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Theorem fwddifval 33697
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 33694 . . . 4 △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))))
2 dmeq 5749 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
32eleq2d 2899 . . . . . 6 (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
42, 3rabeqbidv 3461 . . . . 5 (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5 fveq1 6651 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6 fveq1 6651 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
75, 6oveq12d 7158 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)))
84, 7mpteq12dv 5127 . . . 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 10607 . . . . . 6 ℂ ∈ V
12 elpm2r 8411 . . . . . 6 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
1311, 11, 12mpanl12 701 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
149, 10, 13syl2anc 587 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
159fdmd 6504 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1611a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
1716, 10ssexd 5204 . . . . . 6 (𝜑𝐴 ∈ V)
1815, 17eqeltrd 2914 . . . . 5 (𝜑 → dom 𝐹 ∈ V)
19 rabexg 5210 . . . . 5 (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20 mptexg 6966 . . . . 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 6773 . . 3 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2315eleq2d 2899 . . . . 5 (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2415, 23rabeqbidv 3461 . . . 4 (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2524mpteq1d 5131 . . 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 2857 . 2 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
27 fvoveq1 7163 . . . 4 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28 fveq2 6652 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2927, 28oveq12d 7158 . . 3 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
3029adantl 485 . 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 7147 . . . . 5 (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
3433eleq1d 2898 . . . 4 (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3534elrab 3655 . . 3 (𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
3631, 32, 35sylanbrc 586 . 2 (𝜑𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37 ovexd 7175 . 2 (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)) ∈ V)
3826, 30, 36, 37fvmptd 6757 1 (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  {crab 3134  Vcvv 3469  wss 3908  cmpt 5122  dom cdm 5532  wf 6330  cfv 6334  (class class class)co 7140  pm cpm 8394  cc 10524  1c1 10527   + caddc 10529  cmin 10859  cfwddif 33693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-pm 8396  df-fwddif 33694
This theorem is referenced by: (None)
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