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Theorem dvnres 25995

Description: Multiple derivative version of dvres3a 25978. (Contributed by Mario Carneiro, 11-Feb-2015.)

**Assertion**
Ref	Expression
dvnres	⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑁 ∈ ℕ₀) ∧ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))

Proof of Theorem dvnres Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6869	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	dmeqd 5883	. . . . . . . 8 ⊢ (𝑥 = 0 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘0))
3	2	eqeq1d 2766	. . . . . . 7 ⊢ (𝑥 = 0 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹))
4		fveq2 6869	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0))
5	1	reseq1d 5966	. . . . . . . 8 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
6	4, 5	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = 0 → (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆)))
7	3, 6	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 0 → ((dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))))
8	7	imbi2d 342	. . . . 5 ⊢ (𝑥 = 0 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆)))))
9		fveq2 6869	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
10	9	dmeqd 5883	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
11	10	eqeq1d 2766	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹))
12		fveq2 6869	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛))
13	9	reseq1d 5966	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))
14	12, 13	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)))
15	11, 14	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑛 → ((dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))))
16	15	imbi2d 342	. . . . 5 ⊢ (𝑥 = 𝑛 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)))))
17		fveq2 6869	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
18	17	dmeqd 5883	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
19	18	eqeq1d 2766	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20		fveq2 6869	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)))
21	17	reseq1d 5966	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
22	20, 21	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))
23	19, 22	imbi12d 346	. . . . . 6 ⊢ (𝑥 = (𝑛 + 1) → ((dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))))
24	23	imbi2d 342	. . . . 5 ⊢ (𝑥 = (𝑛 + 1) → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))))
25		fveq2 6869	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
26	25	dmeqd 5883	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑁))
27	26	eqeq1d 2766	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹))
28		fveq2 6869	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁))
29	25	reseq1d 5966	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	28, 29	eqeq12d 2780	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆)))
31	27, 30	imbi12d 346	. . . . . 6 ⊢ (𝑥 = 𝑁 → ((dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))))
32	31	imbi2d 342	. . . . 5 ⊢ (𝑥 = 𝑁 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆)))))
33		recnprss 25968	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
34	33	adantr 484	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → 𝑆 ⊆ ℂ)
35		pmresg 8854	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25988	. . . . . . . 8 ⊢ ((𝑆 ⊆ ℂ ∧ (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
37	34, 35, 36	syl2anc 593	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
38		ssidd 3961	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39		dvn0 25988	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
40	38, 39	sylan 589	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
41	40	reseq1d 5966	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹 ↾ 𝑆))
42	37, 41	eqtr4d 2802	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
43	42	a1d 25	. . . . 5 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆)))
44		cnelprrecn 11168	. . . . . . . . . 10 ⊢ ℂ ∈ {ℝ, ℂ}
45		simplr 778	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝐹 ∈ (ℂ ↑_pm ℂ))
46		simprl 780	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ₀)
47		dvnbss 25992	. . . . . . . . . 10 ⊢ ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
48	44, 45, 46, 47	mp3an2i 1489	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
49		simprr 782	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)
50		ssidd 3961	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51		dvnp1 25989	. . . . . . . . . . . . 13 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
52	50, 45, 46, 51	syl3anc 1392	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
53	52	dmeqd 5883	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
54	49, 53	eqtr3d 2801	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 = dom (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)))
55		dvnf 25991	. . . . . . . . . . . 12 ⊢ ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → ((ℂ D^𝑛 𝐹)‘𝑛):dom ((ℂ D^𝑛 𝐹)‘𝑛)⟶ℂ)
56	44, 45, 46, 55	mp3an2i 1489	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((ℂ D^𝑛 𝐹)‘𝑛):dom ((ℂ D^𝑛 𝐹)‘𝑛)⟶ℂ)
57		cnex 11156	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
58	57, 57	elpm2 8858	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (ℂ ↑_pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
59	58	simprbi 501	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (ℂ ↑_pm ℂ) → dom 𝐹 ⊆ ℂ)
