taylthlem1 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > taylthlem1		Structured version Visualization version GIF version

Theorem taylthlem1 26501

Description: Lemma for taylth 26503. This is the main part of Taylor's theorem, except for the induction step, which is supposed to be proven using L'Hôpital's rule. However, since our proof of L'Hôpital assumes that 𝑆 = ℝ, we can only do this part generically, and for taylth 26503 itself we must restrict to ℝ. (Contributed by Mario Carneiro, 1-Jan-2017.)

**Hypotheses**
Ref	Expression
taylthlem1.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
taylthlem1.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
taylthlem1.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
taylthlem1.d	⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) = 𝐴)
taylthlem1.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
taylthlem1.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
taylthlem1.t	⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
taylthlem1.r	⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁)))
taylthlem1.i	⊢ ((𝜑 ∧ (𝑛 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵))

**Assertion**
Ref	Expression
taylthlem1	⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Distinct variable groups: 𝑥,𝑛,𝑦,𝐴 𝐵,𝑛,𝑥,𝑦 𝑛,𝐹,𝑥,𝑦 𝜑,𝑛,𝑥,𝑦 𝑛,𝑁,𝑥,𝑦 𝑆,𝑛,𝑥,𝑦 𝑇,𝑛,𝑥,𝑦 Allowed substitution hints: 𝑅(𝑥,𝑦,𝑛)

Proof of Theorem taylthlem1 Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylthlem1.n	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
2		elfz1end 13581	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
4		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑁 − 𝑚) = (𝑁 − 1))
5	4	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
6	5	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
7	4	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)))
8	7	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
9	6, 8	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑚 = 1 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)))
10		oveq2 7419	. . . . . . . . 9 ⊢ (𝑚 = 1 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑1))
11	9, 10	oveq12d 7429	. . . . . . . 8 ⊢ (𝑚 = 1 → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚)) = (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1)))
12	11	mpteq2dv 5209	. . . . . . 7 ⊢ (𝑚 = 1 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))))
13	12	oveq1d 7426	. . . . . 6 ⊢ (𝑚 = 1 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) lim_ℂ 𝐵))
14	13	eleq2d 2855	. . . . 5 ⊢ (𝑚 = 1 → (0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) ↔ 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) lim_ℂ 𝐵)))
15	14	imbi2d 343	. . . 4 ⊢ (𝑚 = 1 → ((𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵)) ↔ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) lim_ℂ 𝐵))))
16		oveq2 7419	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑁 − 𝑚) = (𝑁 − 𝑛))
17	16	fveq2d 6886	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛)))
18	17	fveq1d 6884	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥))
19	16	fveq2d 6886	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛)))
20	19	fveq1d 6884	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥))
21	18, 20	oveq12d 7429	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥)))
22		oveq2 7419	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑛))
23	21, 22	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚)) = (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥)) / ((𝑥 − 𝐵)↑𝑛)))
24	23	mpteq2dv 5209	. . . . . . . 8 ⊢ (𝑚 = 𝑛 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥)) / ((𝑥 − 𝐵)↑𝑛))))
25		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦))
26		fveq2 6882	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦))
27	25, 26	oveq12d 7429	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)))
28		oveq1 7418	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐵) = (𝑦 − 𝐵))
29	28	oveq1d 7426	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐵)↑𝑛) = ((𝑦 − 𝐵)↑𝑛))
30	27, 29	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥)) / ((𝑥 − 𝐵)↑𝑛)) = (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛)))
31	30	cbvmptv 5219	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥)) / ((𝑥 − 𝐵)↑𝑛))) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛)))
32	24, 31	eqtrdi 2820	. . . . . . 7 ⊢ (𝑚 = 𝑛 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))))
33	32	oveq1d 7426	. . . . . 6 ⊢ (𝑚 = 𝑛 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) = ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵))
34	33	eleq2d 2855	. . . . 5 ⊢ (𝑚 = 𝑛 → (0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) ↔ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵)))
35	34	imbi2d 343	. . . 4 ⊢ (𝑚 = 𝑛 → ((𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵)) ↔ (𝜑 → 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵))))
36		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑁 − 𝑚) = (𝑁 − (𝑛 + 1)))
37	36	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1))))
38	37	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥))
39	36	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1))))
40	39	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥))
41	38, 40	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)))
42		oveq2 7419	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑(𝑛 + 1)))
43	41, 42	oveq12d 7429	. . . . . . . 8 ⊢ (𝑚 = (𝑛 + 1) → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚)) = (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1))))
