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Theorem dvnres 26158

Description: Multiple derivative version of dvres3a 26141. (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 6882	. . . . . . . . 9 ⊢ (𝑥 = 0 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘0))
2	1	dmeqd 5893	. . . . . . . 8 ⊢ (𝑥 = 0 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘0))
3	2	eqeq1d 2764	. . . . . . 7 ⊢ (𝑥 = 0 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘0) = dom 𝐹))
4		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 0 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0))
5	1	reseq1d 5975	. . . . . . . 8 ⊢ (𝑥 = 0 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
6	4, 5	eqeq12d 2778	. . . . . . 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 347	. . . . . 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 343	. . . . 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 6882	. . . . . . . . 9 ⊢ (𝑥 = 𝑛 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑛))
10	9	dmeqd 5893	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑛))
11	10	eqeq1d 2764	. . . . . . 7 ⊢ (𝑥 = 𝑛 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑛) = dom 𝐹))
12		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑛))
13	9	reseq1d 5975	. . . . . . . 8 ⊢ (𝑥 = 𝑛 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑛) ↾ 𝑆))
14	12, 13	eqeq12d 2778	. . . . . . 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 347	. . . . . 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 343	. . . . 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 6882	. . . . . . . . 9 ⊢ (𝑥 = (𝑛 + 1) → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
18	17	dmeqd 5893	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)))
19	18	eqeq1d 2764	. . . . . . 7 ⊢ (𝑥 = (𝑛 + 1) → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘(𝑛 + 1)))
21	17	reseq1d 5975	. . . . . . . 8 ⊢ (𝑥 = (𝑛 + 1) → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
22	20, 21	eqeq12d 2778	. . . . . . 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 347	. . . . . 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 343	. . . . 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 6882	. . . . . . . . 9 ⊢ (𝑥 = 𝑁 → ((ℂ D^𝑛 𝐹)‘𝑥) = ((ℂ D^𝑛 𝐹)‘𝑁))
26	25	dmeqd 5893	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom ((ℂ D^𝑛 𝐹)‘𝑁))
27	26	eqeq1d 2764	. . . . . . 7 ⊢ (𝑥 = 𝑁 → (dom ((ℂ D^𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D^𝑛 𝐹)‘𝑁) = dom 𝐹))
28		fveq2 6882	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑥) = ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘𝑁))
29	25	reseq1d 5975	. . . . . . . 8 ⊢ (𝑥 = 𝑁 → (((ℂ D^𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D^𝑛 𝐹)‘𝑁) ↾ 𝑆))
30	28, 29	eqeq12d 2778	. . . . . . 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 347	. . . . . 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 343	. . . . 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 26131	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
34	33	adantr 486	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → 𝑆 ⊆ ℂ)
35		pmresg 8880	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆))
36		dvn0 26151	. . . . . . . 8 ⊢ ((𝑆 ⊆ ℂ ∧ (𝐹 ↾ 𝑆) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
37	34, 35, 36	syl2anc 596	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (𝐹 ↾ 𝑆))
38		ssidd 3957	. . . . . . . . 9 ⊢ (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39		dvn0 26151	. . . . . . . . 9 ⊢ ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
40	38, 39	sylan 592	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝐹)‘0) = 𝐹)
41	40	reseq1d 5975	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹 ↾ 𝑆))
42	37, 41	eqtr4d 2800	. . . . . 6 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) → ((𝑆 D^𝑛 (𝐹 ↾ 𝑆))‘0) = (((ℂ D^𝑛 𝐹)‘0) ↾ 𝑆))
43	42	a1d 26	. . . . 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 11218	. . . . . . . . . 10 ⊢ ℂ ∈ {ℝ, ℂ}
45		simplr 781	. . . . . . . . . 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 783	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ₀)
47		dvnbss 26155	. . . . . . . . . 10 ⊢ ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ) ∧ 𝑛 ∈ ℕ₀) → dom ((ℂ D^𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
48	44, 45, 46, 47	mp3an2i 1495	. . . . . . . . 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 785	. . . . . . . . . . 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 3957	. . . . . . . . . . . . 13 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51		dvnp1 26152	. . . . . . . . . . . . 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 1398	. . . . . . . . . . . 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 5893	. . . . . . . . . . 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 2799	. . . . . . . . . 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 26154	. . . . . . . . . . . 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 1495	. . . . . . . . . . 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 11206	. . . . . . . . . . . . . . 15 ⊢ ℂ ∈ V
58	57, 57	elpm2 8884	. . . . . . . . . . . . . 14 ⊢ (𝐹 ∈ (ℂ ↑_pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
59	58	simprbi 503	. . . . . . . . . . . . 13 ⊢ (𝐹 ∈ (ℂ ↑_pm ℂ) → dom 𝐹 ⊆ ℂ)
60	45, 59	syl 18	. . . . . . . . . . . 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 3944	. . . . . . . . . . 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 26128	. . . . . . . . . 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 3968	. . . . . . . . 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 3951	. . . . . . . 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 462	. . . . . . 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 83	. . . . . 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 7424	. . . . . . 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 486	. . . . . . . . 9 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
69	35	adantr 486	. . . . . . . . 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 26152	. . . . . . . . 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 1398	. . . . . . . 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 5975	. . . . . . . . 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 779	. . . . . . . . . 10 ⊢ (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℂ)) ∧ (𝑛 ∈ ℕ₀ ∧ dom ((ℂ D^𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
74		eqid 2762	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
75	74	cnfldtop 25008	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) ∈ Top
76		unicntop 25010	. . . . . . . . . . . . . 14 ⊢ ℂ = ∪ (TopOpen‘ℂ_fld)
77	76	ntrss2 23281	. . . . . . . . . . . . 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 599	. . . . . . . . . . . 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 25007	. . . . . . . . . . . . . . . 16 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
80	79	toponrestid 23145	. . . . . . . . . . . . . . 15 ⊢ (TopOpen‘ℂ_fld) = ((TopOpen‘ℂ_fld) ↾_t ℂ)
81	50, 56, 61, 80, 74	dvbssntr 26127	. . . . . . . . . . . . . 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 3968	. . . . . . . . . . . . 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 3944	. . . . . . . . . . . 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 3951	. . . . . . . . . . 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 23290	. . . . . . . . . . . 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 599	. . . . . . . . . . 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 260	. . . . . . . . . 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 2798	. . . . . . . . . 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 26141	. . . . . . . . . 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 852	. . . . . . . . 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 2800	. . . . . . . 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 2778	. . . . . . 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 249	. . . . . 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 860	. . . . 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 12717	. . . 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 33	. . 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 1135	. 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 412	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ⊆ wss 3902 {cpr 4589 dom cdm 5659 ↾ cres 5661 ⟶wf 6533 ‘cfv 6537 (class class class)co 7416 ↑_pm cpm 8830 ℂcc 11123 ℝcr 11124 0cc0 11125 1c1 11126 + caddc 11128 ℕ₀cn0 12529 TopOpenctopn 17508 ℂ_fldccnfld 21584 Topctop 23117 intcnt 23241 D cdv 26090 D^𝑛 cdvn 26091

