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Theorem dvnres 23445
Description: Multiple derivative version of dvres3a 23429. (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 6088 . . . . . . . . 9 (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0))
21dmeqd 5235 . . . . . . . 8 (𝑥 = 0 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘0))
32eqeq1d 2611 . . . . . . 7 (𝑥 = 0 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹))
4 fveq2 6088 . . . . . . . 8 (𝑥 = 0 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘0))
51reseq1d 5303 . . . . . . . 8 (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
64, 5eqeq12d 2624 . . . . . . 7 (𝑥 = 0 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))
73, 6imbi12d 332 . . . . . 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 328 . . . . 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 6088 . . . . . . . . 9 (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛))
109dmeqd 5235 . . . . . . . 8 (𝑥 = 𝑛 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑛))
1110eqeq1d 2611 . . . . . . 7 (𝑥 = 𝑛 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹))
12 fveq2 6088 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑛))
139reseq1d 5303 . . . . . . . 8 (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))
1412, 13eqeq12d 2624 . . . . . . 7 (𝑥 = 𝑛 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
1511, 14imbi12d 332 . . . . . 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 328 . . . . 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 6088 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1817dmeqd 5235 . . . . . . . 8 (𝑥 = (𝑛 + 1) → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1918eqeq1d 2611 . . . . . . 7 (𝑥 = (𝑛 + 1) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20 fveq2 6088 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)))
2117reseq1d 5303 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
2220, 21eqeq12d 2624 . . . . . . 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 332 . . . . . 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 328 . . . . 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 6088 . . . . . . . . 9 (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁))
2625dmeqd 5235 . . . . . . . 8 (𝑥 = 𝑁 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑁))
2726eqeq1d 2611 . . . . . . 7 (𝑥 = 𝑁 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹))
28 fveq2 6088 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑁))
2925reseq1d 5303 . . . . . . . 8 (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
3028, 29eqeq12d 2624 . . . . . . 7 (𝑥 = 𝑁 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))
3127, 30imbi12d 332 . . . . . 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 328 . . . . 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 23419 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3433adantr 479 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → 𝑆 ⊆ ℂ)
35 pmresg 7749 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
36 dvn0 23438 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
3734, 35, 36syl2anc 690 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
38 ssid 3586 . . . . . . . . . 10 ℂ ⊆ ℂ
3938a1i 11 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
40 dvn0 23438 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4139, 40sylan 486 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4241reseq1d 5303 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹𝑆))
4337, 42eqtr4d 2646 . . . . . 6 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
4443a1d 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)))
45 cnelprrecn 9886 . . . . . . . . . . . . 13 ℂ ∈ {ℝ, ℂ}
4645a1i 11 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ∈ {ℝ, ℂ})
47 simplr 787 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝐹 ∈ (ℂ ↑pm ℂ))
48 simprl 789 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ0)
49 dvnbss 23442 . . . . . . . . . . . 12 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
5046, 47, 48, 49syl3anc 1317 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
51 simprr 791 . . . . . . . . . . . . 13 (((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)
5238a1i 11 . . . . . . . . . . . . . . 15 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → ℂ ⊆ ℂ)
53 dvnp1 23439 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5452, 47, 48, 53syl3anc 1317 . . . . . . . . . . . . . 14 (((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)))
5554dmeqd 5235 . . . . . . . . . . . . 13 (((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)))
5651, 55eqtr3d 2645 . . . . . . . . . . . 12 (((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)))
57 dvnf 23441 . . . . . . . . . . . . . 14 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ)
5846, 47, 48, 57syl3anc 1317 . . . . . . . . . . . . 13 (((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)
59 cnex 9874 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
6059, 59elpm2 7753 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
6160simprbi 478 . . . . . . . . . . . . . . 15 (𝐹 ∈ (ℂ ↑pm ℂ) → dom 𝐹 ⊆ ℂ)
6247, 61syl 17 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ ℂ)
6350, 62sstrd 3577 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ)
6452, 58, 63dvbss 23416 . . . . . . . . . . . 12 (((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))
6556, 64eqsstrd 3601 . . . . . . . . . . 11 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom 𝐹 ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
6650, 65eqssd 3584 . . . . . . . . . 10 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹)
6766expr 640 . . . . . . . . 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))
6867imim1d 79 . . . . . . . 8 (((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))))
69 oveq2 6535 . . . . . . . . . . 11 (((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆) → (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
7034adantr 479 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
7135adantr 479 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
72 dvnp1 23439 . . . . . . . . . . . . 13 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)))