44	43	mpteq2dv 5209	. . . . . . 7 ⊢ (𝑚 = (𝑛 + 1) → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))))
45	44	oveq1d 7426	. . . . . 6 ⊢ (𝑚 = (𝑛 + 1) → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵))
46	45	eleq2d 2855	. . . . 5 ⊢ (𝑚 = (𝑛 + 1) → (0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) ↔ 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵)))
47	46	imbi2d 343	. . . 4 ⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵)) ↔ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵))))
48		oveq2 7419	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝑁 − 𝑚) = (𝑁 − 𝑁))
49	48	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)))
50	49	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥))
51	48	fveq2d 6886	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)))
52	51	fveq1d 6884	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥))
53	50, 52	oveq12d 7429	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)))
54		oveq2 7419	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑁))
55	53, 54	oveq12d 7429	. . . . . . . 8 ⊢ (𝑚 = 𝑁 → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚)) = (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁)))
56	55	mpteq2dv 5209	. . . . . . 7 ⊢ (𝑚 = 𝑁 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))))
57	56	oveq1d 7426	. . . . . 6 ⊢ (𝑚 = 𝑁 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) lim_ℂ 𝐵))
58	57	eleq2d 2855	. . . . 5 ⊢ (𝑚 = 𝑁 → (0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵) ↔ 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) lim_ℂ 𝐵)))
59	58	imbi2d 343	. . . 4 ⊢ (𝑚 = 𝑁 → ((𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥)) / ((𝑥 − 𝐵)↑𝑚))) lim_ℂ 𝐵)) ↔ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) lim_ℂ 𝐵))))
60		taylthlem1.b	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
61		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
62		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) = (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵))
63	61, 62	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)) = ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)))
64		eqid 2769	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))) = (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
65		ovex 7444	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)) ∈ V
66	63, 64, 65	fvmpt 6990	. . . . . . . . . . . 12 ⊢ (𝐵 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)))
67	60, 66	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)))
68		taylthlem1.s	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
69		taylthlem1.f	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
70		taylthlem1.a	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
71	1	nnnn0d 12564	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
72		nn0uz 12899	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
73	71, 72	eleqtrdi 2879	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
74		eluzfz2b 13560	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ 𝑁 ∈ (0...𝑁))
75	73, 74	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
76		taylthlem1.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) = 𝐴)
77	60, 76	eleqtrrd 2872	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
78		taylthlem1.t	. . . . . . . . . . . . 13 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
79	68, 69, 70, 75, 77, 78	dvntaylp0 26500	. . . . . . . . . . . 12 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
80	79	oveq2d 7427	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)) = ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵)))
81		cnex 11180	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℂ ∈ V)
83		elpm2r 8841	. . . . . . . . . . . . . . 15 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
84	82, 68, 69, 70, 83	syl22anc 851	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
85		dvnf 26054	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)
86	68, 84, 71, 85	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)
87	86, 77	ffvelcdmd 7081	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) ∈ ℂ)
88	87	subidd 11556	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵)) = 0)
89	67, 80, 88	3eqtrd 2808	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = 0)
90		funmpt 6575	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
91		ovex 7444	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)) ∈ V
92	91, 64	dmmpti 6680	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))) = 𝐴
93	60, 92	eleqtrrdi 2880	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))))
94		funbrfvb 6935	. . . . . . . . . . 11 ⊢ ((Fun (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))) ∧ 𝐵 ∈ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))) → (((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = 0 ↔ 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0))
95	90, 93, 94	sylancr 598	. . . . . . . . . 10 ⊢ (𝜑 → (((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = 0 ↔ 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0))
96	89, 95	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0)
97		nnm1nn0 12544	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
98	1, 97	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
99		dvnf 26054	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ (𝑁 − 1) ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ)
100	68, 84, 98, 99	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ)