60	45, 59	syl 17	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ ℂ)
61	48, 60	sstrd 3948	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ ℂ)
62	50, 56, 61	dvbss 25965	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D^𝑛 𝐹)‘𝑛))
63	54, 62	eqsstrd 3972	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ dom ((ℂ D^𝑛 𝐹)‘𝑛))
64	48, 63	eqssd 3955	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹)
65	64	expr 460	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ 𝑛 ∈ ℕ₀) → (dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹))
66	65	imim1d 82	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ 𝑛 ∈ ℕ₀) → ((dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)) → (dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))))
67		oveq2 7406	. . . . . . 7 ⊢ (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆) → (𝑆 D ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛)) = (𝑆 D (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)))
68	34	adantr 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
69	35	adantr 484	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
70		dvnp1 25989	. . . . . . . . 9 ⊢ ((𝑆 ⊆ ℂ ∧ (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆) ∧ 𝑛 ∈ ℕ₀) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛)))
71	68, 69, 46, 70	syl3anc 1392	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛)))
72	52	reseq1d 5966	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆) = ((ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ↾ 𝑆))
73		simpll 776	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
74		eqid 2764	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
75	74	cnfldtop 24845	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ Top
76		unicntop 24847	. . . . . . . . . . . . . 14 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
77	76	ntrss2 23119	. . . . . . . . . . . . 13 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ ℂ) → ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D^𝑛 𝐹)‘𝑛))
78	75, 61, 77	sylancr 596	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D^𝑛 𝐹)‘𝑛))
79	74	cnfldtopon 24844	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
80	79	toponrestid 22983	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
81	50, 56, 61, 80, 74	dvbssntr 25964	. . . . . . . . . . . . . 14 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ⊆ ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)))
82	54, 81	eqsstrd 3972	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)))
83	48, 82	sstrd 3948	. . . . . . . . . . . 12 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)))
84	78, 83	eqssd 3955	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
85	76	isopn3 23128	. . . . . . . . . . . 12 ⊢ (((TopOpen‘ℂ_fld) ∈ Top ∧ dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ ℂ) → (dom ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂ_fld) ↔ ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)) = dom ((ℂ D^𝑛 𝐹)‘𝑛)))
86	75, 61, 85	sylancr 596	. . . . . . . . . . 11 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (dom ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂ_fld) ↔ ((int‘(TopOpen‘ℂ_fld))‘dom ((ℂ D^𝑛 𝐹)‘𝑛)) = dom ((ℂ D^𝑛 𝐹)‘𝑛)))
87	84, 86	mpbird 259	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂ_fld))
88	64, 54	eqtr2d 2800	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
89	74	dvres3a 25978	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ ((ℂ D^𝑛 𝐹)‘𝑛):dom ((ℂ D^𝑛 𝐹)‘𝑛)⟶ℂ) ∧ (dom ((ℂ D^𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂ_fld) ∧ dom (ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) = dom ((ℂ D^𝑛 𝐹)‘𝑛))) → (𝑆 D (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)) = ((ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ↾ 𝑆))
90	73, 56, 87, 88, 89	syl22anc 849	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝑆 D (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)) = ((ℂ D ((ℂ D^𝑛 𝐹)‘𝑛)) ↾ 𝑆))
91	72, 90	eqtr4d 2802	. . . . . . . 8 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆) = (𝑆 D (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆)))
92	71, 91	eqeq12d 2780	. . . . . . 7 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆) ↔ (𝑆 D ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛)) = (𝑆 D (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))))
93	67, 92	imbitrrid 248	. . . . . 6 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))
94	66, 93	animpimp2impd 857	. . . . 5 ⊢ (𝑛 ∈ ℕ₀ → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))) → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))))
95	8, 16, 24, 32, 43, 94	nn0ind 12670	. . . 4 ⊢ (𝑁 ∈ ℕ₀ → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))))
96	95	com12 32	. . 3 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝑁 ∈ ℕ₀ → (dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))))
97	96	3impia 1131	. 2 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑁 ∈ ℕ₀) → (dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆)))
98	97	imp 410	1 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑁 ∈ ℕ₀) ∧ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1099 = wceq 1562 ∈ wcel 2144 ⊆ wss 3906 {cpr 4586 dom cdm 5649 ↾ cres 5651 ⟶wf 6519 ‘cfv 6523 (class class class)co 7398 ↑_pm cpm 8811 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ℕ₀cn0 12483 TopOpenctopn 17452 ℂ_fldccnfld 21426 Topctop 22955 intcnt 23079 D cdv 25927 D^𝑛 cdvn 25928

