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Theorem fwddifval 34458
Description: Calculate the value of the forward difference operator at a point. (Contributed by Scott Fenton, 18-May-2020.)
Hypotheses
Ref Expression
fwddifval.1 (𝜑𝐴 ⊆ ℂ)
fwddifval.2 (𝜑𝐹:𝐴⟶ℂ)
fwddifval.3 (𝜑𝑋𝐴)
fwddifval.4 (𝜑 → (𝑋 + 1) ∈ 𝐴)
Assertion
Ref Expression
fwddifval (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))

Proof of Theorem fwddifval
Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fwddif 34455 . . . 4 △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))))
2 dmeq 5810 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
32eleq2d 2826 . . . . . 6 (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
42, 3rabeqbidv 3419 . . . . 5 (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5 fveq1 6768 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6 fveq1 6768 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
75, 6oveq12d 7287 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)))
84, 7mpteq12dv 5170 . . . 4 (𝑓 = 𝐹 → (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
9 fwddifval.2 . . . . 5 (𝜑𝐹:𝐴⟶ℂ)
10 fwddifval.1 . . . . 5 (𝜑𝐴 ⊆ ℂ)
11 cnex 10951 . . . . . 6 ℂ ∈ V
12 elpm2r 8614 . . . . . 6 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
1311, 11, 12mpanl12 699 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
149, 10, 13syl2anc 584 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
159fdmd 6608 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1611a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
1716, 10ssexd 5252 . . . . . 6 (𝜑𝐴 ∈ V)
1815, 17eqeltrd 2841 . . . . 5 (𝜑 → dom 𝐹 ∈ V)
19 rabexg 5259 . . . . 5 (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20 mptexg 7092 . . . . 5 ({𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) ∈ V)
2118, 19, 203syl 18 . . . 4 (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) ∈ V)
221, 8, 14, 21fvmptd3 6893 . . 3 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2315eleq2d 2826 . . . . 5 (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2415, 23rabeqbidv 3419 . . . 4 (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2524mpteq1d 5174 . . 3 (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2622, 25eqtrd 2780 . 2 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
27 fvoveq1 7292 . . . 4 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28 fveq2 6769 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2927, 28oveq12d 7287 . . 3 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
3029adantl 482 . 2 ((𝜑𝑥 = 𝑋) → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
31 fwddifval.3 . . 3 (𝜑𝑋𝐴)
32 fwddifval.4 . . 3 (𝜑 → (𝑋 + 1) ∈ 𝐴)
33 oveq1 7276 . . . . 5 (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
3433eleq1d 2825 . . . 4 (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3534elrab 3626 . . 3 (𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
3631, 32, 35sylanbrc 583 . 2 (𝜑𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37 ovexd 7304 . 2 (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)) ∈ V)
3826, 30, 36, 37fvmptd 6877 1 (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110  {crab 3070  Vcvv 3431  wss 3892  cmpt 5162  dom cdm 5589  wf 6427  cfv 6431  (class class class)co 7269  pm cpm 8597  cc 10868  1c1 10871   + caddc 10873  cmin 11203  cfwddif 34454
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7580  ax-cnex 10926
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6389  df-fun 6433  df-fn 6434  df-f 6435  df-f1 6436  df-fo 6437  df-f1o 6438  df-fv 6439  df-ov 7272  df-oprab 7273  df-mpo 7274  df-pm 8599  df-fwddif 34455
This theorem is referenced by: (None)
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