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Theorem fwddifnval 35630
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 35628 . . . 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 7409 . . . . . . . 8 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43adantr 480 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5 dmeq 5893 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
65eleq2d 2811 . . . . . . . 8 (𝑓 = 𝐹 → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
76adantl 481 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → ((𝑦 + 𝑘) ∈ dom 𝑓 ↔ (𝑦 + 𝑘) ∈ dom 𝐹))
84, 7raleqbidv 3334 . . . . . 6 ((𝑛 = 𝑁𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
98rabbidv 3432 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} = {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
10 oveq1 7408 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
1110adantr 480 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 7408 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312oveq2d 7417 . . . . . . . . 9 (𝑛 = 𝑁 → (-1↑(𝑛𝑘)) = (-1↑(𝑁𝑘)))
14 fveq1 6880 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
1513, 14oveqan12d 7420 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))
1611, 15oveq12d 7419 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
1716adantr 480 . . . . . 6 (((𝑛 = 𝑁𝑓 = 𝐹) ∧ 𝑘 ∈ (0...𝑛)) → ((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
184, 17sumeq12dv 15649 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
199, 18mpteq12dv 5229 . . . 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 481 . . 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 11187 . . . . 5 ℂ ∈ V
25 elpm2r 8835 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2624, 24, 25mpanl12 699 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
2722, 23, 26syl2anc 583 . . 3 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
2824mptrabex 7218 . . . 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 7552 . 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 7424 . . . . . 6 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
3231oveq2d 7417 . . . . 5 (𝑥 = 𝑋 → ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘))))
3332oveq2d 7417 . . . 4 (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3433sumeq2sdv 15647 . . 3 (𝑥 = 𝑋 → Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3534adantl 481 . 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 6718 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
3938adantr 480 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
4037, 39eleqtrrd 2828 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
4140ralrimiva 3138 . . 3 (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42 oveq1 7408 . . . . . 6 (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
4342eleq1d 2810 . . . . 5 (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
4443ralbidv 3169 . . . 4 (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4544elrab 3675 . . 3 (𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹} ↔ (𝑋 ∈ ℂ ∧ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4636, 41, 45sylanbrc 582 . 2 (𝜑𝑋 ∈ {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
47 sumex 15631 . . 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 6995 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 205  wa 395   = wceq 1533  wcel 2098  wral 3053  {crab 3424  Vcvv 3466  wss 3940  cmpt 5221  dom cdm 5666  wf 6529  cfv 6533  (class class class)co 7401  cmpo 7403  pm cpm 8817  cc 11104  0cc0 11106  1c1 11107   + caddc 11109   · cmul 11111  cmin 11441  -cneg 11442  0cn0 12469  ...cfz 13481  cexp 14024  Ccbc 14259  Σcsu 15629  n cfwddifn 35627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5275  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3959  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-tr 5256  df-id 5564  df-eprel 5570  df-po 5578  df-so 5579  df-fr 5621  df-we 5623  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-pred 6290  df-ord 6357  df-on 6358  df-lim 6359  df-suc 6360  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-om 7849  df-1st 7968  df-2nd 7969  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-er 8699  df-pm 8819  df-en 8936  df-dom 8937  df-sdom 8938  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-nn 12210  df-n0 12470  df-z 12556  df-uz 12820  df-fz 13482  df-seq 13964  df-sum 15630  df-fwddifn 35628
This theorem is referenced by:  fwddifn0  35631  fwddifnp1  35632
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