Theorem fwddifval 36632

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 36629	. . . 4 ⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
2		dmeq 5895	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	eleq2d 2849	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
4	2, 3	rabeqbidv 3434	. . . . 5 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5		fveq1 6882	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6		fveq1 6882	. . . . . 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 5199	. . . 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 11182	. . . . . 6 ⊢ ℂ ∈ V
12		elpm2r 8843	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
13	11, 11, 12	mpanl12 714	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
14	9, 10, 13	syl2anc 595	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
15	9	fdmd 6718	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
17	16, 10	ssexd 5296	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
18	15, 17	eqeltrd 2863	. . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V)
19		rabexg 5309	. . . . 5 ⊢ (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20		mptexg 7221	. . . . 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 19	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))) ∈ V)
22	1, 8, 14, 21	fvmptd3 7015	. . 3 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
23	15	eleq2d 2849	. . . . 5 ⊢ (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
24	15, 23	rabeqbidv 3434	. . . 4 ⊢ (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
25	24	mpteq1d 5202	. . 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 2798	. 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 6883	. . . 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 486	. 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 2848	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
35	34	elrab 3651	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
36	31, 32, 35	sylanbrc 594	. 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 6999	1 ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1570   ∈ wcel 2143  {crab 3416  Vcvv 3455   ⊆ wss 3906   ↦ cmpt 5193  dom cdm 5663  ⟶wf 6534  ‘cfv 6538  (class class class)co 7412   ↑_pm cpm 8826  ℂcc 11099  1c1 11102   + caddc 11104   − cmin 11442   △ cfwddif 36628

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-cnex 11157

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-oprab 7416  df-mpo 7417  df-pm 8828  df-fwddif 36629

This theorem is referenced by: (None)