid 9991  ax-rnegex 9992  ax-rrecex 9993  ax-cnre 9994  ax-pre-lttri 9995  ax-pre-lttrn 9996  ax-pre-ltadd 9997  ax-pre-mulgt0 9998
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-fal 1487  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-om 7051  df-1st 7153  df-2nd 7154  df-wrecs 7392  df-recs 7453  df-rdg 7491  df-er 7727  df-pm 7845  df-en 7941  df-dom 7942  df-sdom 7943  df-pnf 10061  df-mnf 10062  df-xr 10063  df-ltxr 10064  df-le 10065  df-sub 10253  df-neg 10254  df-nn 11006  df-n0 11278  df-z 11363  df-uz 11673  df-fz 12312  df-seq 12785  df-sum 14398  df-fwddifn 32243
This theorem is referenced by:  fwddifn0  32246  fwddifnp1  32247
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