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Theorem dvnres 23913
Description: Multiple derivative version of dvres3a 23897. (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 6353 . . . . . . . . 9 (𝑥 = 0 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘0))
21dmeqd 5481 . . . . . . . 8 (𝑥 = 0 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘0))
32eqeq1d 2762 . . . . . . 7 (𝑥 = 0 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘0) = dom 𝐹))
4 fveq2 6353 . . . . . . . 8 (𝑥 = 0 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘0))
51reseq1d 5550 . . . . . . . 8 (𝑥 = 0 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆))
64, 5eqeq12d 2775 . . . . . . 7 (𝑥 = 0 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘0) = (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆)))
73, 6imbi12d 333 . . . . . 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 329 . . . . 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 6353 . . . . . . . . 9 (𝑥 = 𝑛 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑛))
109dmeqd 5481 . . . . . . . 8 (𝑥 = 𝑛 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑛))
1110eqeq1d 2762 . . . . . . 7 (𝑥 = 𝑛 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑛) = dom 𝐹))
12 fveq2 6353 . . . . . . . 8 (𝑥 = 𝑛 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑛))
139reseq1d 5550 . . . . . . . 8 (𝑥 = 𝑛 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆))
1412, 13eqeq12d 2775 . . . . . . 7 (𝑥 = 𝑛 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
1511, 14imbi12d 333 . . . . . 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 329 . . . . 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 6353 . . . . . . . . 9 (𝑥 = (𝑛 + 1) → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1817dmeqd 5481 . . . . . . . 8 (𝑥 = (𝑛 + 1) → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)))
1918eqeq1d 2762 . . . . . . 7 (𝑥 = (𝑛 + 1) → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹))
20 fveq2 6353 . . . . . . . 8 (𝑥 = (𝑛 + 1) → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)))
2117reseq1d 5550 . . . . . . . 8 (𝑥 = (𝑛 + 1) → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘(𝑛 + 1)) ↾ 𝑆))
2220, 21eqeq12d 2775 . . . . . . 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 333 . . . . . 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 329 . . . . 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 6353 . . . . . . . . 9 (𝑥 = 𝑁 → ((ℂ D𝑛 𝐹)‘𝑥) = ((ℂ D𝑛 𝐹)‘𝑁))
2625dmeqd 5481 . . . . . . . 8 (𝑥 = 𝑁 → dom ((ℂ D𝑛 𝐹)‘𝑥) = dom ((ℂ D𝑛 𝐹)‘𝑁))
2726eqeq1d 2762 . . . . . . 7 (𝑥 = 𝑁 → (dom ((ℂ D𝑛 𝐹)‘𝑥) = dom 𝐹 ↔ dom ((ℂ D𝑛 𝐹)‘𝑁) = dom 𝐹))
28 fveq2 6353 . . . . . . . 8 (𝑥 = 𝑁 → ((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = ((𝑆 D𝑛 (𝐹𝑆))‘𝑁))
2925reseq1d 5550 . . . . . . . 8 (𝑥 = 𝑁 → (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆))
3028, 29eqeq12d 2775 . . . . . . 7 (𝑥 = 𝑁 → (((𝑆 D𝑛 (𝐹𝑆))‘𝑥) = (((ℂ D𝑛 𝐹)‘𝑥) ↾ 𝑆) ↔ ((𝑆 D𝑛 (𝐹𝑆))‘𝑁) = (((ℂ D𝑛 𝐹)‘𝑁) ↾ 𝑆)))
3127, 30imbi12d 333 . . . . . 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 329 . . . . 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 23887 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
3433adantr 472 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → 𝑆 ⊆ ℂ)
35 pmresg 8053 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
36 dvn0 23906 . . . . . . . 8 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
3734, 35, 36syl2anc 696 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((𝑆 D𝑛 (𝐹𝑆))‘0) = (𝐹𝑆))
38 ssid 3765 . . . . . . . . . 10 ℂ ⊆ ℂ
3938a1i 11 . . . . . . . . 9 (𝑆 ∈ {ℝ, ℂ} → ℂ ⊆ ℂ)
40 dvn0 23906 . . . . . . . . 9 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4139, 40sylan 489 . . . . . . . 8 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → ((ℂ D𝑛 𝐹)‘0) = 𝐹)
4241reseq1d 5550 . . . . . . 7 ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) → (((ℂ D𝑛 𝐹)‘0) ↾ 𝑆) = (𝐹𝑆))
4337, 42eqtr4d 2797 . . . . . 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 10241 . . . . . . . . . . . . 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 809 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝐹 ∈ (ℂ ↑pm ℂ))
48 simprl 811 . . . . . . . . . . . 12 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑛 ∈ ℕ0)
49 dvnbss 23910 . . . . . . . . . . . 12 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → dom ((ℂ D𝑛 𝐹)‘𝑛) ⊆ dom 𝐹)
5046, 47, 48, 49syl3anc 1477 . . . . . . . . . . 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 813 . . . . . . . . . . . . 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 23907 . . . . . . . . . . . . . . 15 ((ℂ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = (ℂ D ((ℂ D𝑛 𝐹)‘𝑛)))
5452, 47, 48, 53syl3anc 1477 . . . . . . . . . . . . . 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 5481 . . . . . . . . . . . . 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 2796 . . . . . . . . . . . 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 23909 . . . . . . . . . . . . . 14 ((ℂ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ) ∧ 𝑛 ∈ ℕ0) → ((ℂ D𝑛 𝐹)‘𝑛):dom ((ℂ D𝑛 𝐹)‘𝑛)⟶ℂ)
5846, 47, 48, 57syl3anc 1477 . . . . . . . . . . . . 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 10229 . . . . . . . . . . . . . . . . 17 ℂ ∈ V
6059, 59elpm2 8057 . . . . . . . . . . . . . . . 16 (𝐹 ∈ (ℂ ↑pm ℂ) ↔ (𝐹:dom 𝐹⟶ℂ ∧ dom 𝐹 ⊆ ℂ))
