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Theorem taylthlem1 24127

Description: Lemma for taylth 24129. 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 24129 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 12371	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
3	1, 2	sylib 208	. . 3 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
4		oveq2 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑁 − 𝑚) = (𝑁 − 1))
5	4	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
6	5	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
7	4	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)))
8	7	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
9	6, 8	oveq12d 6668	. . . . . . . . 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 6658	. . . . . . . . 9 ⊢ (𝑚 = 1 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑1))
11	9, 10	oveq12d 6668	. . . . . . . 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 4745	. . . . . . 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 6665	. . . . . 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 2687	. . . . 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 330	. . . 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 6658	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑁 − 𝑚) = (𝑁 − 𝑛))
17	16	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛)))
18	17	fveq1d 6193	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥))
19	16	fveq2d 6195	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛)))
20	19	fveq1d 6193	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥))
21	18, 20	oveq12d 6668	. . . . . . . . . 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 6658	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑛))
23	21, 22	oveq12d 6668	. . . . . . . . 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 4745	. . . . . . . 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 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦))
26		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦))
27	25, 26	oveq12d 6668	. . . . . . . . . 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 6657	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐵) = (𝑦 − 𝐵))
29	28	oveq1d 6665	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐵)↑𝑛) = ((𝑦 − 𝐵)↑𝑛))
30	27, 29	oveq12d 6668	. . . . . . . . 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 4750	. . . . . . . 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	syl6eq 2672	. . . . . . 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 6665	. . . . . 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 2687	. . . . 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 330	. . . 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 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑁 − 𝑚) = (𝑁 − (𝑛 + 1)))
37	36	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1))))
38	37	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥))
39	36	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1))))
40	39	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥))
41	38, 40	oveq12d 6668	. . . . . . . . 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 6658	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑(𝑛 + 1)))
43	41, 42	oveq12d 6668	. . . . . . . 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 4745	. . . . . . 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 6665	. . . . . 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 2687	. . . . 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 330	. . . 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 6658	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝑁 − 𝑚) = (𝑁 − 𝑁))
49	48	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)))
50	49	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥))
51	48	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)))
52	51	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥))
53	50, 52	oveq12d 6668	. . . . . . . . 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 6658	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑁))
55	53, 54	oveq12d 6668	. . . . . . . 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 4745	. . . . . . 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 6665	. . . . . 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 2687	. . . . 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 330	. . . 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 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
62		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) = (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵))
63	61, 62	oveq12d 6668	. . . . . . . . . . . . 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 2622	. . . . . . . . . . . . 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 6678	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)) ∈ V
66	63, 64, 65	fvmpt 6282	. . . . . . . . . . . 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 17	. . . . . . . . . . 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 11351	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
72		nn0uz 11722	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
73	71, 72	syl6eleq 2711	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
74		eluzfz2b 12350	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ 𝑁 ∈ (0...𝑁))
75	73, 74	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
76		taylthlem1.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) = 𝐴)
77	60, 76	eleqtrrd 2704	. . . . . . . . . . . . 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 24126	. . . . . . . . . . . 12 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
80	79	oveq2d 6666	. . . . . . . . . . 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 10017	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℂ ∈ V)
83		elpm2r 7875	. . . . . . . . . . . . . . 15 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
84	82, 68, 69, 70, 83	syl22anc 1327	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
85		dvnf 23690	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)
