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Theorem dvnres 25840

Description: Multiple derivative version of dvres3a 25822. (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 6861	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	dmeqd 5872	. . . . . . . 8 ⊢ (𝑥 = 0 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘0))
3	2	eqeq1d 2732	. . . . . . 7 ⊢ (𝑥 = 0 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹))
4		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0))
5	1	reseq1d 5952	. . . . . . . 8 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
6	4, 5	eqeq12d 2746	. . . . . . 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 344	. . . . . 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 340	. . . . 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 6861	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
10	9	dmeqd 5872	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
11	10	eqeq1d 2732	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹))
12		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛))
13	9	reseq1d 5952	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))
14	12, 13	eqeq12d 2746	. . . . . . 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 344	. . . . . 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 340	. . . . 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 6861	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
18	17	dmeqd 5872	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
19	18	eqeq1d 2732	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)))
21	17	reseq1d 5952	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
22	20, 21	eqeq12d 2746	. . . . . . 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 344	. . . . . 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 340	. . . . 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 6861	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
26	25	dmeqd 5872	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑁))
27	26	eqeq1d 2732	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹))
28		fveq2 6861	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁))
29	25	reseq1d 5952	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	28, 29	eqeq12d 2746	. . . . . . 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 344	. . . . . 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 340	. . . . 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 25812	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → 𝑆 ⊆ ℂ)
35		pmresg 8846	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25833	. . . . . . . 8 ⊢ ((𝑆 ⊆ ℂ ∧ (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
37	34, 35, 36	syl2anc 584	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
38		ssidd 3973	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39		dvn0 25833	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
40	38, 39	sylan 580	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
41	40	reseq1d 5952	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹 ↾ 𝑆))
42	37, 41	eqtr4d 2768	. . . . . 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 768	. . . . . . . . . 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 770	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ₀)
47		dvnbss 25837	. . . . . . . . . 10 ⊢ ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
48	44, 45, 46, 47	mp3an2i 1468	. . . . . . . . 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 772	. . . . . . . . . . 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 3973	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51		dvnp1 25834	. . . . . . . . . . . . 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 1373	. . . . . . . . . . . 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 5872	. . . . . . . . . . 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 2767	. . . . . . . . . 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 25836	. . . . . . . . . . . 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 1468	. . . . . . . . . . 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 8850	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (ℂ ↑_pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
59	58	simprbi 496	. . . . . . . . . . . . 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 3960	. . . . . . . . . . 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 25809	. . . . . . . . . 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 3984	. . . . . . . . 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 3967	. . . . . . . 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 456	. . . . . . 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 7398	. . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
69	35	adantr 480	. . . . . . . . 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 25834	. . . . . . . . 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 1373	. . . . . . . 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 5952	. . . . . . . . 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 766	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
74		eqid 2730	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
75	74	cnfldtop 24678	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ Top
76		unicntop 24680	. . . . . . . . . . . . . 14 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
77	76	ntrss2 22951	. . . . . . . . . . . . 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 587	. . . . . . . . . . . 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 24677	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
80	79	toponrestid 22815	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
81	50, 56, 61, 80, 74	dvbssntr 25808	. . . . . . . . . . . . . 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 3984	. . . . . . . . . . . . 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 3960	. . . . . . . . . . . 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 3967	. . . . . . . . . . 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 22960	. . . . . . . . . . . 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 587	. . . . . . . . . . 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 257	. . . . . . . . . 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 2766	. . . . . . . . . 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 25822	. . . . . . . . . 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 838	. . . . . . . . 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 2768	. . . . . . . 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 2746	. . . . . . 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 246	. . . . . 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 846	. . . . 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 12636	. . . 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 1117	. 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 406	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ⊆ wss 3917 {cpr 4594 dom cdm 5641 ↾ cres 5643 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 ↑_pm cpm 8803 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 ℕ₀cn0 12449 TopOpenctopn 17391 ℂ_fldccnfld 21271 Topctop 22787 intcnt 22911 D cdv 25771 D^𝑛 cdvn 25772

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 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 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 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-er 8674 df-map 8804 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fi 9369 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-icc 13320 df-fz 13476 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-struct 17124 df-slot 17159 df-ndx 17171 df-base 17187 df-plusg 17240 df-mulr 17241 df-starv 17242 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-rest 17392 df-topn 17393 df-topgen 17413 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cnp 23122 df-haus 23209 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-xms 24215 df-ms 24216 df-limc 25774 df-dv 25775 df-dvn 25776

This theorem is referenced by: cpnres 25846

W3C validator