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Theorem taylthlem1 26693

Description: Lemma for taylth 26695. 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 26695 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 13681	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
3	1, 2	sylib 221	. . 3 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
4		oveq2 7426	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑁 − 𝑚) = (𝑁 − 1))
5	4	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
6	5	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
7	4	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)))
8	7	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
9	6, 8	oveq12d 7436	. . . . . . . . 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 7426	. . . . . . . . 9 ⊢ (𝑚 = 1 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑1))
11	9, 10	oveq12d 7436	. . . . . . . 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 5199	. . . . . . 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 7433	. . . . . 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 2847	. . . . 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 7426	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑁 − 𝑚) = (𝑁 − 𝑛))
17	16	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛)))
18	17	fveq1d 6885	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥))
19	16	fveq2d 6887	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛)))
20	19	fveq1d 6885	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥))
21	18, 20	oveq12d 7436	. . . . . . . . . 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 7426	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑛))
23	21, 22	oveq12d 7436	. . . . . . . . 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 5199	. . . . . . . 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 6883	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦))
26		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦))
27	25, 26	oveq12d 7436	. . . . . . . . . 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 7425	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐵) = (𝑦 − 𝐵))
29	28	oveq1d 7433	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐵)↑𝑛) = ((𝑦 − 𝐵)↑𝑛))
30	27, 29	oveq12d 7436	. . . . . . . . 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 5209	. . . . . . . 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 2812	. . . . . . 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 7433	. . . . . 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 2847	. . . . 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 7426	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑁 − 𝑚) = (𝑁 − (𝑛 + 1)))
37	36	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1))))
38	37	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥))
39	36	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1))))
40	39	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥))
41	38, 40	oveq12d 7436	. . . . . . . . 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 7426	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑(𝑛 + 1)))
43	41, 42	oveq12d 7436	. . . . . . . 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 5199	. . . . . . 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 7433	. . . . . 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 2847	. . . . 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 7426	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝑁 − 𝑚) = (𝑁 − 𝑁))
49	48	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)))
50	49	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥))
51	48	fveq2d 6887	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)))
52	51	fveq1d 6885	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥))
53	50, 52	oveq12d 7436	. . . . . . . . 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 7426	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑁))
55	53, 54	oveq12d 7436	. . . . . . . 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 5199	. . . . . . 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 7433	. . . . . 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 2847	. . . . 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 6883	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
62		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) = (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵))
63	61, 62	oveq12d 7436	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . . 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 7451	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)) ∈ V
66	63, 64, 65	fvmpt 6991	. . . . . . . . . . . 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 12660	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
72		nn0uz 12996	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
73	71, 72	eleqtrdi 2871	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
74		eluzfz2b 13659	. . . . . . . . . . . . . 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 2864	. . . . . . . . . . . . 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 26692	. . . . . . . . . . . 12 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
80	79	oveq2d 7434	. . . . . . . . . . 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 11274	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℂ ∈ V)
83		elpm2r 8858	. . . . . . . . . . . . . . 15 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
84	82, 68, 69, 70, 83	syl22anc 852	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
85		dvnf 26240	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)
87	86, 77	ffvelcdmd 7083	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) ∈ ℂ)
88	87	subidd 11650	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵)) = 0)
89	67, 80, 88	3eqtrd 2800	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = 0)
90		funmpt 6576	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