87	86, 77	ffvelrnd 6360	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) ∈ ℂ)
88	87	subidd 10380	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵)) = 0)
89	67, 80, 88	3eqtrd 2660	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = 0)
90		funmpt 5926	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
91		ovex 6678	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)) ∈ V
92	91, 64	dmmpti 6023	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))) = 𝐴
93	60, 92	syl6eleqr 2712	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))))
94		funbrfvb 6238	. . . . . . . . . . 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 695	. . . . . . . . . 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 222	. . . . . . . . 9 ⊢ (𝜑 → 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0)
97		nnm1nn0 11334	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
98	1, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
99		dvnf 23690	. . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ)
101		dvnbss 23691	. . . . . . . . . . . . . . . . 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 1326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ dom 𝐹)
103		fdm 6051	. . . . . . . . . . . . . . . . 17 ⊢ (𝐹:𝐴⟶ℂ → dom 𝐹 = 𝐴)
104	69, 103	syl 17	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom 𝐹 = 𝐴)
105	102, 104	sseqtrd 3641	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ 𝐴)
106		fzo0end 12560	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
107		elfzofz 12485	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ (0..^𝑁) → (𝑁 − 1) ∈ (0...𝑁))
108	1, 106, 107	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...𝑁))
109		dvn2bss 23693	. . . . . . . . . . . . . . . . 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)))
110	68, 84, 108, 109	syl3anc 1326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
111	76, 110	eqsstr3d 3640	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
112	105, 111	eqssd 3620	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = 𝐴)
113	112	feq2d 6031	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ ↔ ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ))
114	100, 113	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ)
115	114	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
116	76	feq2d 6031	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ ↔ ((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ))
117	86, 116	mpbid 222	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ)
118	117	ffvelrnda 6359	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) ∈ ℂ)
119	1	nncnd 11036	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
120		1cnd 10056	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
121	119, 120	npcand 10396	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
122	121	fveq2d 6195	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
123		recnprss 23668	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
124	68, 123	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
125		dvnp1 23688	. . . . . . . . . . . . . 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))))
126	124, 84, 98, 125	syl3anc 1326	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
127	122, 126	eqtr3d 2658	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
128	117	feqmptd 6249	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
129	114	feqmptd 6249	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦)))
130	129	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = (𝑆 D (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦))))
131	127, 128, 130	3eqtr3rd 2665	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
132	70, 124	sstrd 3613	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
133	132	sselda 3603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
134		1nn0 11308	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
135	134	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℕ₀)
136		elpm2r 7875	. . . . . . . . . . . . . . . . . . . 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))
137	82, 68, 114, 70, 136	syl22anc 1327	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆))
138		dvn1 23689	. . . . . . . . . . . . . . . . . . 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))))
139	124, 137, 138	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
140	126, 122	eqtr3d 2658	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
141	139, 140	eqtrd 2656	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = ((𝑆 D^𝑛 𝐹)‘𝑁))
142	141	dmeqd 5326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
143	77, 142	eleqtrrd 2704	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1))
144		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵)
145	68, 114, 70, 135, 143, 144	taylpf 24120	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ)
146	120, 119	pncan3d 10395	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
147	146	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
148	147, 78	syl6reqr 2675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 = ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))
149	148	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 𝑇) = (ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵)))
150	149	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)))
151	146	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
152	151	dmeqd 5326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
153	77, 152	eleqtrrd 2704	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))))
154	68, 69, 70, 98, 135, 153	dvntaylp 24125	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵))
155	150, 154	eqtrd 2656	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵))
156	155	feq1d 6030	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1)):ℂ⟶ℂ ↔ (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ))
157	145, 156	mpbird 247	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)):ℂ⟶ℂ)
158	157	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
159	133, 158	syldan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
160		0nn0 11307	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