7370, 71, 48, 72syl3anc 1317 . . . . . . . . . . . 12 (((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)))
7454reseq1d 5303 . . . . . . . . . . . . 13 (((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))
75 simpll 785 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
76 eqid 2609 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
7776cnfldtop 22345 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) ∈ Top
7876cnfldtopon 22344 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
7978toponunii 20495 . . . . . . . . . . . . . . . . . 18 ℂ = (TopOpen‘ℂfld)
8079ntrss2 20619 . . . . . . . . . . . . . . . . 17 (((TopOpen‘ℂfld) ∈ Top ∧ dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ ℂ) → ((int‘(TopOpen‘ℂfld))‘dom ((ℂ D𝑛 𝐹)‘𝑛)) ⊆ dom ((ℂ D𝑛 𝐹)‘𝑛))
8177, 63, 80sylancr 693 . . . . . . . . . . . . . . . 16 (((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))
8279restid 15866 . . . . . . . . . . . . . . . . . . . . 21 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
8377, 82ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
8483eqcomi 2618 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8552, 58, 63, 84, 76dvbssntr 23415 . . . . . . . . . . . . . . . . . 18 (((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)))
8656, 85eqsstrd 3601 . . . . . . . . . . . . . . . . 17 (((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)))
8750, 86sstrd 3577 . . . . . . . . . . . . . . . 16 (((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)))
8881, 87eqssd 3584 . . . . . . . . . . . . . . 15 (((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))
8979isopn3 20628 . . . . . . . . . . . . . . . 16 (((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)))
9077, 63, 89sylancr 693 . . . . . . . . . . . . . . 15 (((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)))
9188, 90mpbird 245 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → dom ((ℂ D𝑛 𝐹)‘𝑛) ∈ (TopOpen‘ℂfld))
9266, 56eqtr2d 2644 . . . . . . . . . . . . . 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)) = dom ((ℂ D𝑛 𝐹)‘𝑛))
9376dvres3a 23429 . . . . . . . . . . . . . 14 (((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))
9475, 58, 91, 92, 93syl22anc 1318 . . . . . . . . . . . . 13 (((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))
9574, 94eqtr4d 2646 . . . . . . . . . . . 12 (((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)))
9673, 95eqeq12d 2624 . . . . . . . . . . 11 (((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))))
9769, 96syl5ibr 234 . . . . . . . . . 10 (((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)))
9897expr 640 . . . . . . . . 9 (((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))))
9998a2d 29 . . . . . . . 8 (((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)) → (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))))
10068, 99syld 45 . . . . . . 7 (((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 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))))
101100expcom 449 . . . . . 6 (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)) → (dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹 → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆)))))
102101a2d 29 . . . . 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)))))
1038, 16, 24, 32, 44, 102nn0ind 11307 . . . 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))))
104103com12 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))))
1051043impia 1252 . 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)))
106105imp 443 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wss 3539  {cpr 4126  dom cdm 5028  cres 5030  wf 5786  cfv 5790  (class class class)co 6527  pm cpm 7723  cc 9791  cr 9792  0cc0 9793  1c1 9794   + caddc 9796  0cn0 11142  t crest 15853  TopOpenctopn 15854  fldccnfld 19516  Topctop 20465  intcnt 20579   D cdv 23378   D𝑛 cdvn 23379
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825  ax-inf2 8399  ax-cnex 9849  ax-resscn 9850  ax-1cn 9851  ax-icn 9852  ax-addcl 9853  ax-addrcl 9854  ax-mulcl 9855  ax-mulrcl 9856  ax-mulcom 9857  ax-addass 9858  ax-mulass 9859  ax-distr 9860  ax-i2m1 9861  ax-1ne0 9862  ax-1rid 9863  ax-rnegex 9864  ax-rrecex 9865  ax-cnre 9866  ax-pre-lttri 9867  ax-pre-lttrn 9868  ax-pre-ltadd 9869  ax-pre-mulgt0 9870  ax-pre-sup 9871
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-nel 2782  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-pss 3555  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129  df-op 4131  df-uni 4367  df-int 4405  df-iun 4451  df-iin 4452  df-br 4578  df-opab 4638  df-mpt 4639  df-tr 4675  df-eprel 4939  df-id 4943  df-po 4949  df-so 4950  df-fr 4987  df-we 4989  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-pred 5583  df-ord 5629  df-on 5630  df-lim 5631  df-suc 5632  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-om 6936  df-1st 7037  df-2nd 7038  df-wrecs 7272  df-recs 7333  df-rdg 7371  df-1o 7425  df-oadd 7429  df-er 7607  df-map 7724  df-pm 7725  df-en 7820  df-dom 7821  df-sdom 7822  df-fin 7823  df-fi 8178  df-sup 8209  df-inf 8210  df-pnf 9933  df-mnf 9934  df-xr 9935  df-ltxr 9936  df-le 9937  df-sub 10120  df-neg 10121  df-div 10537  df-nn 10871  df-2 10929  df-3 10930  df-4 10931  df-5 10932  df-6 10933  df-7 10934  df-8 10935  df-9 10936  df-n0 11143  df-z 11214  df-dec 11329  df-uz 11523  df-q 11624  df-rp 11668  df-xneg 11781  df-xadd 11782  df-xmul 11783  df-icc 12012  df-fz 12156  df-seq 12622  df-exp 12681  df-cj 13636  df-re 13637  df-im 13638  df-sqrt 13772  df-abs 13773  df-struct 15646  df-ndx 15647  df-slot 15648  df-base 15649  df-plusg 15730  df-mulr 15731  df-starv 15732  df-tset 15736  df-ple 15737  df-ds 15740  df-unif 15741  df-rest 15855  df-topn 15856  df-topgen 15876  df-psmet 19508  df-xmet 19509  df-met 19510  df-bl 19511  df-mopn 19512  df-fbas 19513  df-fg 19514  df-cnfld 19517  df-top 20469  df-bases 20470  df-topon 20471  df-topsp 20472  df-cld 20581  df-ntr 20582  df-cls 20583  df-nei 20660  df-lp 20698  df-perf 20699  df-cnp 20790  df-haus 20877  df-fil 21408  df-fm 21500  df-flim 21501  df-flf 21502  df-xms 21883  df-ms 21884  df-limc 23381  df-dv 23382  df-dvn 23383
This theorem is referenced by:  cpnres  23451
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