101		dvnbss 26055	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ (𝑁 − 1) ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ dom 𝐹)
102	68, 84, 98, 101	syl3anc 1396	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ dom 𝐹)
103	69, 102	fssdmd 6725	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ 𝐴)
104		fzo0end 13786	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
105		elfzofz 13703	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ (0..^𝑁) → (𝑁 − 1) ∈ (0...𝑁))
106	1, 104, 105	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...𝑁))
107		dvn2bss 26057	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ (𝑁 − 1) ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
108	68, 84, 106, 107	syl3anc 1396	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
109	76, 108	eqsstrrd 3980	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
110	103, 109	eqssd 3962	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = 𝐴)
111	110	feq2d 6690	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ ↔ ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ))
112	100, 111	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ)
113	112	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
114	76	feq2d 6690	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ ↔ ((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ))
115	86, 114	mpbid 235	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ)
116	115	ffvelcdmda 7080	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) ∈ ℂ)
117	1	nncnd 12248	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
118		1cnd 11201	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
119	117, 118	npcand 11572	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
120	119	fveq2d 6886	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
121		recnprss 26031	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
122	68, 121	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
123		dvnp1 26052	. . . . . . . . . . . . . 14 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ (𝑁 − 1) ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
124	122, 84, 98, 123	syl3anc 1396	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
125	120, 124	eqtr3d 2806	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
126	115	feqmptd 6950	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
127	112	feqmptd 6950	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦)))
128	127	oveq2d 7427	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = (𝑆 D (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦))))
129	125, 126, 128	3eqtr3rd 2813	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
130	70, 122	sstrd 3955	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
131	130	sselda 3945	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
132		1nn0 12519	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℕ₀)
134		elpm2r 8841	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆))
135	82, 68, 112, 70, 134	syl22anc 851	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆))
136		dvn1 26053	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑆 ⊆ ℂ ∧ ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
137	122, 135, 136	syl2anc 595	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
138	124, 120	eqtr3d 2806	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
139	137, 138	eqtrd 2804	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = ((𝑆 D^𝑛 𝐹)‘𝑁))
140	139	dmeqd 5896	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
141	77, 140	eleqtrrd 2872	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1))
142		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵)
143	68, 112, 70, 133, 141, 142	taylpf 26494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ)
144	118, 117	pncan3d 11571	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
145	144	oveq1d 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
146	78, 145	eqtr4id 2823	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 = ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))
147	146	oveq2d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 𝑇) = (ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵)))
148	147	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)))
149	144	fveq2d 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
150	149	dmeqd 5896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
151	77, 150	eleqtrrd 2872	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))))
152	68, 69, 70, 98, 133, 151	dvntaylp 26499	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵))
153	148, 152	eqtrd 2804	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵))
154	153	feq1d 6688	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1)):ℂ⟶ℂ ↔ (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ))
155	143, 154	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)):ℂ⟶ℂ)
156	155	ffvelcdmda 7080	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
157	131, 156	syldan 602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
158		0nn0 12518	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
159	158	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℕ₀)
160		elpm2r 8841	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
161	82, 68, 115, 70, 160	syl22anc 851	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
162		dvn0 26051	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑆 ⊆ ℂ ∧ ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑁))
163	122, 161, 162	syl2anc 595	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑁))
