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Theorem dvnres 24509
 Description: Multiple derivative version of dvres3a 24492. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
dvnres (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑁 ∈ ℕ0) ∧ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))

Proof of Theorem dvnres
Dummy variables 𝑛 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 6642 . . . . . . . . 9 (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0))
21dmeqd 5746 . . . . . . . 8 (𝑥 = 0 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘0))
32eqeq1d 2822 . . . . . . 7 (𝑥 = 0 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹))
4 fveq2 6642 . . . . . . . 8 (𝑥 = 0 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘0))
51reseq1d 5824 . . . . . . . 8 (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
64, 5eqeq12d 2836 . . . . . . 7 (𝑥 = 0 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))
73, 6imbi12d 347 . . . . . 6 (𝑥 = 0 → ((dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))))
87imbi2d 343 . . . . 5 (𝑥 = 0 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))))
9 fveq2 6642 . . . . . . . . 9 (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛))
109dmeqd 5746 . . . . . . . 8 (𝑥 = 𝑛 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑛))
1110eqeq1d 2822 . . . . . . 7 (𝑥 = 𝑛 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹))
12 fveq2 6642 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑛))
139reseq1d 5824 . . . . . . . 8 (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))
1412, 13eqeq12d 2836 . . . . . . 7 (𝑥 = 𝑛 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
1511, 14imbi12d 347 . . . . . 6 (𝑥 = 𝑛 → ((dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))))
1615imbi2d 343 . . . . 5 (𝑥 = 𝑛 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))))
17 fveq2 6642 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1817dmeqd 5746 . . . . . . . 8 (𝑥 = (𝑛 + 1) → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1918eqeq1d 2822 . . . . . . 7 (𝑥 = (𝑛 + 1) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20 fveq2 6642 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)))
2117reseq1d 5824 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
2220, 21eqeq12d 2836 . . . . . . 7 (𝑥 = (𝑛 + 1) → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))
2319, 22imbi12d 347 . . . . . 6 (𝑥 = (𝑛 + 1) → ((dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))))
2423imbi2d 343 . . . . 5 (𝑥 = (𝑛 + 1) → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))))
25 fveq2 6642 . . . . . . . . 9 (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁))
2625dmeqd 5746 . . . . . . . 8 (𝑥 = 𝑁 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑁))
2726eqeq1d 2822 . . . . . . 7 (𝑥 = 𝑁 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹))
28 fveq2 6642 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑁))
2925reseq1d 5824 . . . . . . . 8 (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
3028, 29eqeq12d 2836 . . . . . . 7 (𝑥 = 𝑁 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))
3127, 30imbi12d 347 . . . . . 6 (𝑥 = 𝑁 → ((dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆)) ↔ (dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))))
3231imbi2d 343 . . . . 5 (𝑥 = 𝑁 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆))) ↔ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))))
33 recnprss 24482 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3433adantr 483 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → 𝑆 ⊆ ℂ)
35 pmresg 8408 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
36 dvn0 24502 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
3734, 35, 36syl2anc 586 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
38 ssidd 3965 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39 dvn0 24502 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4038, 39sylan 582 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4140reseq1d 5824 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹𝑆))
4237, 41eqtr4d 2858 . . . . . 6 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
4342a1d 25 . . . . 5 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))
44 cnelprrecn 10604 . . . . . . . . . 10 ℂ ∈ {ℝ, ℂ}
45 simplr 767 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝐹 ∈ (ℂ ↑pm ℂ))
46 simprl 769 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ0)
47 dvnbss 24506 . . . . . . . . . 10 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
4844, 45, 46, 47mp3an2i 1462 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
49 simprr 771 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)
50 ssidd 3965 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51 dvnp1 24503 . . . . . . . . . . . . 13 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5250, 45, 46, 51syl3anc 1367 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5352dmeqd 5746 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5449, 53eqtr3d 2857 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 = dom (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
55 dvnf 24505 . . . . . . . . . . . 12 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ)
5644, 45, 46, 55mp3an2i 1462 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ)
57 cnex 10592 . . . . . . . . . . . . . . 15 ℂ ∈ V
5857, 57elpm2 8412 . . . . . . . . . . . . . 14 (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
5958simprbi 499 . . . . . . . . . . . . 13 (𝐹 ∈ (ℂ ↑pm ℂ) → dom 𝐹 ⊆ ℂ)
6045, 59syl 17 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ ℂ)
6148, 60sstrd 3952 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ)
6250, 56, 61dvbss 24479 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
6354, 62eqsstrd 3980 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
6448, 63eqssd 3959 . . . . . . . 8 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹)
6564expr 459 . . . . . . 7 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ 𝑛 ∈ ℕ0) → (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹))
6665imim1d 82 . . . . . 6 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ 𝑛 ∈ ℕ0) → ((dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)) → (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))))
67 oveq2 7137 . . . . . . 7 (((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆) → (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
