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Theorem fwddifnval 33521
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 33519 . . . 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 7153 . . . . . . . 8 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43adantr 481 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5 dmeq 5765 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
65eleq2d 2895 . . . . . . . 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 3399 . . . . . 6 ((𝑛 = 𝑁𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
98rabbidv 3478 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} = {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
10 oveq1 7152 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
1110adantr 481 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 7152 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312oveq2d 7161 . . . . . . . . 9 (𝑛 = 𝑁 → (-1↑(𝑛𝑘)) = (-1↑(𝑁𝑘)))
14 fveq1 6662 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
1513, 14oveqan12d 7164 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))
1611, 15oveq12d 7163 . . . . . . 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 15051 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
199, 18mpteq12dv 5142 . . . 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 10606 . . . . 5 ℂ ∈ V
25 elpm2r 8413 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2624, 24, 25mpanl12 698 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
2722, 23, 26syl2anc 584 . . 3 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
2824mptrabex 6979 . . . 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, 29ovmpod 7291 . 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 7168 . . . . . 6 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
3231oveq2d 7161 . . . . 5 (𝑥 = 𝑋 → ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘))))
3332oveq2d 7161 . . . 4 (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3433sumeq2sdv 15049 . . 3 (𝑥 = 𝑋 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3534adantl 482 . 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)
3822fdmd 6516 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
3938adantr 481 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
4037, 39eleqtrrd 2913 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
4140ralrimiva 3179 . . 3 (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42 oveq1 7152 . . . . . 6 (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
4342eleq1d 2894 . . . . 5 (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
4443ralbidv 3194 . . . 4 (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4544elrab 3677 . . 3 (𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↔ (𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4636, 41, 45sylanbrc 583 . 2 (𝜑𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
47 sumex 15032 . . 3 Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V
4847a1i 11 . 2 (𝜑 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))) ∈ V)
4930, 35, 46, 48fvmptd 6767 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 207  wa 396   = wceq 1528  wcel 2105  wral 3135  {crab 3139  Vcvv 3492  wss 3933  cmpt 5137  dom cdm 5548  wf 6344  cfv 6348  (class class class)co 7145  cmpo 7147  pm cpm 8396  cc 10523  0cc0 10525  1c1 10526   + caddc 10528   · cmul 10530  cmin 10858  -cneg 10859  0cn0 11885  ...cfz 12880  cexp 13417  Ccbc 13650  Σcsu 15030  n cfwddifn 33518
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-fal 1541  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-er 8278  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-nn 11627  df-n0 11886  df-z 11970  df-uz 12232  df-fz 12881  df-seq 13358  df-sum 15031  df-fwddifn 33519
This theorem is referenced by:  fwddifn0  33522  fwddifnp1  33523
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