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Theorem dvnres 24131
Description: Multiple derivative version of dvres3a 24115. (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 6446 . . . . . . . . 9 (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0))
21dmeqd 5571 . . . . . . . 8 (𝑥 = 0 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘0))
32eqeq1d 2779 . . . . . . 7 (𝑥 = 0 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹))
4 fveq2 6446 . . . . . . . 8 (𝑥 = 0 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘0))
51reseq1d 5641 . . . . . . . 8 (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
64, 5eqeq12d 2792 . . . . . . 7 (𝑥 = 0 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))
73, 6imbi12d 336 . . . . . 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 332 . . . . 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 6446 . . . . . . . . 9 (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛))
109dmeqd 5571 . . . . . . . 8 (𝑥 = 𝑛 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑛))
1110eqeq1d 2779 . . . . . . 7 (𝑥 = 𝑛 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹))
12 fveq2 6446 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑛))
139reseq1d 5641 . . . . . . . 8 (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))
1412, 13eqeq12d 2792 . . . . . . 7 (𝑥 = 𝑛 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
1511, 14imbi12d 336 . . . . . 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 332 . . . . 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 6446 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1817dmeqd 5571 . . . . . . . 8 (𝑥 = (𝑛 + 1) → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1918eqeq1d 2779 . . . . . . 7 (𝑥 = (𝑛 + 1) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20 fveq2 6446 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)))
2117reseq1d 5641 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
2220, 21eqeq12d 2792 . . . . . . 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 336 . . . . . 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 332 . . . . 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 6446 . . . . . . . . 9 (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁))
2625dmeqd 5571 . . . . . . . 8 (𝑥 = 𝑁 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑁))
2726eqeq1d 2779 . . . . . . 7 (𝑥 = 𝑁 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹))
28 fveq2 6446 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑁))
2925reseq1d 5641 . . . . . . . 8 (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
3028, 29eqeq12d 2792 . . . . . . 7 (𝑥 = 𝑁 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))
3127, 30imbi12d 336 . . . . . 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 332 . . . . 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 24105 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3433adantr 474 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → 𝑆 ⊆ ℂ)
35 pmresg 8168 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
36 dvn0 24124 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
3734, 35, 36syl2anc 579 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
38 ssidd 3842 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
39 dvn0 24124 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4038, 39sylan 575 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4140reseq1d 5641 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹𝑆))
4237, 41eqtr4d 2816 . . . . . 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 10365 . . . . . . . . . . 11 ℂ ∈ {ℝ, ℂ}
4544a1i 11 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ∈ {ℝ, ℂ})
46 simplr 759 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝐹 ∈ (ℂ ↑pm ℂ))
47 simprl 761 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ0)
48 dvnbss 24128 . . . . . . . . . 10 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
4945, 46, 47, 48syl3anc 1439 . . . . . . . . 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)
50 simprr 763 . . . . . . . . . . 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)
51 ssidd 3842 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
52 dvnp1 24125 . . . . . . . . . . . . 13 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5351, 46, 47, 52syl3anc 1439 . . . . . . . . . . . 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)))
5453dmeqd 5571 . . . . . . . . . . 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)))
5550, 54eqtr3d 2815 . . . . . . . . . 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)))
56 dvnf 24127 . . . . . . . . . . . 12 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ)
5745, 46, 47, 56syl3anc 1439 . . . . . . . . . . 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)
58 cnex 10353 . . . . . . . . . . . . . . 15 ℂ ∈ V
5958, 58elpm2 8172 . . . . . . . . . . . . . 14 (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
6059simprbi 492 . . . . . . . . . . . . 13 (𝐹 ∈ (ℂ ↑pm ℂ) → dom 𝐹 ⊆ ℂ)
6146, 60syl 17 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ ℂ)
6249, 61sstrd 3830 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ)
6351, 57, 62dvbss 24102 . . . . . . . . . 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))
6455, 63eqsstrd 3857 . . . . . . . . 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))
6549, 64eqssd 3837 . . . . . . . 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)