161	160	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℕ₀)
162		elpm2r 7875	. . . . . . . . . . . . . . . . . . 19 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
163	82, 68, 117, 70, 162	syl22anc 1327	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
164		dvn0 23687	. . . . . . . . . . . . . . . . . 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))
165	124, 163, 164	syl2anc 693	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑁))
166	165	dmeqd 5326	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
167	77, 166	eleqtrrd 2704	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0))
168		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵)
169	68, 117, 70, 161, 167, 168	taylpf 24120	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ)
170	119	addid2d 10237	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
171	170	oveq1d 6665	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
172	171, 78	syl6eqr 2674	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
173	172	oveq2d 6666	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
174	173	fveq1d 6193	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = ((ℂ D^𝑛 𝑇)‘𝑁))
175	170	fveq2d 6195	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
176	175	dmeqd 5326	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
177	77, 176	eleqtrrd 2704	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)))
178	68, 69, 70, 71, 161, 177	dvntaylp 24125	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵))
179	174, 178	eqtr3d 2658	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵))
180	179	feq1d 6030	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ ↔ (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ))
181	169, 180	mpbird 247	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ)
182	181	ffvelrnda 6359	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
183	133, 182	syldan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
184	124	sselda 3603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
185	184, 158	syldan 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
186	184, 182	syldan 487	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
187		eqid 2622	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
188	187	cnfldtopon 22586	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
189		toponmax 20730	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
190	188, 189	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
191		df-ss 3588	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆)
192	124, 191	sylib 208	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆)
193		ssid 3624	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ⊆ ℂ
194	193	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℂ ⊆ ℂ)
195		mapsspm 7891	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ↑_𝑚 ℂ) ⊆ (ℂ ↑_pm ℂ)
196	68, 69, 70, 71, 77, 78	taylpf 24120	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
197	81, 81	elmap 7886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ (ℂ ↑_𝑚 ℂ) ↔ 𝑇:ℂ⟶ℂ)
198	196, 197	sylibr 224	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_𝑚 ℂ))
199	195, 198	sseldi 3601	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
200		dvnp1 23688	. . . . . . . . . . . . . . . 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))))
201	194, 199, 98, 200	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))))
202	121	fveq2d 6195	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = ((ℂ D^𝑛 𝑇)‘𝑁))
203	201, 202	eqtr3d 2658	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = ((ℂ D^𝑛 𝑇)‘𝑁))
204	157	feqmptd 6249	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))
205	204	oveq2d 6666	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))
206	181	feqmptd 6249	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
207	203, 205, 206	3eqtr3d 2664	. . . . . . . . . . . . 13 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
208	187, 68, 190, 192, 158, 182, 207	dvmptres3 23719	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
209		eqid 2622	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
210		resttopon 20965	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
211	188, 124, 210	sylancr 695	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
212		topontop 20718	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
213	211, 212	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
214		toponuni 20719	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
215	211, 214	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
216	70, 215	sseqtrd 3641	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
217		eqid 2622	. . . . . . . . . . . . . . 15 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
218	217	ntrss2 20861	. . . . . . . . . . . . . 14 ⊢ ((((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top ∧ 𝐴 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)) → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
219	213, 216, 218	syl2anc 693	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
220	140	dmeqd 5326	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
221	220, 76	eqtrd 2656	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = 𝐴)
222	124, 114, 70, 209, 187	dvbssntr 23664	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
223	221, 222	eqsstr3d 3640	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
224	219, 223	eqssd 3620	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) = 𝐴)
225	68, 185, 186, 208, 70, 209, 187, 224	dvmptres2 23725	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
226	68, 115, 118, 131, 159, 183, 225	dvmptsub 23730	. . . . . . . . . 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))))
227	226	breqd 4664	. . . . . . . . 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))
228	96, 227	mpbird 247	. . . . . . . 8 ⊢ (𝜑 → 𝐵(𝑆 D (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))0)
229		eqid 2622	. . . . . . . . 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)))
230	115, 159	subcld 10392	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) ∈ ℂ)