6160simprbi 483 . . . . . . . . . . . . . . 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 3754 . . . . . . . . . . . . 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 23884 . . . . . . . . . . . 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 3780 . . . . . . . . . . 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 3761 . . . . . . . . . 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 644 . . . . . . . . 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 82 . . . . . . . 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 6822 . . . . . . . . . . 11 (((𝑆 D𝑛 (𝐹𝑆))‘𝑛) = (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆) → (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)) = (𝑆 D (((ℂ D𝑛 𝐹)‘𝑛) ↾ 𝑆)))
7034adantr 472 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ⊆ ℂ)
7135adantr 472 . . . . . . . . . . . . 13 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → (𝐹𝑆) ∈ (ℂ ↑pm 𝑆))
72 dvnp1 23907 . . . . . . . . . . . . 13 ((𝑆 ⊆ ℂ ∧ (𝐹𝑆) ∈ (ℂ ↑pm 𝑆) ∧ 𝑛 ∈ ℕ0) → ((𝑆 D𝑛 (𝐹𝑆))‘(𝑛 + 1)) = (𝑆 D ((𝑆 D𝑛 (𝐹𝑆))‘𝑛)))
7370, 71, 48, 72syl3anc 1477 . . . . . . . . . . . 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 5550 . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . 14 (((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑pm ℂ)) ∧ (𝑛 ∈ ℕ0 ∧ dom ((ℂ D𝑛 𝐹)‘(𝑛 + 1)) = dom 𝐹)) → 𝑆 ∈ {ℝ, ℂ})
76 eqid 2760 . . . . . . . . . . . . . . . . . 18 (TopOpen‘ℂfld) = (TopOpen‘ℂfld)
7776cnfldtop 22808 . . . . . . . . . . . . . . . . 17 (TopOpen‘ℂfld) ∈ Top
7876cnfldtopon 22807 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
7978toponunii 20943 . . . . . . . . . . . . . . . . . 18 ℂ = (TopOpen‘ℂfld)
8079ntrss2 21083 . . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . . 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 16316 . . . . . . . . . . . . . . . . . . . . 21 ((TopOpen‘ℂfld) ∈ Top → ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld))
8377, 82ax-mp 5 . . . . . . . . . . . . . . . . . . . 20 ((TopOpen‘ℂfld) ↾t ℂ) = (TopOpen‘ℂfld)
8483eqcomi 2769 . . . . . . . . . . . . . . . . . . 19 (TopOpen‘ℂfld) = ((TopOpen‘ℂfld) ↾t ℂ)
8552, 58, 63, 84, 76dvbssntr 23883 . . . . . . . . . . . . . . . . . 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 3780 . . . . . . . . . . . . . . . . 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 3754 . . . . . . . . . . . . . . . 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 3761 . . . . . . . . . . . . . . 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 21092 . . . . . . . . . . . . . . . 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 698 . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . 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 2795 . . . . . . . . . . . . . 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 23897 . . . . . . . . . . . . . 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 1478 . . . . . . . . . . . . 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 2797 . . . . . . . . . . . 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 2775 . . . . . . . . . . 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 236 . . . . . . . . . 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 644 . . . . . . . . 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 47 . . . . . . 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 450 . . . . . 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 11684 . . . 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 1110 . 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 444 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 196  wa 383  w3a 1072   = wceq 1632  wcel 2139  wss 3715  {cpr 4323  dom cdm 5266  cres 5268  wf 6045  cfv 6049  (class class class)co 6814  pm cpm 8026  cc 10146  cr 10147  0cc0 10148  1c1 10149   + caddc 10151  0cn0 11504  t crest 16303  TopOpenctopn 16304  fldccnfld 19968  Topctop 20920  intcnt 21043   D cdv 23846   D𝑛 cdvn 23847
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-inf2 8713  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225  ax-pre-sup 10226
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-iin 4675  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-map 8027  df-pm 8028  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-fi 8484  df-sup 8515  df-inf 8516  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-div 10897  df-nn 11233  df-2 11291  df-3 11292  df-4 11293  df-5 11294  df-6 11295  df-7 11296  df-8 11297  df-9 11298  df-n0 11505  df-z 11590  df-dec 11706  df-uz 11900  df-q 12002  df-rp 12046  df-xneg 12159  df-xadd 12160  df-xmul 12161  df-icc 12395  df-fz 12540  df-seq 13016  df-exp 13075  df-cj 14058  df-re 14059  df-im 14060  df-sqrt 14194  df-abs 14195  df-struct 16081  df-ndx 16082  df-slot 16083  df-base 16085  df-plusg 16176  df-mulr 16177  df-starv 16178  df-tset 16182  df-ple 16183  df-ds 16186  df-unif 16187  df-rest 16305  df-topn 16306  df-topgen 16326  df-psmet 19960  df-xmet 19961  df-met 19962  df-bl 19963  df-mopn 19964  df-fbas 19965  df-fg 19966  df-cnfld 19969  df-top 20921  df-topon 20938  df-topsp 20959  df-bases 20972  df-cld 21045  df-ntr 21046  df-cls 21047  df-nei 21124  df-lp 21162  df-perf 21163  df-cnp 21254  df-haus 21341  df-fil 21871  df-fm 21963  df-flim 21964  df-flf 21965  df-xms 22346  df-ms 22347  df-limc 23849  df-dv 23850  df-dvn 23851
This theorem is referenced by:  cpnres  23919
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