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Theorem dvnres 24455
Description: Multiple derivative version of dvres3a 24439. (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 6663 . . . . . . . . 9 (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0))
21dmeqd 5767 . . . . . . . 8 (𝑥 = 0 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘0))
32eqeq1d 2820 . . . . . . 7 (𝑥 = 0 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹))
4 fveq2 6663 . . . . . . . 8 (𝑥 = 0 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘0))
51reseq1d 5845 . . . . . . . 8 (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
64, 5eqeq12d 2834 . . . . . . 7 (𝑥 = 0 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))
73, 6imbi12d 346 . . . . . 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 342 . . . . 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 6663 . . . . . . . . 9 (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛))
109dmeqd 5767 . . . . . . . 8 (𝑥 = 𝑛 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑛))
1110eqeq1d 2820 . . . . . . 7 (𝑥 = 𝑛 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹))
12 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑛))
139reseq1d 5845 . . . . . . . 8 (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))
1412, 13eqeq12d 2834 . . . . . . 7 (𝑥 = 𝑛 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
1511, 14imbi12d 346 . . . . . 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 342 . . . . 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 6663 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1817dmeqd 5767 . . . . . . . 8 (𝑥 = (𝑛 + 1) → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1918eqeq1d 2820 . . . . . . 7 (𝑥 = (𝑛 + 1) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20 fveq2 6663 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)))
2117reseq1d 5845 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
2220, 21eqeq12d 2834 . . . . . . 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 346 . . . . . 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 342 . . . . 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 6663 . . . . . . . . 9 (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁))
2625dmeqd 5767 . . . . . . . 8 (𝑥 = 𝑁 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑁))
2726eqeq1d 2820 . . . . . . 7 (𝑥 = 𝑁 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹))
28 fveq2 6663 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑁))
2925reseq1d 5845 . . . . . . . 8 (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
3028, 29eqeq12d 2834 . . . . . . 7 (𝑥 = 𝑁 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))
3127, 30imbi12d 346 . . . . . 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 342 . . . . 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 24429 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3433adantr 481 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → 𝑆 ⊆ ℂ)
35 pmresg 8423 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
36 dvn0 24448 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
3734, 35, 36syl2anc 584 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
38 ssidd 3987 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39 dvn0 24448 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4038, 39sylan 580 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4140reseq1d 5845 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹𝑆))
4237, 41eqtr4d 2856 . . . . . 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 10618 . . . . . . . . . 10 ℂ ∈ {ℝ, ℂ}
45 simplr 765 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝐹 ∈ (ℂ ↑pm ℂ))
46 simprl 767 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ0)
47 dvnbss 24452 . . . . . . . . . 10 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
4844, 45, 46, 47mp3an2i 1457 . . . . . . . . 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 769 . . . . . . . . . . 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 3987 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
51 dvnp1 24449 . . . . . . . . . . . . 13 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5250, 45, 46, 51syl3anc 1363 . . . . . . . . . . . 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 5767 . . . . . . . . . . 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 2855 . . . . . . . . . 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 24451 . . . . . . . . . . . 12 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ)
5644, 45, 46, 55mp3an2i 1457 . . . . . . . . . . 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 10606 . . . . . . . . . . . . . . 15 ℂ ∈ V
5857, 57elpm2 8427 . . . . . . . . . . . . . 14 (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
5958simprbi 497 . . . . . . . . . . . . 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 3974 . . . . . . . . . . 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 24426 . . . . . . . . . 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 4002 . . . . . . . . 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 3981 . . . . . . . 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 457 . . . . . . 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 7153 . . . . . . 7 (((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆) → (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
6834adantr 481 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
6935adantr 481 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
70 dvnp1 24449 . . . . . . . . 9 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)))
7168, 69, 46, 70syl3anc 1363 . . . . . . . 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 5845 . . . . . . . . 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 763 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
74 eqid 2818 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
7574cnfldtop 23319 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ Top
76 unicntop 23321 . . . . . . . . . . . . . 14 ℂ = (TopOpen‘ℂfld)
7776ntrss2 21593 . . . . . . . . . . . . 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 587 . . . . . . . . . . . 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 23318 . . . . . . . . . . . . . . . 16 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
8079toponrestid 21457 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8150, 56, 61, 80, 74dvbssntr 24425 . . . . . . . . . . . . . 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 4002 . . . . . . . . . . . . 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 3974 . . . . . . . . . . . 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 3981 . . . . . . . . . . 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 21602 . . . . . . . . . . . 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 587 . . . . . . . . . . 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 258 . . . . . . . . . 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 2854 . . . . . . . . . 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 24439 . . . . . . . . . 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 834 . . . . . . . . 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 2856 . . . . . . . 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 2834 . . . . . . 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 247 . . . . . 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 840 . . . . 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 12065 . . . 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 1109 . 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 407 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wss 3933  {cpr 4559  dom cdm 5548  cres 5550  wf 6344  cfv 6348  (class class class)co 7145  pm cpm 8396  cc 10523  cr 10524  0cc0 10525  1c1 10526   + caddc 10528  0cn0 11885  TopOpenctopn 16683  fldccnfld 20473  Topctop 21429  intcnt 21553   D cdv 24388   D𝑛 cdvn 24389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-inf2 9092  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-oadd 8095  df-er 8278  df-map 8397  df-pm 8398  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-fi 8863  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-z 11970  df-dec 12087  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-icc 12733  df-fz 12881  df-seq 13358  df-exp 13418  df-cj 14446  df-re 14447  df-im 14448  df-sqrt 14582  df-abs 14583  df-struct 16473  df-ndx 16474  df-slot 16475  df-base 16477  df-plusg 16566  df-mulr 16567  df-starv 16568  df-tset 16572  df-ple 16573  df-ds 16575  df-unif 16576  df-rest 16684  df-topn 16685  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-met 20467  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-cnfld 20474  df-top 21430  df-topon 21447  df-topsp 21469  df-bases 21482  df-cld 21555  df-ntr 21556  df-cls 21557  df-nei 21634  df-lp 21672  df-perf 21673  df-cnp 21764  df-haus 21851  df-fil 22382  df-fm 22474  df-flim 22475  df-flf 22476  df-xms 22857  df-ms 22858  df-limc 24391  df-dv 24392  df-dvn 24393
This theorem is referenced by:  cpnres  24461
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