231		eqid 2622	. . . . . . . . . 10 ⊢ (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)))
232	230, 231	fmptd 6385	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))):𝐴⟶ℂ)
233	209, 187, 229, 124, 232, 70	eldv 23662	. . . . . . . 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))))
234	228, 233	mpbid 222	. . . . . . 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)))
235	234	simprd 479	. . . . . 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))
236		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
237		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
238		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
239	237, 238	oveq12d 6668	. . . . . . . . . . . . 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)))
240		ovex 6678	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ V
241	239, 231, 240	fvmpt 6282	. . . . . . . . . . . 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)))
242		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
243		fveq2 6191	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵))
244	242, 243	oveq12d 6668	. . . . . . . . . . . . . . 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)))
245		ovex 6678	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)) ∈ V
246	244, 231, 245	fvmpt 6282	. . . . . . . . . . . . . 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)))
247	60, 246	syl 17	. . . . . . . . . . . . 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)))
248	68, 69, 70, 108, 77, 78	dvntaylp0 24126	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
249	248	oveq2d 6666	. . . . . . . . . . . . 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)))
250	114, 60	ffvelrnd 6360	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) ∈ ℂ)
251	250	subidd 10380	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵)) = 0)
252	247, 249, 251	3eqtrd 2660	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵) = 0)
253	241, 252	oveqan12rd 6670	. . . . . . . . . . 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))
254	114	ffvelrnda 6359	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
255	132	sselda 3603	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
256	157	ffvelrnda 6359	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
257	255, 256	syldan 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
258	254, 257	subcld 10392	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ ℂ)
259	258	subid1d 10381	. . . . . . . . . . 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)))
260	253, 259	eqtr2d 2657	. . . . . . . . . 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)))
261	236, 260	sylan2 491	. . . . . . . . 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)))
262	132	ssdifssd 3748	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
263	262	sselda 3603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
264	132, 60	sseldd 3604	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℂ)
265	264	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
266	263, 265	subcld 10392	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
267	266	exp1d 13003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑1) = (𝑥 − 𝐵))
268	261, 267	oveq12d 6668	. . . . . . . 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)))
269	268	mpteq2dva 4744	. . . . . . 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))))
270	269	oveq1d 6665	. . . . . 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))
271	235, 270	eleqtrrd 2704	. . . . 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))
272	271	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)))
273		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))
274	273	expr 643	. . . . . 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)))
275	274	expcom 451	. . . . 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))))
276	275	a2d 29	. . . 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))))
277	15, 35, 47, 59, 272, 276	fzind2 12586	. . 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)))
278	3, 277	mpcom 38	. 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))
279	119	subidd 10380	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
280	279	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = ((𝑆 D^𝑛 𝐹)‘0))
281		dvn0 23687	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
282	124, 84, 281	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
283	280, 282	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = 𝐹)
284	283	fveq1d 6193	. . . . . . 7 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) = (𝐹‘𝑥))
285	279	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = ((ℂ D^𝑛 𝑇)‘0))
286		dvn0 23687	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
287	193, 199, 286	sylancr 695	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
288	285, 287	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = 𝑇)
289	288	fveq1d 6193	. . . . . . 7 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥) = (𝑇‘𝑥))
290	284, 289	oveq12d 6668	. . . . . 6 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) = ((𝐹‘𝑥) − (𝑇‘𝑥)))
291	290	oveq1d 6665	. . . . 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)))
292	291	mpteq2dv 4745	. . . 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))))
293		taylthlem1.r	. . . 4 ⊢ 𝑅 = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝐹‘𝑥) − (𝑇‘𝑥)) / ((𝑥 − 𝐵)↑𝑁)))
294	292, 293	syl6eqr 2674	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) / ((𝑥 − 𝐵)↑𝑁))) = 𝑅)
295	294	oveq1d 6665	. 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))
296	278, 295	eleqtrd 2703	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 {csn 4177 {cpr 4179 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ↑_pm cpm 7858 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤ_≥cuz 11687 ...cfz 12326 ..^cfzo 12465 ↑cexp 12860 ↾_t crest 16081 TopOpenctopn 16082 ℂ_fldccnfld 19746 Topctop 20698 TopOnctopon 20715 intcnt 20821 lim_ℂ climc 23626 D cdv 23627 D^𝑛 cdvn 23628 Tayl ctayl 24107

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-limc 23630 df-dv 23631 df-dvn 23632 df-tayl 24109

This theorem is referenced by: taylth 24129

W3C validator