6665expr 450 . . . . . . 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))
6766imim1d 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))))
68 oveq2 6930 . . . . . . 7 (((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆) → (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
6934adantr 474 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
7035adantr 474 . . . . . . . . 9 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
71 dvnp1 24125 . . . . . . . . 9 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)))
7269, 70, 47, 71syl3anc 1439 . . . . . . . 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)))
7353reseq1d 5641 . . . . . . . . 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))
74 simpll 757 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
75 eqid 2777 . . . . . . . . . . . . . 14 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
7675cnfldtop 22995 . . . . . . . . . . . . 13 (TopOpen‘ℂfld) ∈ Top
7775cnfldtopon 22994 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
7877toponunii 21128 . . . . . . . . . . . . . 14 ℂ = (TopOpen‘ℂfld)
7978ntrss2 21269 . . . . . . . . . . . . 13 (((TopOpen‘ℂfld) ∈ Top ∧ dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ) → ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
8076, 62, 79sylancr 581 . . . . . . . . . . . 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))
8177toponrestid 21133 . . . . . . . . . . . . . . 15 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8251, 57, 62, 81, 75dvbssntr 24101 . . . . . . . . . . . . . 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)))
8355, 82eqsstrd 3857 . . . . . . . . . . . . 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)))
8449, 83sstrd 3830 . . . . . . . . . . . 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)))
8580, 84eqssd 3837 . . . . . . . . . . 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))
8678isopn3 21278 . . . . . . . . . . . 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)))
8776, 62, 86sylancr 581 . . . . . . . . . . 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)))
8885, 87mpbird 249 . . . . . . . . . 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))
8965, 55eqtr2d 2814 . . . . . . . . . 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))
9075dvres3a 24115 . . . . . . . . . 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))
9174, 57, 88, 89, 90syl22anc 829 . . . . . . . . 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))
9273, 91eqtr4d 2816 . . . . . . . 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)))
9372, 92eqeq12d 2792 . . . . . . 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))))
9468, 93syl5ibr 238 . . . . . 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)))
9567, 94animpimp2impd 835 . . . . 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)))))
968, 16, 24, 32, 43, 95nn0ind 11824 . . . 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))))
9796com12 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))))
98973impia 1106 . 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)))
9998imp 397 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 198  wa 386  w3a 1071   = wceq 1601  wcel 2106  wss 3791  {cpr 4399  dom cdm 5355  cres 5357  wf 6131  cfv 6135  (class class class)co 6922  pm cpm 8141  cc 10270  cr 10271  0cc0 10272  1c1 10273   + caddc 10275  0cn0 11642  TopOpenctopn 16468  fldccnfld 20142  Topctop 21105  intcnt 21229   D cdv 24064   D𝑛 cdvn 24065
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2054  ax-8 2108  ax-9 2115  ax-10 2134  ax-11 2149  ax-12 2162  ax-13 2333  ax-ext 2753  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-inf2 8835  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349  ax-pre-sup 10350
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2550  df-eu 2586  df-clab 2763  df-cleq 2769  df-clel 2773  df-nfc 2920  df-ne 2969  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rmo 3097  df-rab 3098  df-v 3399  df-sbc 3652  df-csb 3751  df-dif 3794  df-un 3796  df-in 3798  df-ss 3805  df-pss 3807  df-nul 4141  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-tp 4402  df-op 4404  df-uni 4672  df-int 4711  df-iun 4755  df-iin 4756  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-oadd 7847  df-er 8026  df-map 8142  df-pm 8143  df-en 8242  df-dom 8243  df-sdom 8244  df-fin 8245  df-fi 8605  df-sup 8636  df-inf 8637  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-z 11729  df-dec 11846  df-uz 11993  df-q 12096  df-rp 12138  df-xneg 12257  df-xadd 12258  df-xmul 12259  df-icc 12494  df-fz 12644  df-seq 13120  df-exp 13179  df-cj 14246  df-re 14247  df-im 14248  df-sqrt 14382  df-abs 14383  df-struct 16257  df-ndx 16258  df-slot 16259  df-base 16261  df-plusg 16351  df-mulr 16352  df-starv 16353  df-tset 16357  df-ple 16358  df-ds 16360  df-unif 16361  df-rest 16469  df-topn 16470  df-topgen 16490  df-psmet 20134  df-xmet 20135  df-met 20136  df-bl 20137  df-mopn 20138  df-fbas 20139  df-fg 20140  df-cnfld 20143  df-top 21106  df-topon 21123  df-topsp 21145  df-bases 21158  df-cld 21231  df-ntr 21232  df-cls 21233  df-nei 21310  df-lp 21348  df-perf 21349  df-cnp 21440  df-haus 21527  df-fil 22058  df-fm 22150  df-flim 22151  df-flf 22152  df-xms 22533  df-ms 22534  df-limc 24067  df-dv 24068  df-dvn 24069
This theorem is referenced by:  cpnres  24137
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