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1817 ax-4 1831 ax-5 1932 ax-6 1989 ax-7 2030 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2214 ax-ext 2736 ax-rep 5229 ax-sep 5248 ax-nul 5258 ax-pow 5324 ax-pr 5392 ax-un 7720 ax-inf2 9598 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1565 df-fal 1575 df-ex 1802 df-nf 1806 df-sb 2093 df-mo 2568 df-eu 2598 df-clab 2743 df-cleq 2756 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3458 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5544 df-eprel 5549 df-po 5557 df-so 5558 df-fr 5602 df-we 5604 df-xp 5655 df-rel 5656 df-cnv 5657 df-co 5658 df-dm 5659 df-rn 5660 df-res 5661 df-ima 5662 df-pred 6290 df-ord 6351 df-on 6352 df-lim 6353 df-suc 6354 df-iota 6479 df-fun 6525 df-fn 6526 df-f 6527 df-f1 6528 df-fo 6529 df-f1o 6530 df-fv 6531 df-riota 7355 df-ov 7401 df-oprab 7402 df-mpo 7403 df-om 7849 df-1st 7972 df-2nd 7973 df-frecs 8264 df-wrecs 8295 df-recs 8344 df-rdg 8383 df-1o 8439 df-er 8680 df-map 8812 df-pm 8813 df-en 8930 df-dom 8931 df-sdom 8932 df-fin 8933 df-fi 9359 df-sup 9390 df-inf 9391 df-pnf 11220 df-mnf 11221 df-xr 11222 df-ltxr 11223 df-le 11224 df-sub 11418 df-neg 11419 df-div 11847 df-nn 12213 df-2 12282 df-3 12283 df-4 12284 df-5 12285 df-6 12286 df-7 12287 df-8 12288 df-9 12289 df-n0 12484 df-z 12571 df-dec 12691 df-uz 12842 df-q 12952 df-rp 12996 df-xneg 13116 df-xadd 13117 df-xmul 13118 df-icc 13358 df-fz 13515 df-seq 14017 df-exp 14077 df-cj 15128 df-re 15129 df-im 15130 df-sqrt 15264 df-abs 15265 df-struct 17185 df-slot 17220 df-ndx 17232 df-base 17248 df-plusg 17301 df-mulr 17302 df-starv 17303 df-tset 17307 df-ple 17308 df-ds 17310 df-unif 17311 df-rest 17453 df-topn 17454 df-topgen 17474 df-psmet 21418 df-xmet 21419 df-met 21420 df-bl 21421 df-mopn 21422 df-fbas 21423 df-fg 21424 df-cnfld 21427 df-top 22956 df-topon 22973 df-topsp 22995 df-bases 23008 df-cld 23081 df-ntr 23082 df-cls 23083 df-nei 23160 df-lp 23198 df-perf 23199 df-cnp 23290 df-haus 23377 df-fil 23908 df-fm 24000 df-flim 24001 df-flf 24002 df-xms 24382 df-ms 24383 df-limc 25930 df-dv 25931 df-dvn 25932

This theorem is referenced by: cpnres 26001

W3C validator