Theorem fwddifval 37118

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 37115	. . . 4 ⊢ △ = (𝑓 ∈ (ℂ ↑pm ℂ) ↦ (𝑥 ∈ {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} ↦ ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥))))
2		dmeq 5893	. . . . . 6 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
3	2	eleq2d 2850	. . . . . 6 ⊢ (𝑓 = 𝐹 → ((𝑦 + 1) ∈ dom 𝑓 ↔ (𝑦 + 1) ∈ dom 𝐹))
4	2, 3	rabeqbidv 3433	. . . . 5 ⊢ (𝑓 = 𝐹 → {𝑦 ∈ dom 𝑓 ∣ (𝑦 + 1) ∈ dom 𝑓} = {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹})
5		fveq1 6910	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘(𝑥 + 1)) = (𝐹‘(𝑥 + 1)))
6		fveq1 6910	. . . . . 6 ⊢ (𝑓 = 𝐹 → (𝑓‘𝑥) = (𝐹‘𝑥))
7	5, 6	oveq12d 7464	. . . . 5 ⊢ (𝑓 = 𝐹 → ((𝑓‘(𝑥 + 1)) − (𝑓‘𝑥)) = ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)))
8	4, 7	mpteq12dv 5196	. . . 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 11311	. . . . . 6 ⊢ ℂ ∈ V
12		elpm2r 8896	. . . . . 6 ⊢ (((ℂ ∈ V ∧ ℂ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ)) → 𝐹 ∈ (ℂ ↑pm ℂ))
13	11, 11, 12	mpanl12 715	. . . . 5 ⊢ ((𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℂ) → 𝐹 ∈ (ℂ ↑pm ℂ))
14	9, 10, 13	syl2anc 596	. . . 4 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑pm ℂ))
15	9	fdmd 6745	. . . . . 6 ⊢ (𝜑 → dom 𝐹 = 𝐴)
16	11	a1i 11	. . . . . . 7 ⊢ (𝜑 → ℂ ∈ V)
17	16, 10	ssexd 5291	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ V)
18	15, 17	eqeltrd 2864	. . . . 5 ⊢ (𝜑 → dom 𝐹 ∈ V)
19		rabexg 5304	. . . . 5 ⊢ (dom 𝐹 ∈ V → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ∈ V)
20		mptexg 7253	. . . . 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 7043	. . 3 ⊢ (𝜑 → ( △ ‘𝐹) = (𝑥 ∈ {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} ↦ ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥))))
23	15	eleq2d 2850	. . . . 5 ⊢ (𝜑 → ((𝑦 + 1) ∈ dom 𝐹 ↔ (𝑦 + 1) ∈ 𝐴))
24	15, 23	rabeqbidv 3433	. . . 4 ⊢ (𝜑 → {𝑦 ∈ dom 𝐹 ∣ (𝑦 + 1) ∈ dom 𝐹} = {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
25	24	mpteq1d 5199	. . 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 7469	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘(𝑥 + 1)) = (𝐹‘(𝑋 + 1)))
28		fveq2 6911	. . . 4 ⊢ (𝑥 = 𝑋 → (𝐹‘𝑥) = (𝐹‘𝑋))
29	27, 28	oveq12d 7464	. . 3 ⊢ (𝑥 = 𝑋 → ((𝐹‘(𝑥 + 1)) − (𝐹‘𝑥)) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))
30	29	adantl 487	. 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 7453	. . . . 5 ⊢ (𝑦 = 𝑋 → (𝑦 + 1) = (𝑋 + 1))
34	33	eleq1d 2849	. . . 4 ⊢ (𝑦 = 𝑋 → ((𝑦 + 1) ∈ 𝐴 ↔ (𝑋 + 1) ∈ 𝐴))
35	34	elrab 3648	. . 3 ⊢ (𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴} ↔ (𝑋 ∈ 𝐴 ∧ (𝑋 + 1) ∈ 𝐴))
36	31, 32, 35	sylanbrc 595	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑦 ∈ 𝐴 ∣ (𝑦 + 1) ∈ 𝐴})
37		ovexd 7481	. 2 ⊢ (𝜑 → ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)) ∈ V)
38	26, 30, 36, 37	fvmptd 7027	1 ⊢ (𝜑 → (( △ ‘𝐹)‘𝑋) = ((𝐹‘(𝑋 + 1)) − (𝐹‘𝑋)))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {crab 3416  Vcvv 3454   ⊆ wss 3900   ↦ cmpt 5190  dom cdm 5656  ⟶wf 6557  ‘cfv 6561  (class class class)co 7446   ↑_pm cpm 8879  ℂcc 11228  1c1 11231   + caddc 11233   − cmin 11571   △ cfwddif 37114

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778  ax-cnex 11286

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-pm 8881  df-fwddif 37115

This theorem is used by: (None)