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-inf2 9623 ax-cnex 11181 ax-resscn 11182 ax-1cn 11183 ax-icn 11184 ax-addcl 11185 ax-addrcl 11186 ax-mulcl 11187 ax-mulrcl 11188 ax-mulcom 11189 ax-addass 11190 ax-mulass 11191 ax-distr 11192 ax-i2m1 11193 ax-1ne0 11194 ax-1rid 11195 ax-rnegex 11196 ax-rrecex 11197 ax-cnre 11198 ax-pre-lttri 11199 ax-pre-lttrn 11200 ax-pre-ltadd 11201 ax-pre-mulgt0 11202 ax-pre-sup 11203

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-1o 8458 df-er 8699 df-map 8831 df-pm 8832 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-fi 9384 df-sup 9415 df-inf 9416 df-pnf 11270 df-mnf 11271 df-xr 11272 df-ltxr 11273 df-le 11274 df-sub 11468 df-neg 11469 df-div 11897 df-nn 12259 df-2 12328 df-3 12329 df-4 12330 df-5 12331 df-6 12332 df-7 12333 df-8 12334 df-9 12335 df-n0 12530 df-z 12617 df-dec 12738 df-uz 12889 df-q 12999 df-rp 13043 df-xneg 13163 df-xadd 13164 df-xmul 13165 df-icc 13405 df-fz 13562 df-seq 14066 df-exp 14126 df-cj 15186 df-re 15187 df-im 15188 df-sqrt 15322 df-abs 15323 df-struct 17241 df-slot 17276 df-ndx 17288 df-base 17304 df-plusg 17357 df-mulr 17358 df-starv 17359 df-tset 17363 df-ple 17364 df-ds 17366 df-unif 17367 df-rest 17509 df-topn 17510 df-topgen 17530 df-psmet 21576 df-xmet 21577 df-met 21578 df-bl 21579 df-mopn 21580 df-fbas 21581 df-fg 21582 df-cnfld 21585 df-top 23118 df-topon 23135 df-topsp 23157 df-bases 23170 df-cld 23243 df-ntr 23244 df-cls 23245 df-nei 23322 df-lp 23360 df-perf 23361 df-cnp 23452 df-haus 23539 df-fil 24071 df-fm 24163 df-flim 24164 df-flf 24165 df-xms 24545 df-ms 24546 df-limc 26093 df-dv 26094 df-dvn 26095

This theorem is used by: cpnres 26164

W3C validator