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Theorem fwddifnval 32607
 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 32605 . . . 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 6801 . . . . . . . 8 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43adantr 466 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5 dmeq 5462 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
65eleq2d 2836 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
76adantl 467 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
84, 7raleqbidv 3301 . . . . . 6 ((𝑛 = 𝑁𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
98rabbidv 3339 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} = {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
10 oveq1 6800 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
1110adantr 466 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 6800 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312oveq2d 6809 . . . . . . . . 9 (𝑛 = 𝑁 → (-1↑(𝑛𝑘)) = (-1↑(𝑁𝑘)))
14 fveq1 6331 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
1513, 14oveqan12d 6812 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))
1611, 15oveq12d 6811 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
1716adantr 466 . . . . . 6 (((𝑛 = 𝑁𝑓 = 𝐹) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
184, 17sumeq12dv 14645 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
199, 18mpteq12dv 4867 . . . 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 467 . . 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 10219 . . . . 5 ℂ ∈ V
25 elpm2r 8027 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2624, 24, 25mpanl12 682 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
2722, 23, 26syl2anc 573 . . 3 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
2824mptrabex 6632 . . . 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 6935 . 2 (𝜑 → (𝑁n 𝐹) = (𝑥 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↦ Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))))
31 fvoveq1 6816 . . . . . 6 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
3231oveq2d 6809 . . . . 5 (𝑥 = 𝑋 → ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘))))
3332oveq2d 6809 . . . 4 (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3433sumeq2sdv 14643 . . 3 (𝑥 = 𝑋 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3534adantl 467 . 2 ((𝜑𝑥 = 𝑋) → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
36 fwddifnval.4 . . 3 (𝜑𝑋 ∈ ℂ)
37 fwddifnval.5 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ 𝐴)
38 fdm 6191 . . . . . . 7 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
3922, 38syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
4039adantr 466 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
4137, 40eleqtrrd 2853 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
4241ralrimiva 3115 . . 3 (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
43 oveq1 6800 . . . . . 6 (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
4443eleq1d 2835 . . . . 5 (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
4544ralbidv 3135 . . . 4 (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4645elrab 3515 . . 3 (𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↔ (𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4736, 42, 46sylanbrc 572 . 2 (𝜑𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
48 sumex 14626 . . 3 Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V
4948a1i 11 . 2 (𝜑 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V)
5030, 35, 47, 49fvmptd 6430 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 382   = wceq 1631   ∈ wcel 2145  ∀wral 3061  {crab 3065  Vcvv 3351   ⊆ wss 3723   ↦ cmpt 4863  dom cdm 5249  ⟶wf 6027  ‘cfv 6031  (class class class)co 6793   ↦ cmpt2 6795   ↑pm cpm 8010  ℂcc 10136  0cc0 10138  1c1 10139   + caddc 10141   · cmul 10143   − cmin 10468  -cneg 10469  ℕ0cn0 11494  ...cfz 12533  ↑cexp 13067  Ccbc 13293  Σcsu 14624   △n cfwddifn 32604 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-cnex 10194  ax-resscn 10195  ax-1cn 10196  ax-icn 10197  ax-addcl 10198  ax-addrcl 10199  ax-mulcl 10200  ax-mulrcl 10201  ax-mulcom 10202  ax-addass 10203  ax-mulass 10204  ax-distr 10205  ax-i2m1 10206  ax-1ne0 10207  ax-1rid 10208  ax-rnegex 10209  ax-rrecex 10210  ax-cnre 10211  ax-pre-lttri 10212  ax-pre-lttrn 10213  ax-pre-ltadd 10214  ax-pre-mulgt0 10215 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-fal 1637  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-nel 3047  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-riota 6754  df-ov 6796  df-oprab 6797  df-mpt2 6798  df-om 7213  df-1st 7315  df-2nd 7316  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-pm 8012  df-en 8110  df-dom 8111  df-sdom 8112  df-pnf 10278  df-mnf 10279  df-xr 10280  df-ltxr 10281  df-le 10282  df-sub 10470  df-neg 10471  df-nn 11223  df-n0 11495  df-z 11580  df-uz 11889  df-fz 12534  df-seq 13009  df-sum 14625  df-fwddifn 32605 This theorem is referenced by:  fwddifn0  32608  fwddifnp1  32609
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