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Theorem fwddifval 32858
 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 32855 . . . 4 △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))))
2 dmeq 5569 . . . . . 6 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
32eleq2d 2845 . . . . . 6 (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
42, 3rabeqbidv 3392 . . . . 5 (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5 fveq1 6445 . . . . . 6 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6 fveq1 6445 . . . . . 6 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
75, 6oveq12d 6940 . . . . 5 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)))
84, 7mpteq12dv 4969 . . . 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 10353 . . . . . 6 ℂ ∈ V
12 elpm2r 8158 . . . . . 6 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
1311, 11, 12mpanl12 692 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
149, 10, 13syl2anc 579 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
159fdmd 6300 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1611a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
1716, 10ssexd 5042 . . . . . 6 (𝜑𝐴 ∈ V)
1815, 17eqeltrd 2859 . . . . 5 (𝜑 → dom 𝐹 ∈ V)
19 rabexg 5048 . . . . 5 (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20 mptexg 6756 . . . . 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 6564 . . 3 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2315eleq2d 2845 . . . . 5 (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2415, 23rabeqbidv 3392 . . . 4 (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2524mpteq1d 4973 . . 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 2814 . 2 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
27 fvoveq1 6945 . . . 4 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28 fveq2 6446 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
2927, 28oveq12d 6940 . . 3 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
3029adantl 475 . 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 6929 . . . . 5 (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
3433eleq1d 2844 . . . 4 (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3534elrab 3572 . . 3 (𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
3631, 32, 35sylanbrc 578 . 2 (𝜑𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37 ovexd 6956 . 2 (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)) ∈ V)
3826, 30, 36, 37fvmptd 6548 1 (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 386   = wceq 1601   ∈ wcel 2107  {crab 3094  Vcvv 3398   ⊆ wss 3792   ↦ cmpt 4965  dom cdm 5355  ⟶wf 6131  ‘cfv 6135  (class class class)co 6922   ↑pm cpm 8141  ℂcc 10270  1c1 10273   + caddc 10275   − cmin 10606   △ cfwddif 32854 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-id 5261  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-pm 8143  df-fwddif 32855 This theorem is referenced by: (None)
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