91		ovex 7451	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)) ∈ V
92	91, 64	dmmpti 6681	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))) = 𝐴
93	60, 92	eleqtrrdi 2872	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))))
94		funbrfvb 6936	. . . . . . . . . . 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 599	. . . . . . . . . 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 12640	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
98	1, 97	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
99		dvnf 26240	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ)
101		dvnbss 26241	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ dom 𝐹)
103	69, 102	fssdmd 6726	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ 𝐴)
104		fzo0end 13886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
105		elfzofz 13803	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ (0..^𝑁) → (𝑁 − 1) ∈ (0...𝑁))
106	1, 104, 105	3syl 19	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...𝑁))
107		dvn2bss 26243	. . . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
109	76, 108	eqsstrrd 3966	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
110	103, 109	eqssd 3948	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = 𝐴)
111	110	feq2d 6691	. . . . . . . . . . . . 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 7082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
114	76	feq2d 6691	. . . . . . . . . . . . 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 7082	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) ∈ ℂ)
117	1	nncnd 12344	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
118		1cnd 11295	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
119	117, 118	npcand 11666	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
120	119	fveq2d 6887	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
121		recnprss 26217	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
122	68, 121	syl 18	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
123		dvnp1 26238	. . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
125	120, 124	eqtr3d 2798	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
126	115	feqmptd 6951	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
127	112	feqmptd 6951	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦)))
128	127	oveq2d 7434	. . . . . . . . . . . 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 2805	. . . . . . . . . . 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 3941	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
131	130	sselda 3931	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
132		1nn0 12615	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℕ₀)
134		elpm2r 8858	. . . . . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆))
136		dvn1 26239	. . . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
138	124, 120	eqtr3d 2798	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
139	137, 138	eqtrd 2796	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = ((𝑆 D^𝑛 𝐹)‘𝑁))
140	139	dmeqd 5887	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
141	77, 140	eleqtrrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1))
142		eqid 2761	. . . . . . . . . . . . . . 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 26686	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ)
144	118, 117	pncan3d 11665	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
145	144	oveq1d 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
146	78, 145	eqtr4id 2815	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 = ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))
147	146	oveq2d 7434	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 𝑇) = (ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵)))
148	147	fveq1d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)))
149	144	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
150	149	dmeqd 5887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
151	77, 150	eleqtrrd 2864	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))))
152	68, 69, 70, 98, 133, 151	dvntaylp 26691	. . . . . . . . . . . . . . . 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 2796	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵))
154	153	feq1d 6689	. . . . . . . . . . . . . 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 7082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
157	131, 156	syldan 603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
158		0nn0 12614	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
159	158	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℕ₀)
160		elpm2r 8858	. . . . . . . . . . . . . . . . . . 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 852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
162		dvn0 26237	. . . . . . . . . . . . . . . . . 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 596	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑁))
164	163	dmeqd 5887	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
165	77, 164	eleqtrrd 2864	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0))
166		eqid 2761	. . . . . . . . . . . . . . 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 26686	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ)
168	117	addlidd 11504	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
169	168	oveq1d 7433	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
170	169, 78	eqtr4di 2814	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
171	170	oveq2d 7434	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
172	171	fveq1d 6885	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = ((ℂ D^𝑛 𝑇)‘𝑁))
173	168	fveq2d 6887	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
174	173	dmeqd 5887	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
175	77, 174	eleqtrrd 2864	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)))
