Theorem fwddifval 35435

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 35432	. . . 4 ⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
2		dmeq 5904	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	eleq2d 2818	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
4	2, 3	rabeqbidv 3448	. . . . 5 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5		fveq1 6891	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6		fveq1 6891	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
7	5, 6	oveq12d 7430	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)))
8	4, 7	mpteq12dv 5240	. . . 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 11194	. . . . . 6 ⊢ ℂ ∈ V
12		elpm2r 8842	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
13	11, 11, 12	mpanl12 699	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
14	9, 10, 13	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
15	9	fdmd 6729	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
17	16, 10	ssexd 5325	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
18	15, 17	eqeltrd 2832	. . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V)
19		rabexg 5332	. . . . 5 ⊢ (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20		mptexg 7226	. . . . 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 7022	. . 3 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
23	15	eleq2d 2818	. . . . 5 ⊢ (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
24	15, 23	rabeqbidv 3448	. . . 4 ⊢ (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
25	24	mpteq1d 5244	. . 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 7435	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28		fveq2 6892	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
29	27, 28	oveq12d 7430	. . 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 7419	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
34	33	eleq1d 2817	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
35	34	elrab 3684	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
36	31, 32, 35	sylanbrc 582	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37		ovexd 7447	. 2 ⊢ (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)) ∈ V)
38	26, 30, 36, 37	fvmptd 7006	1 ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  {crab 3431  Vcvv 3473   ⊆ wss 3949   ↦ cmpt 5232  dom cdm 5677  ⟶wf 6540  ‘cfv 6544  (class class class)co 7412   ↑_pm cpm 8824  ℂcc 11111  1c1 11114   + caddc 11116   − cmin 11449   △ cfwddif 35431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7728  ax-cnex 11169

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-pm 8826  df-fwddif 35432

This theorem is referenced by: (None)