164	163	dmeqd 5896	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
165	77, 164	eleqtrrd 2872	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0))
166		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵)
167	68, 115, 70, 159, 165, 166	taylpf 26494	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ)
168	117	addlidd 11410	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
169	168	oveq1d 7426	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
170	169, 78	eqtr4di 2822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
171	170	oveq2d 7427	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
172	171	fveq1d 6884	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = ((ℂ D^𝑛 𝑇)‘𝑁))
173	168	fveq2d 6886	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
174	173	dmeqd 5896	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
175	77, 174	eleqtrrd 2872	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)))
176	68, 69, 70, 71, 159, 175	dvntaylp 26499	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵))
177	172, 176	eqtr3d 2806	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵))
178	177	feq1d 6688	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ ↔ (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ))
179	167, 178	mpbird 260	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ)
180	179	ffvelcdmda 7080	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
181	131, 180	syldan 602	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
182	122	sselda 3945	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
183	182, 156	syldan 602	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
184	182, 180	syldan 602	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
185		eqid 2769	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
186	185	cnfldtopon 24907	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
187		toponmax 23051	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
188	186, 187	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
189		dfss2 3931	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆)
190	122, 189	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆)
191		ssid 3967	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ⊆ ℂ
192	191	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℂ ⊆ ℂ)
193		mapsspm 8873	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ↑_m ℂ) ⊆ (ℂ ↑_pm ℂ)
194	68, 69, 70, 71, 77, 78	taylpf 26494	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
195	81, 81	elmap 8868	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ (ℂ ↑_m ℂ) ↔ 𝑇:ℂ⟶ℂ)
196	194, 195	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_m ℂ))
197	193, 196	sselid 3943	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
198		dvnp1 26052	. . . . . . . . . . . . . . . 16 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ) ∧ (𝑁 − 1) ∈ ℕ₀) → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))))
199	192, 197, 98, 198	syl3anc 1396	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))))
200	119	fveq2d 6886	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = ((ℂ D^𝑛 𝑇)‘𝑁))
201	199, 200	eqtr3d 2806	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = ((ℂ D^𝑛 𝑇)‘𝑁))
202	155	feqmptd 6950	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))
203	202	oveq2d 7427	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))
204	179	feqmptd 6950	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
205	201, 203, 204	3eqtr3d 2812	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
206	185, 68, 188, 190, 156, 180, 205	dvmptres3 26083	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
207		eqid 2769	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
208		resttopon 23286	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
209	186, 122, 208	sylancr 598	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
210		topontop 23038	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
211	209, 210	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
212		toponuni 23039	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
213	209, 212	syl 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
214	70, 213	sseqtrd 3981	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
215		eqid 2769	. . . . . . . . . . . . . . 15 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
216	215	ntrss2 23182	. . . . . . . . . . . . . 14 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top ∧ 𝐴 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)) → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
217	211, 214, 216	syl2anc 595	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
218	138	dmeqd 5896	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
219	218, 76	eqtrd 2804	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = 𝐴)
220	122, 112, 70, 207, 185	dvbssntr 26027	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
221	219, 220	eqsstrrd 3980	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
222	217, 221	eqssd 3962	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) = 𝐴)
223	68, 183, 184, 206, 70, 207, 185, 222	dvmptres2 26089	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
224	68, 113, 116, 129, 157, 181, 223	dvmptsub 26094	. . . . . . . . . 10 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))) = (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))))
225	224	breqd 5124	. . . . . . . . 9 ⊢ (𝜑 → (𝐵(𝑆 D (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))0 ↔ 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0))
