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Theorem fwddifnval 36145
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 36143 . . . 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 7439 . . . . . . . 8 (𝑛 = 𝑁 → (0...𝑛) = (0...𝑁))
43adantr 480 . . . . . . 7 ((𝑛 = 𝑁𝑓 = 𝐹) → (0...𝑛) = (0...𝑁))
5 dmeq 5917 . . . . . . . . 9 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
65eleq2d 2825 . . . . . . . 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 3344 . . . . . 6 ((𝑛 = 𝑁𝑓 = 𝐹) → (∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓 ↔ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹))
98rabbidv 3441 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑛)(𝑦 + 𝑘) ∈ dom 𝑓} = {𝑦 ∈ ℂ ∣ ∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹})
10 oveq1 7438 . . . . . . . . 9 (𝑛 = 𝑁 → (𝑛C𝑘) = (𝑁C𝑘))
1110adantr 480 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → (𝑛C𝑘) = (𝑁C𝑘))
12 oveq1 7438 . . . . . . . . . 10 (𝑛 = 𝑁 → (𝑛𝑘) = (𝑁𝑘))
1312oveq2d 7447 . . . . . . . . 9 (𝑛 = 𝑁 → (-1↑(𝑛𝑘)) = (-1↑(𝑁𝑘)))
14 fveq1 6906 . . . . . . . . 9 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 𝑘)) = (𝐹‘(𝑥 + 𝑘)))
1513, 14oveqan12d 7450 . . . . . . . 8 ((𝑛 = 𝑁𝑓 = 𝐹) → ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))))
1611, 15oveq12d 7449 . . . . . . 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 15739 . . . . 5 ((𝑛 = 𝑁𝑓 = 𝐹) → Σ𝑘 ∈ (0...𝑛)((𝑛C𝑘) · ((-1↑(𝑛𝑘)) · (𝑓‘(𝑥 + 𝑘)))) = Σ𝑘 ∈ (0...𝑁)((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))))
199, 18mpteq12dv 5239 . . . 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 11234 . . . . 5 ℂ ∈ V
25 elpm2r 8884 . . . . 5 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
2624, 24, 25mpanl12 702 . . . 4 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
2722, 23, 26syl2anc 584 . . 3 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
2824mptrabex 7245 . . . 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 7585 . 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 7454 . . . . . 6 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 𝑘)) = (𝐹‘(𝑋 + 𝑘)))
3231oveq2d 7447 . . . . 5 (𝑥 = 𝑋 → ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘))) = ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘))))
3332oveq2d 7447 . . . 4 (𝑥 = 𝑋 → ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑥 + 𝑘)))) = ((𝑁C𝑘) · ((-1↑(𝑁𝑘)) · (𝐹‘(𝑋 + 𝑘)))))
3433sumeq2sdv 15736 . . 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 6747 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
3938adantr 480 . . . . 5 ((𝜑𝑘 ∈ (0...𝑁)) → dom 𝐹 = 𝐴)
4037, 39eleqtrrd 2842 . . . 4 ((𝜑𝑘 ∈ (0...𝑁)) → (𝑋 + 𝑘) ∈ dom 𝐹)
4140ralrimiva 3144 . . 3 (𝜑 → ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹)
42 oveq1 7438 . . . . . 6 (𝑦 = 𝑋 → (𝑦 + 𝑘) = (𝑋 + 𝑘))
4342eleq1d 2824 . . . . 5 (𝑦 = 𝑋 → ((𝑦 + 𝑘) ∈ dom 𝐹 ↔ (𝑋 + 𝑘) ∈ dom 𝐹))
4443ralbidv 3176 . . . 4 (𝑦 = 𝑋 → (∀𝑘 ∈ (0...𝑁)(𝑦 + 𝑘) ∈ dom 𝐹 ↔ ∀𝑘 ∈ (0...𝑁)(𝑋 + 𝑘) ∈ dom 𝐹))
4544elrab 3695 . . 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 15721 . . 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 7023 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 206  wa 395   = wceq 1537  wcel 2106  wral 3059  {crab 3433  Vcvv 3478  wss 3963  cmpt 5231  dom cdm 5689  wf 6559  cfv 6563  (class class class)co 7431  cmpo 7433  pm cpm 8866  cc 11151  0cc0 11153  1c1 11154   + caddc 11156   · cmul 11158  cmin 11490  -cneg 11491  0cn0 12524  ...cfz 13544  cexp 14099  Ccbc 14338  Σcsu 15719  n cfwddifn 36142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-pm 8868  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-n0 12525  df-z 12612  df-uz 12877  df-fz 13545  df-seq 14040  df-sum 15720  df-fwddifn 36143
This theorem is referenced by:  fwddifn0  36146  fwddifnp1  36147
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