176	68, 69, 70, 71, 159, 175	dvntaylp 26691	. . . . . . . . . . . . . . . 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 2798	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵))
178	177	feq1d 6689	. . . . . . . . . . . . . 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 7082	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
181	131, 180	syldan 603	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
182	122	sselda 3931	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
183	182, 156	syldan 603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
184	182, 180	syldan 603	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
185		eqid 2761	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
186	185	cnfldtopon 25094	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
187		toponmax 23237	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
188	186, 187	mp1i 14	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
189		dfss2 3917	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆)
190	122, 189	sylib 221	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆)
191		ssid 3953	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ⊆ ℂ
192	191	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℂ ⊆ ℂ)
193		mapsspm 8897	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ↑_m ℂ) ⊆ (ℂ ↑_pm ℂ)
194	68, 69, 70, 71, 77, 78	taylpf 26686	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
195	81, 81	elmap 8892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ (ℂ ↑_m ℂ) ↔ 𝑇:ℂ⟶ℂ)
196	194, 195	sylibr 237	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_m ℂ))
197	193, 196	sselid 3929	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
198		dvnp1 26238	. . . . . . . . . . . . . . . 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 1398	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))))
200	119	fveq2d 6887	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = ((ℂ D^𝑛 𝑇)‘𝑁))
201	199, 200	eqtr3d 2798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = ((ℂ D^𝑛 𝑇)‘𝑁))
202	155	feqmptd 6951	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))
203	202	oveq2d 7434	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))
204	179	feqmptd 6951	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
205	201, 203, 204	3eqtr3d 2804	. . . . . . . . . . . . 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 26269	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
207		eqid 2761	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
208		resttopon 23472	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
209	186, 122, 208	sylancr 599	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
210		topontop 23224	. . . . . . . . . . . . . . 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 23225	. . . . . . . . . . . . . . . 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 3967	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
215		eqid 2761	. . . . . . . . . . . . . . 15 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
216	215	ntrss2 23368	. . . . . . . . . . . . . 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 596	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
218	138	dmeqd 5887	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
219	218, 76	eqtrd 2796	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = 𝐴)
220	122, 112, 70, 207, 185	dvbssntr 26213	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
221	219, 220	eqsstrrd 3966	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
222	217, 221	eqssd 3948	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) = 𝐴)
223	68, 183, 184, 206, 70, 207, 185, 222	dvmptres2 26275	. . . . . . . . . . 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 26280	. . . . . . . . . 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 5114	. . . . . . . . 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 2761	. . . . . . . . 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 11662	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) ∈ ℂ)
229	228	fmpttd 7113	. . . . . . . . 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 26211	. . . . . . . 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 501	. . . . . 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 4078	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
234		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
235		fveq2 6883	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
236	234, 235	oveq12d 7436	. . . . . . . . . . . . 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 2761	. . . . . . . . . . . . 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 7451	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ V
239	236, 237, 238	fvmpt 6991	. . . . . . . . . . . 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 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
241		fveq2 6883	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵))
242	240, 241	oveq12d 7436	. . . . . . . . . . . . . . 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 7451	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)) ∈ V
244	242, 237, 243	fvmpt 6991	. . . . . . . . . . . . . 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 26692	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
247	246	oveq2d 7434	. . . . . . . . . . . . 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 7083	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) ∈ ℂ)
249	248	subidd 11650	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵)) = 0)
250	245, 247, 249	3eqtrd 2800	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵) = 0)
251	239, 250	oveqan12rd 7438	. . . . . . . . . . 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 7082	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
253	130	sselda 3931	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
254	155	ffvelcdmda 7082	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
255	253, 254	syldan 603	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
