Theorem fwddifval 36185

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 36182	. . . 4 ⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
2		dmeq 5888	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	eleq2d 2821	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
4	2, 3	rabeqbidv 3439	. . . . 5 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5		fveq1 6880	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6		fveq1 6880	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
7	5, 6	oveq12d 7428	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)))
8	4, 7	mpteq12dv 5212	. . . 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 11215	. . . . . 6 ⊢ ℂ ∈ V
12		elpm2r 8864	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
13	11, 11, 12	mpanl12 702	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
14	9, 10, 13	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
15	9	fdmd 6721	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
17	16, 10	ssexd 5299	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
18	15, 17	eqeltrd 2835	. . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V)
19		rabexg 5312	. . . . 5 ⊢ (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20		mptexg 7218	. . . . 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 7014	. . 3 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
23	15	eleq2d 2821	. . . . 5 ⊢ (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
24	15, 23	rabeqbidv 3439	. . . 4 ⊢ (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
25	24	mpteq1d 5215	. . 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 2771	. 2 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
27		fvoveq1 7433	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28		fveq2 6881	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
29	27, 28	oveq12d 7428	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))
30	29	adantl 481	. 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 7417	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
34	33	eleq1d 2820	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
35	34	elrab 3676	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
36	31, 32, 35	sylanbrc 583	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37		ovexd 7445	. 2 ⊢ (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)) ∈ V)
38	26, 30, 36, 37	fvmptd 6998	1 ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2109  {crab 3420  Vcvv 3464   ⊆ wss 3931   ↦ cmpt 5206  dom cdm 5659  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410   ↑_pm cpm 8846  ℂcc 11132  1c1 11135   + caddc 11137   − cmin 11471   △ cfwddif 36181

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-rep 5254  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3365  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-pm 8848  df-fwddif 36182

This theorem is referenced by: (None)