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	⊢ (𝜑 → 𝐴 ⊆ ℂ)
fwddifval.2	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
fwddifval.3	⊢ (𝜑 → 𝑋 ∈ 𝐴)
fwddifval.4	⊢ (𝜑 → (𝑋 + 1) ∈ 𝐴)

Assertion

Ref	Expression
fwddifval	⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))

Proof of Theorem fwddifval

Dummy variables 𝑥 𝑦 𝑓 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fwddif 32855	. . . 4 ⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
2		dmeq 5569	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	eleq2d 2845	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
4	2, 3	rabeqbidv 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 → (𝑓‘𝑥) = (𝐹‘𝑥))
7	5, 6	oveq12d 6940	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)))
8	4, 7	mpteq12dv 4969	. . . 4 ⊢ (𝑓 = 𝐹 → (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
9		fwddifval.2	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
10		fwddifval.1	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
11		cnex 10353	. . . . . 6 ⊢ ℂ ∈ V
12		elpm2r 8158	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
13	11, 11, 12	mpanl12 692	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
14	9, 10, 13	syl2anc 579	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
15	9	fdmd 6300	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
17	16, 10	ssexd 5042	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
18	15, 17	eqeltrd 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)) − (𝐹‘𝑥))) ∈ V)
21	18, 19, 20	3syl 18	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))) ∈ V)
22	1, 8, 14, 21	fvmptd3 6564	. . 3 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
23	15	eleq2d 2845	. . . . 5 ⊢ (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
24	15, 23	rabeqbidv 3392	. . . 4 ⊢ (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
25	24	mpteq1d 4973	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))) = (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
26	22, 25	eqtrd 2814	. 2 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
27		fvoveq1 6945	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28		fveq2 6446	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
29	27, 28	oveq12d 6940	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))
30	29	adantl 475	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))
31		fwddifval.3	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
32		fwddifval.4	. . 3 ⊢ (𝜑 → (𝑋 + 1) ∈ 𝐴)
33		oveq1 6929	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
34	33	eleq1d 2844	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
35	34	elrab 3572	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
36	31, 32, 35	sylanbrc 578	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37		ovexd 6956	. 2 ⊢ (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)) ∈ V)
38	26, 30, 36, 37	fvmptd 6548	1 ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))

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)