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Theorem dvnres 25893

Description: Multiple derivative version of dvres3a 25875. (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 6835	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	dmeqd 5855	. . . . . . . 8 ⊢ (𝑥 = 0 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘0))
3	2	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = 0 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹))
4		fveq2 6835	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0))
5	1	reseq1d 5938	. . . . . . . 8 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
6	4, 5	eqeq12d 2753	. . . . . . 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 6835	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
10	9	dmeqd 5855	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
11	10	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹))
12		fveq2 6835	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛))
13	9	reseq1d 5938	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))
14	12, 13	eqeq12d 2753	. . . . . . 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 6835	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
18	17	dmeqd 5855	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
19	18	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20		fveq2 6835	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)))
21	17	reseq1d 5938	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
22	20, 21	eqeq12d 2753	. . . . . . 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 6835	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
26	25	dmeqd 5855	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑁))
27	26	eqeq1d 2739	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹))
28		fveq2 6835	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁))
29	25	reseq1d 5938	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	28, 29	eqeq12d 2753	. . . . . . 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 25865	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → 𝑆 ⊆ ℂ)
35		pmresg 8812	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25886	. . . . . . . 8 ⊢ ((𝑆 ⊆ ℂ ∧ (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
37	34, 35, 36	syl2anc 585	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
38		ssidd 3958	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39		dvn0 25886	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
40	38, 39	sylan 581	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
41	40	reseq1d 5938	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹 ↾ 𝑆))
42	37, 41	eqtr4d 2775	. . . . . 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 11123	. . . . . . . . . 10 ⊢ ℂ ∈ {ℝ, ℂ}
45		simplr 769	. . . . . . . . . 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 771	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ₀)
47		dvnbss 25890	. . . . . . . . . 10 ⊢ ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
48	44, 45, 46, 47	mp3an2i 1469	. . . . . . . . 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 773	. . . . . . . . . . 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 3958	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51		dvnp1 25887	. . . . . . . . . . . . 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 1374	. . . . . . . . . . . 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 5855	. . . . . . . . . . 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 2774	. . . . . . . . . 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 25889	. . . . . . . . . . . 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 1469	. . . . . . . . . . 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 11111	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
58	57, 57	elpm2 8816	. . . . . . . . . . . . . 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 3945	. . . . . . . . . . 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 25862	. . . . . . . . . 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 3969	. . . . . . . . 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 3952	. . . . . . . 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 7368	. . . . . . 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 25887	. . . . . . . . 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 1374	. . . . . . . 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 5938	. . . . . . . . 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 767	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
74		eqid 2737	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
75	74	cnfldtop 24731	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ Top
76		unicntop 24733	. . . . . . . . . . . . . 14 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
77	76	ntrss2 23005	. . . . . . . . . . . . 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 588	. . . . . . . . . . . 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 24730	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
80	79	toponrestid 22869	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
81	50, 56, 61, 80, 74	dvbssntr 25861	. . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . 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 3945	. . . . . . . . . . . 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 3952	. . . . . . . . . . 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 23014	. . . . . . . . . . . 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 588	. . . . . . . . . . 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 2773	. . . . . . . . . 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 25875	. . . . . . . . . 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 839	. . . . . . . . 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 2775	. . . . . . . 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 2753	. . . . . . 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 847	. . . . 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 12591	. . . 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 1118	. 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 1087 = wceq 1542 ∈ wcel 2114 ⊆ wss 3902 {cpr 4583 dom cdm 5625 ↾ cres 5627 ⟶wf 6489 ‘cfv 6493 (class class class)co 7360 ↑_pm cpm 8768 ℂcc 11028 ℝcr 11029 0cc0 11030 1c1 11031 + caddc 11033 ℕ₀cn0 12405 TopOpenctopn 17345 ℂ_fldccnfld 21313 Topctop 22841 intcnt 22965 D cdv 25824 D^𝑛 cdvn 25825

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5225 ax-sep 5242 ax-nul 5252 ax-pow 5311 ax-pr 5378 ax-un 7682 ax-inf2 9554 ax-cnex 11086 ax-resscn 11087 ax-1cn 11088 ax-icn 11089 ax-addcl 11090 ax-addrcl 11091 ax-mulcl 11092 ax-mulrcl 11093 ax-mulcom 11094 ax-addass 11095 ax-mulass 11096 ax-distr 11097 ax-i2m1 11098 ax-1ne0 11099 ax-1rid 11100 ax-rnegex 11101 ax-rrecex 11102 ax-cnre 11103 ax-pre-lttri 11104 ax-pre-lttrn 11105 ax-pre-ltadd 11106 ax-pre-mulgt0 11107 ax-pre-sup 11108

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3062 df-rmo 3351 df-reu 3352 df-rab 3401 df-v 3443 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4287 df-if 4481 df-pw 4557 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4904 df-iun 4949 df-iin 4950 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-er 8637 df-map 8769 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-fi 9318 df-sup 9349 df-inf 9350 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12150 df-2 12212 df-3 12213 df-4 12214 df-5 12215 df-6 12216 df-7 12217 df-8 12218 df-9 12219 df-n0 12406 df-z 12493 df-dec 12612 df-uz 12756 df-q 12866 df-rp 12910 df-xneg 13030 df-xadd 13031 df-xmul 13032 df-icc 13272 df-fz 13428 df-seq 13929 df-exp 13989 df-cj 15026 df-re 15027 df-im 15028 df-sqrt 15162 df-abs 15163 df-struct 17078 df-slot 17113 df-ndx 17125 df-base 17141 df-plusg 17194 df-mulr 17195 df-starv 17196 df-tset 17200 df-ple 17201 df-ds 17203 df-unif 17204 df-rest 17346 df-topn 17347 df-topgen 17367 df-psmet 21305 df-xmet 21306 df-met 21307 df-bl 21308 df-mopn 21309 df-fbas 21310 df-fg 21311 df-cnfld 21314 df-top 22842 df-topon 22859 df-topsp 22881 df-bases 22894 df-cld 22967 df-ntr 22968 df-cls 22969 df-nei 23046 df-lp 23084 df-perf 23085 df-cnp 23176 df-haus 23263 df-fil 23794 df-fm 23886 df-flim 23887 df-flf 23888 df-xms 24268 df-ms 24269 df-limc 25827 df-dv 25828 df-dvn 25829

This theorem is referenced by: cpnres 25899

W3C validator