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Theorem dvnres 25982

Description: Multiple derivative version of dvres3a 25964. (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 6907	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	dmeqd 5919	. . . . . . . 8 ⊢ (𝑥 = 0 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘0))
3	2	eqeq1d 2737	. . . . . . 7 ⊢ (𝑥 = 0 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹))
4		fveq2 6907	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0))
5	1	reseq1d 5999	. . . . . . . 8 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
6	4, 5	eqeq12d 2751	. . . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
10	9	dmeqd 5919	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
11	10	eqeq1d 2737	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹))
12		fveq2 6907	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛))
13	9	reseq1d 5999	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))
14	12, 13	eqeq12d 2751	. . . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
18	17	dmeqd 5919	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
19	18	eqeq1d 2737	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20		fveq2 6907	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)))
21	17	reseq1d 5999	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
22	20, 21	eqeq12d 2751	. . . . . . 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 6907	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
26	25	dmeqd 5919	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑁))
27	26	eqeq1d 2737	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹))
28		fveq2 6907	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁))
29	25	reseq1d 5999	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	28, 29	eqeq12d 2751	. . . . . . 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 25954	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
34	33	adantr 480	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → 𝑆 ⊆ ℂ)
35		pmresg 8909	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 25975	. . . . . . . 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 4019	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39		dvn0 25975	. . . . . . . . 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 5999	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹 ↾ 𝑆))
42	37, 41	eqtr4d 2778	. . . . . 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 11246	. . . . . . . . . 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 25979	. . . . . . . . . 10 ⊢ ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
48	44, 45, 46, 47	mp3an2i 1465	. . . . . . . . 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 4019	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51		dvnp1 25976	. . . . . . . . . . . . 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 1370	. . . . . . . . . . . 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 5919	. . . . . . . . . . 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 2777	. . . . . . . . . 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 25978	. . . . . . . . . . . 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 1465	. . . . . . . . . . 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 11234	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
58	57, 57	elpm2 8913	. . . . . . . . . . . . . 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 4006	. . . . . . . . . . 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 25951	. . . . . . . . . 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 4034	. . . . . . . . 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 4013	. . . . . . . 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 7439	. . . . . . 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 25976	. . . . . . . . 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 1370	. . . . . . . 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 5999	. . . . . . . . 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 2735	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
75	74	cnfldtop 24820	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ Top
76		unicntop 24822	. . . . . . . . . . . . . 14 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
77	76	ntrss2 23081	. . . . . . . . . . . . 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 24819	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
80	79	toponrestid 22943	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
81	50, 56, 61, 80, 74	dvbssntr 25950	. . . . . . . . . . . . . 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 4034	. . . . . . . . . . . . 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 4006	. . . . . . . . . . . 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 4013	. . . . . . . . . . 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 23090	. . . . . . . . . . . 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 2776	. . . . . . . . . 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 25964	. . . . . . . . . 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 2778	. . . . . . . 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 2751	. . . . . . 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 12711	. . . 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 1116	. 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 1537 ∈ wcel 2106 ⊆ wss 3963 {cpr 4633 dom cdm 5689 ↾ cres 5691 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ↑_pm cpm 8866 ℂcc 11151 ℝcr 11152 0cc0 11153 1c1 11154 + caddc 11156 ℕ₀cn0 12524 TopOpenctopn 17468 ℂ_fldccnfld 21382 Topctop 22915 intcnt 23041 D cdv 25913 D^𝑛 cdvn 25914

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-inf2 9679 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-pre-sup 11231

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-iin 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fi 9449 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-q 12989 df-rp 13033 df-xneg 13152 df-xadd 13153 df-xmul 13154 df-icc 13391 df-fz 13545 df-seq 14040 df-exp 14100 df-cj 15135 df-re 15136 df-im 15137 df-sqrt 15271 df-abs 15272 df-struct 17181 df-slot 17216 df-ndx 17228 df-base 17246 df-plusg 17311 df-mulr 17312 df-starv 17313 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-rest 17469 df-topn 17470 df-topgen 17490 df-psmet 21374 df-xmet 21375 df-met 21376 df-bl 21377 df-mopn 21378 df-fbas 21379 df-fg 21380 df-cnfld 21383 df-top 22916 df-topon 22933 df-topsp 22955 df-bases 22969 df-cld 23043 df-ntr 23044 df-cls 23045 df-nei 23122 df-lp 23160 df-perf 23161 df-cnp 23252 df-haus 23339 df-fil 23870 df-fm 23962 df-flim 23963 df-flf 23964 df-xms 24346 df-ms 24347 df-limc 25916 df-dv 25917 df-dvn 25918

This theorem is referenced by: cpnres 25988

W3C validator