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Theorem fwddifval 31273
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 31270 . . . . 5 △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥))))
21a1i 11 . . . 4 (𝜑 → △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓𝑥)))))
3 dmeq 5233 . . . . . . 7 (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
43eleq2d 2672 . . . . . . 7 (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
53, 4rabeqbidv 3167 . . . . . 6 (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
6 fveq1 6087 . . . . . . 7 (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
7 fveq1 6087 . . . . . . 7 (𝑓 = 𝐹 → (𝑓𝑥) = (𝐹𝑥))
86, 7oveq12d 6545 . . . . . 6 (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)))
95, 8mpteq12dv 4657 . . . . 5 (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))))
109adantl 480 . . . 4 ((𝜑𝑓 = 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))))
11 fwddifval.2 . . . . 5 (𝜑𝐹:𝐴⟶ℂ)
12 fwddifval.1 . . . . 5 (𝜑𝐴 ⊆ ℂ)
13 cnex 9874 . . . . . 6 ℂ ∈ V
14 elpm2r 7739 . . . . . 6 (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
1513, 13, 14mpanl12 713 . . . . 5 ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
1611, 12, 15syl2anc 690 . . . 4 (𝜑𝐹 ∈ (ℂ ↑pm ℂ))
17 fdm 5950 . . . . . . 7 (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
1811, 17syl 17 . . . . . 6 (𝜑 → dom 𝐹 = 𝐴)
1913a1i 11 . . . . . . 7 (𝜑 → ℂ ∈ V)
2019, 12ssexd 4728 . . . . . 6 (𝜑𝐴 ∈ V)
2118, 20eqeltrd 2687 . . . . 5 (𝜑 → dom 𝐹 ∈ V)
22 rabexg 4734 . . . . 5 (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
23 mptexg 6367 . . . . 5 ({𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) ∈ V)
2421, 22, 233syl 18 . . . 4 (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) ∈ V)
252, 10, 16, 24fvmptd 6182 . . 3 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2618eleq2d 2672 . . . . 5 (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
2718, 26rabeqbidv 3167 . . . 4 (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
2827mpteq1d 4660 . . 3 (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
2925, 28eqtrd 2643 . 2 (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹𝑥))))
30 oveq1 6534 . . . . 5 (𝑥 = 𝑋 → (𝑥 + 1) = (𝑋 + 1))
3130fveq2d 6092 . . . 4 (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
32 fveq2 6088 . . . 4 (𝑥 = 𝑋 → (𝐹𝑥) = (𝐹𝑋))
3331, 32oveq12d 6545 . . 3 (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
3433adantl 480 . 2 ((𝜑𝑥 = 𝑋) → ((𝐹‘(𝑥 + 1)) − (𝐹𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
35 fwddifval.3 . . 3 (𝜑𝑋𝐴)
36 fwddifval.4 . . 3 (𝜑 → (𝑋 + 1) ∈ 𝐴)
37 oveq1 6534 . . . . 5 (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
3837eleq1d 2671 . . . 4 (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
3938elrab 3330 . . 3 (𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
4035, 36, 39sylanbrc 694 . 2 (𝜑𝑋 ∈ {𝑦𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
41 ovex 6555 . . 3 ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)) ∈ V
4241a1i 11 . 2 (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)) ∈ V)
4329, 34, 40, 42fvmptd 6182 1 (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1474  wcel 1976  {crab 2899  Vcvv 3172  wss 3539  cmpt 4637  dom cdm 5028  wf 5786  cfv 5790  (class class class)co 6527  pm cpm 7723  cc 9791  1c1 9794   + caddc 9796  cmin 10118  cfwddif 31269
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-cnex 9849
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-pm 7725  df-fwddif 31270
This theorem is referenced by: (None)
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