256	252, 255	subcld 11662	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ ℂ)
257	256	subid1d 11651	. . . . . . . . . . 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 2797	. . . . . . . . . 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 605	. . . . . . . . 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 4094	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
261	260	sselda 3931	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
262	130, 60	sseldd 3932	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℂ)
263	262	adantr 486	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
264	261, 263	subcld 11662	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
265	264	exp1d 14277	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑1) = (𝑥 − 𝐵))
266	259, 265	oveq12d 7436	. . . . . . . 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 5198	. . . . . . 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 7433	. . . . . 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 2864	. . . . 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 462	. . . . . 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 419	. . . . 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 13916	. . 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 11650	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
278	277	fveq2d 6887	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = ((𝑆 D^𝑛 𝐹)‘0))
279		dvn0 26237	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
280	122, 84, 279	syl2anc 596	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
281	278, 280	eqtrd 2796	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = 𝐹)
282	281	fveq1d 6885	. . . . . . 7 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) = (𝐹‘𝑥))
283	277	fveq2d 6887	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = ((ℂ D^𝑛 𝑇)‘0))
284		dvn0 26237	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
285	191, 197, 284	sylancr 599	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
286	283, 285	eqtrd 2796	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = 𝑇)
287	286	fveq1d 6885	. . . . . . 7 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥) = (𝑇‘𝑥))
288	282, 287	oveq12d 7436	. . . . . 6 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) = ((𝐹‘𝑥) − (𝑇‘𝑥)))
289	288	oveq1d 7433	. . . . 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 5199	. . . 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 2814	. . 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 7433	. 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 2863	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 ∈ wcel 2145 Vcvv 3451 ∖ cdif 3896 ∩ cin 3898 ⊆ wss 3899 {csn 4584 {cpr 4586 ∪ cuni 4867 class class class wbr 5103 ↦ cmpt 5186 dom cdm 5651 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7418 ↑_m cmap 8840 ↑_pm cpm 8841 ℂcc 11191 ℝcr 11192 0cc0 11193 1c1 11194 + caddc 11196 − cmin 11534 / cdiv 11966 ℕcn 12328 ℕ₀cn0 12599 ℤ_≥cuz 12958 ...cfz 13632 ..^cfzo 13781 ↑cexp 14197 ↾_t crest 17584 TopOpenctopn 17585 ℂ_fldccnfld 21671 Topctop 23204 TopOnctopon 23221 intcnt 23328 lim_ℂ climc 26175 D cdv 26176 D^𝑛 cdvn 26177 Tayl ctayl 26673

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-rep 5232 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7749 ax-inf2 9635 ax-cnex 11249 ax-resscn 11250 ax-1cn 11251 ax-icn 11252 ax-addcl 11253 ax-addrcl 11254 ax-mulcl 11255 ax-mulrcl 11256 ax-mulcom 11257 ax-addass 11258 ax-mulass 11259 ax-distr 11260 ax-i2m1 11261 ax-1ne0 11262 ax-1rid 11263 ax-rnegex 11264 ax-rrecex 11265 ax-cnre 11266 ax-pre-lttri 11267 ax-pre-lttrn 11268 ax-pre-ltadd 11269 ax-pre-mulgt0 11270 ax-pre-sup 11271 ax-addf 11272

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-tp 4589 df-op 4591 df-uni 4868 df-int 4908 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-se 5605 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 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 7375 df-ov 7421 df-oprab 7422 df-mpo 7423 df-of 7691 df-om 7876 df-1st 7999 df-2nd 8000 df-supp 8171 df-frecs 8292 df-wrecs 8323 df-recs 8372 df-rdg 8411 df-1o 8469 df-2o 8470 df-er 8710 df-map 8842 df-pm 8843 df-ixp 8919 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-fsupp 9347 df-fi 9396 df-sup 9427 df-inf 9428 df-oi 9497 df-card 10013 df-pnf 11338 df-mnf 11339 df-xr 11340 df-ltxr 11341 df-le 11342 df-sub 11536 df-neg 11537 df-div 11967 df-nn 12329 df-2 12398 df-3 12399 df-4 12400 df-5 12401 df-6 12402 df-7 12403 df-8 12404 df-9 12405 df-n0 12600 df-xnn0 12673 df-z 12687 df-dec 12808 df-uz 12959 df-q 13069 df-rp 13114 df-xneg 13234 df-xadd 13235 df-xmul 13236 df-icc 13476 df-fz 13633 df-fzo 13782 df-seq 14138 df-exp 14198 df-fac 14411 df-hash 14468 df-cj 15259 df-re 15260 df-im 15261 df-sqrt 15395 df-abs 15396 df-clim 15648 df-sum 15847 df-struct 17318 df-sets 17335 df-slot 17353 df-ndx 17365 df-base 17381 df-ress 17402 df-plusg 17434 df-mulr 17435 df-starv 17436 df-sca 17437 df-vsca 17438 df-ip 17439 df-tset 17440 df-ple 17441 df-ds 17443 df-unif 17444 df-hom 17445 df-cco 17446 df-rest 17586 df-topn 17587 df-0g 17605 df-gsum 17606 df-topgen 17607 df-pt 17608 df-prds 17611 df-xrs 17667 df-qtop 17672 df-imas 17673 df-xps 17675 df-mre 17749 df-mrc 17750 df-acs 17752 df-mgm 18809 df-sgrp 18901 df-mnd 18917 df-submnd 18972 df-grp 19140 df-minusg 19141 df-mulg 19271 df-cntz 19524 df-cmn 19989 df-abl 19990 df-mgp 20354 df-ur 20401 df-ring 20454 df-cring 20455 df-psmet 21663 df-xmet 21664 df-met 21665 df-bl 21666 df-mopn 21667 df-fbas 21668 df-fg 21669 df-cnfld 21672 df-top 23205 df-topon 23222 df-topsp 23244 df-bases 23257 df-cld 23330 df-ntr 23331 df-cls 23332 df-nei 23409 df-lp 23447 df-perf 23448 df-cn 23538 df-cnp 23539 df-haus 23626 df-tx 23874 df-hmeo 24067 df-fil 24158 df-fm 24250 df-flim 24251 df-flf 24252 df-tsms 24439 df-xms 24632 df-ms 24633 df-tms 24634 df-cncf 25192 df-limc 26179 df-dv 26180 df-dvn 26181 df-tayl 26675

This theorem is used by: taylth 26695

W3C validator