226	96, 225	mpbird 260	. . . . . . . 8 ⊢ (𝜑 → 𝐵(𝑆 D (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))0)
227		eqid 2769	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵)))
228	113, 157	subcld 11568	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) ∈ ℂ)
229	228	fmpttd 7111	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))):𝐴⟶ℂ)
230	207, 185, 227, 122, 229, 70	eldv 26025	. . . . . . . 8 ⊢ (𝜑 → (𝐵(𝑆 D (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))0 ↔ (𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ∧ 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵))) lim_ℂ 𝐵))))
231	226, 230	mpbid 235	. . . . . . 7 ⊢ (𝜑 → (𝐵 ∈ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ∧ 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵))) lim_ℂ 𝐵)))
232	231	simprd 500	. . . . . 6 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵))) lim_ℂ 𝐵))
233		eldifi 4093	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
234		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
235		fveq2 6882	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
236	234, 235	oveq12d 7429	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)))
237		eqid 2769	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))
238		ovex 7444	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ V
239	236, 237, 238	fvmpt 6990	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)))
240		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
241		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵))
242	240, 241	oveq12d 7429	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝐵 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)))
243		ovex 7444	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)) ∈ V
244	242, 237, 243	fvmpt 6990	. . . . . . . . . . . . . 14 ⊢ (𝐵 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)))
245	60, 244	syl 18	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)))
246	68, 69, 70, 106, 77, 78	dvntaylp0 26500	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
247	246	oveq2d 7427	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵)))
248	112, 60	ffvelcdmd 7081	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) ∈ ℂ)
249	248	subidd 11556	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵)) = 0)
250	245, 247, 249	3eqtrd 2808	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵) = 0)
251	239, 250	oveqan12rd 7431	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) = (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) − 0))
252	112	ffvelcdmda 7080	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
253	130	sselda 3945	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
254	155	ffvelcdmda 7080	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
255	253, 254	syldan 602	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
256	252, 255	subcld 11568	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ ℂ)
257	256	subid1d 11557	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) − 0) = ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)))
258	251, 257	eqtr2d 2805	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) = (((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)))
259	233, 258	sylan2 604	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) = (((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)))
260	130	ssdifssd 4109	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
261	260	sselda 3945	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
262	130, 60	sseldd 3946	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℂ)
263	262	adantr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
264	261, 263	subcld 11568	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
265	264	exp1d 14176	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑1) = (𝑥 − 𝐵))
266	259, 265	oveq12d 7429	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1)) = ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵)))
267	266	mpteq2dva 5208	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵))))
268	267	oveq1d 7426	. . . . . 6 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝑥) − ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵)) / (𝑥 − 𝐵))) lim_ℂ 𝐵))
269	232, 268	eleqtrrd 2872	. . . . 5 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) lim_ℂ 𝐵))
270	269	a1i 11	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘1) → (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) / ((𝑥 − 𝐵)↑1))) lim_ℂ 𝐵)))
271		taylthlem1.i	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑛 ∈ (1..^𝑁) ∧ 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵))) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵))
272	271	expr 461	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (1..^𝑁)) → (0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵)))
273	272	expcom 418	. . . . 5 ⊢ (𝑛 ∈ (1..^𝑁) → (𝜑 → (0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵) → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵))))
274	273	a2d 30	. . . 4 ⊢ (𝑛 ∈ (1..^𝑁) → ((𝜑 → 0 ∈ ((𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦)) / ((𝑦 − 𝐵)↑𝑛))) lim_ℂ 𝐵)) → (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑛 + 1)))) lim_ℂ 𝐵))))
275	15, 35, 47, 59, 270, 274	fzind2 13816	. . 3 ⊢ (𝑁 ∈ (1...𝑁) → (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) lim_ℂ 𝐵)))
276	3, 275	mpcom 39	. 2 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) lim_ℂ 𝐵))
277	117	subidd 11556	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