6834adantr 483 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
6935adantr 483 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
70 dvnp1 24503 . . . . . . . . 9 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)))
7168, 69, 46, 70syl3anc 1367 . . . . . . . 8 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)))
7252reseq1d 5824 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆) = ((ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ↾ 𝑆))
73 simpll 765 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
74 eqid 2820 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
7574cnfldtop 23364 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ Top
76 unicntop 23366 . . . . . . . . . . . . . 14 ℂ = (TopOpen‘ℂfld)
7776ntrss2 21637 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ Top ∧ dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ) → ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
7875, 61, 77sylancr 589 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
7974cnfldtopon 23363 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
8079toponrestid 21501 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8150, 56, 61, 80, 74dvbssntr 24478 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ⊆ ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)))
8254, 81eqsstrd 3980 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)))
8348, 82sstrd 3952 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)))
8478, 83eqssd 3959 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) = dom ((ℂ D𝑛 𝐹)‘𝑛))
8576isopn3 21646 . . . . . . . . . . . 12 (((TopOpen‘ℂfld) ∈ Top ∧ dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ) → (dom ((ℂ D𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂfld) ↔ ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) = dom ((ℂ D𝑛 𝐹)‘𝑛)))
8675, 61, 85sylancr 589 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (dom ((ℂ D𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂfld) ↔ ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) = dom ((ℂ D𝑛 𝐹)‘𝑛)))
8784, 86mpbird 259 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂfld))
8864, 54eqtr2d 2856 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) = dom ((ℂ D𝑛 𝐹)‘𝑛))
8974dvres3a 24492 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ) ∧ (dom ((ℂ D𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂfld) ∧ dom (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) = dom ((ℂ D𝑛 𝐹)‘𝑛))) → (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)) = ((ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ↾ 𝑆))
9073, 56, 87, 88, 89syl22anc 836 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)) = ((ℂ D ((ℂ D𝑛 𝐹)‘𝑛)) ↾ 𝑆))
9172, 90eqtr4d 2858 . . . . . . . 8 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
9271, 91eqeq12d 2836 . . . . . . 7 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆) ↔ (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))))
9367, 92syl5ibr 248 . . . . . 6 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))
9466, 93animpimp2impd 842 . . . . 5 (𝑛 ∈ ℕ0 → (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))) → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))))
958, 16, 24, 32, 43, 94nn0ind 12052 . . . 4 (𝑁 ∈ ℕ0 → ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))))
9695com12 32 . . 3 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (𝑁 ∈ ℕ0 → (dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))))
97963impia 1113 . 2 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑁 ∈ ℕ0) → (dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))
9897imp 409 1 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑁 ∈ ℕ0) ∧ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537   ∈ wcel 2114   ⊆ wss 3909  {cpr 4541  dom cdm 5527   ↾ cres 5529  ⟶wf 6323  ‘cfv 6327  (class class class)co 7129   ↑pm cpm 8381  ℂcc 10509  ℝcr 10510  0cc0 10511  1c1 10512   + caddc 10514  ℕ0cn0 11872  TopOpenctopn 16670  ℂfldccnfld 20517  Topctop 21473  intcnt 21597   D cdv 24441   D𝑛 cdvn 24442 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5162  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302  ax-un 7435  ax-inf2 9078  ax-cnex 10567  ax-resscn 10568  ax-1cn 10569  ax-icn 10570  ax-addcl 10571  ax-addrcl 10572  ax-mulcl 10573  ax-mulrcl 10574  ax-mulcom 10575  ax-addass 10576  ax-mulass 10577  ax-distr 10578  ax-i2m1 10579  ax-1ne0 10580  ax-1rid 10581  ax-rnegex 10582  ax-rrecex 10583  ax-cnre 10584  ax-pre-lttri 10585  ax-pre-lttrn 10586  ax-pre-ltadd 10587  ax-pre-mulgt0 10588  ax-pre-sup 10589 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3472  df-sbc 3749  df-csb 3857  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4811  df-int 4849  df-iun 4893  df-iin 4894  df-br 5039  df-opab 5101  df-mpt 5119  df-tr 5145  df-id 5432  df-eprel 5437  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6286  df-fun 6329  df-fn 6330  df-f 6331  df-f1 6332  df-fo 6333  df-f1o 6334  df-fv 6335  df-riota 7087  df-ov 7132  df-oprab 7133  df-mpo 7134  df-om 7555  df-1st 7663  df-2nd 7664  df-wrecs 7921  df-recs 7982  df-rdg 8020  df-1o 8076  df-oadd 8080  df-er 8263  df-map 8382  df-pm 8383  df-en 8484  df-dom 8485  df-sdom 8486  df-fin 8487  df-fi 8849  df-sup 8880  df-inf 8881  df-pnf 10651  df-mnf 10652  df-xr 10653  df-ltxr 10654  df-le 10655  df-sub 10846  df-neg 10847  df-div 11272  df-nn 11613  df-2 11675  df-3 11676  df-4 11677  df-5 11678  df-6 11679  df-7 11680  df-8 11681  df-9 11682  df-n0 11873  df-z 11957  df-dec 12074  df-uz 12219  df-q 12324  df-rp 12365  df-xneg 12482  df-xadd 12483  df-xmul 12484  df-icc 12720  df-fz 12873  df-seq 13350  df-exp 13411  df-cj 14434  df-re 14435  df-im 14436  df-sqrt 14570  df-abs 14571  df-struct 16460  df-ndx 16461  df-slot 16462  df-base 16464  df-plusg 16553  df-mulr 16554  df-starv 16555  df-tset 16559  df-ple 16560  df-ds 16562  df-unif 16563  df-rest 16671  df-topn 16672  df-topgen 16692  df-psmet 20509  df-xmet 20510  df-met 20511  df-bl 20512  df-mopn 20513  df-fbas 20514  df-fg 20515  df-cnfld 20518  df-top 21474  df-topon 21491  df-topsp 21513  df-bases 21526  df-cld 21599  df-ntr 21600  df-cls 21601  df-nei 21678  df-lp 21716  df-perf 21717  df-cnp 21808  df-haus 21895  df-fil 22426  df-fm 22518  df-flim 22519  df-flf 22520  df-xms 22902  df-ms 22903  df-limc 24444  df-dv 24445  df-dvn 24446 This theorem is referenced by:  cpnres  24515
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