278	277	fveq2d 6886	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = ((𝑆 D^𝑛 𝐹)‘0))
279		dvn0 26051	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
280	122, 84, 279	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
281	278, 280	eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = 𝐹)
282	281	fveq1d 6884	. . . . . . 7 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) = (𝐹‘𝑥))
283	277	fveq2d 6886	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = ((ℂ D^𝑛 𝑇)‘0))
284		dvn0 26051	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
285	191, 197, 284	sylancr 598	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
286	283, 285	eqtrd 2804	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = 𝑇)
287	286	fveq1d 6884	. . . . . . 7 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥) = (𝑇‘𝑥))
288	282, 287	oveq12d 7429	. . . . . 6 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) = ((𝐹‘𝑥) − (𝑇‘𝑥)))
289	288	oveq1d 7426	. . . . 5 ⊢ (𝜑 → (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁)) = (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁)))
290	289	mpteq2dv 5209	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))))
291		taylthlem1.r	. . . 4 ⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁)))
292	290, 291	eqtr4di 2822	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) = 𝑅)
293	292	oveq1d 7426	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) lim_ℂ 𝐵) = (𝑅 lim_ℂ 𝐵))
294	276, 293	eleqtrd 2871	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 = wceq 1567 ∈ wcel 2149 Vcvv 3463 ∖ cdif 3910 ∩ cin 3912 ⊆ wss 3913 {csn 4594 {cpr 4596 ∪ cuni 4876 class class class wbr 5113 ↦ cmpt 5196 dom cdm 5662 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7411 ↑_m cmap 8823 ↑_pm cpm 8824 ℂcc 11097 ℝcr 11098 0cc0 11099 1c1 11100 + caddc 11102 − cmin 11440 / cdiv 11870 ℕcn 12232 ℕ₀cn0 12503 ℤ_≥cuz 12861 ...cfz 13534 ..^cfzo 13681 ↑cexp 14096 ↾_t crest 17472 TopOpenctopn 17473 ℂ_fldccnfld 21490 Topctop 23018 TopOnctopon 23035 intcnt 23142 lim_ℂ climc 25989 D cdv 25990 D^𝑛 cdvn 25991 Tayl ctayl 26481

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1822 ax-4 1836 ax-5 1937 ax-6 1994 ax-7 2035 ax-8 2151 ax-9 2159 ax-10 2182 ax-11 2198 ax-12 2219 ax-ext 2741 ax-rep 5242 ax-sep 5261 ax-nul 5271 ax-pow 5337 ax-pr 5405 ax-un 7733 ax-inf2 9609 ax-cnex 11155 ax-resscn 11156 ax-1cn 11157 ax-icn 11158 ax-addcl 11159 ax-addrcl 11160 ax-mulcl 11161 ax-mulrcl 11162 ax-mulcom 11163 ax-addass 11164 ax-mulass 11165 ax-distr 11166 ax-i2m1 11167 ax-1ne0 11168 ax-1rid 11169 ax-rnegex 11170 ax-rrecex 11171 ax-cnre 11172 ax-pre-lttri 11173 ax-pre-lttrn 11174 ax-pre-ltadd 11175 ax-pre-mulgt0 11176 ax-pre-sup 11177 ax-addf 11178

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1102 df-3an 1103 df-tru 1570 df-fal 1580 df-ex 1807 df-nf 1811 df-sb 2098 df-mo 2573 df-eu 2603 df-clab 2748 df-cleq 2761 df-clel 2844 df-nfc 2918 df-ne 2965 df-nel 3071 df-ral 3086 df-rex 3096 df-rmo 3376 df-reu 3377 df-rab 3424 df-v 3465 df-sbc 3754 df-csb 3862 df-dif 3916 df-un 3918 df-in 3920 df-ss 3930 df-pss 3933 df-nul 4295 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-tp 4599 df-op 4601 df-uni 4877 df-int 4917 df-iun 4962 df-iin 4963 df-br 5114 df-opab 5178 df-mpt 5197 df-tr 5223 df-id 5557 df-eprel 5562 df-po 5570 df-so 5571 df-fr 5615 df-se 5616 df-we 5617 df-xp 5668 df-rel 5669 df-cnv 5670 df-co 5671 df-dm 5672 df-rn 5673 df-res 5674 df-ima 5675 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7368 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7675 df-om 7862 df-1st 7985 df-2nd 7986 df-supp 8156 df-frecs 8277 df-wrecs 8308 df-recs 8357 df-rdg 8396 df-1o 8452 df-2o 8453 df-er 8693 df-map 8825 df-pm 8826 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9321 df-fi 9370 df-sup 9401 df-inf 9402 df-oi 9471 df-card 9924 df-pnf 11244 df-mnf 11245 df-xr 11246 df-ltxr 11247 df-le 11248 df-sub 11442 df-neg 11443 df-div 11871 df-nn 12233 df-2 12302 df-3 12303 df-4 12304 df-5 12305 df-6 12306 df-7 12307 df-8 12308 df-9 12309 df-n0 12504 df-xnn0 12577 df-z 12591 df-dec 12711 df-uz 12862 df-q 12972 df-rp 13016 df-xneg 13136 df-xadd 13137 df-xmul 13138 df-icc 13378 df-fz 13535 df-fzo 13682 df-seq 14037 df-exp 14097 df-fac 14309 df-hash 14366 df-cj 15149 df-re 15150 df-im 15151 df-sqrt 15285 df-abs 15286 df-clim 15538 df-sum 15737 df-struct 17206 df-sets 17223 df-slot 17241 df-ndx 17253 df-base 17269 df-ress 17290 df-plusg 17322 df-mulr 17323 df-starv 17324 df-sca 17325 df-vsca 17326 df-ip 17327 df-tset 17328 df-ple 17329 df-ds 17331 df-unif 17332 df-hom 17333 df-cco 17334 df-rest 17474 df-topn 17475 df-0g 17493 df-gsum 17494 df-topgen 17495 df-pt 17496 df-prds 17499 df-xrs 17555 df-qtop 17560 df-imas 17561 df-xps 17563 df-mre 17637 df-mrc 17638 df-acs 17640 df-mgm 18697 df-sgrp 18776 df-mnd 18792 df-submnd 18841 df-grp 19002 df-minusg 19003 df-mulg 19133 df-cntz 19386 df-cmn 19851 df-abl 19852 df-mgp 20216 df-ur 20263 df-ring 20316 df-cring 20317 df-psmet 21482 df-xmet 21483 df-met 21484 df-bl 21485 df-mopn 21486 df-fbas 21487 df-fg 21488 df-cnfld 21491 df-top 23019 df-topon 23036 df-topsp 23058 df-bases 23071 df-cld 23144 df-ntr 23145 df-cls 23146 df-nei 23223 df-lp 23261 df-perf 23262 df-cn 23352 df-cnp 23353 df-haus 23440 df-tx 23687 df-hmeo 23880 df-fil 23971 df-fm 24063 df-flim 24064 df-flf 24065 df-tsms 24252 df-xms 24445 df-ms 24446 df-tms 24447 df-cncf 25005 df-limc 25993 df-dv 25994 df-dvn 25995 df-tayl 26483

This theorem is referenced by: taylth 26503

W3C validator