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Theorem taylthlem1 26424

Description: Lemma for taylth 26426. 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 26426 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 13553	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
3	1, 2	sylib 220	. . 3 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
4		oveq2 7399	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑁 − 𝑚) = (𝑁 − 1))
5	4	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
6	5	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
7	4	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)))
8	7	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
9	6, 8	oveq12d 7409	. . . . . . . . 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 7399	. . . . . . . . 9 ⊢ (𝑚 = 1 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑1))
11	9, 10	oveq12d 7409	. . . . . . . 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 5191	. . . . . . 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 7406	. . . . . 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 342	. . . 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 7399	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑁 − 𝑚) = (𝑁 − 𝑛))
17	16	fveq2d 6866	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛)))
18	17	fveq1d 6864	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥))
19	16	fveq2d 6866	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛)))
20	19	fveq1d 6864	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥))
21	18, 20	oveq12d 7409	. . . . . . . . . 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 7399	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑛))
23	21, 22	oveq12d 7409	. . . . . . . . 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 5191	. . . . . . . 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 6862	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦))
26		fveq2 6862	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦))
27	25, 26	oveq12d 7409	. . . . . . . . . 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 7398	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐵) = (𝑦 − 𝐵))
29	28	oveq1d 7406	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐵)↑𝑛) = ((𝑦 − 𝐵)↑𝑛))
30	27, 29	oveq12d 7409	. . . . . . . . 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 5201	. . . . . . . 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 7406	. . . . . 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 342	. . . 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 7399	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑁 − 𝑚) = (𝑁 − (𝑛 + 1)))
37	36	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1))))
38	37	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥))
39	36	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1))))
40	39	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥))
41	38, 40	oveq12d 7409	. . . . . . . . 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 7399	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑(𝑛 + 1)))
43	41, 42	oveq12d 7409	. . . . . . . 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 5191	. . . . . . 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 7406	. . . . . 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 342	. . . 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 7399	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝑁 − 𝑚) = (𝑁 − 𝑁))
49	48	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)))
50	49	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥))
51	48	fveq2d 6866	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)))
52	51	fveq1d 6864	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥))
53	50, 52	oveq12d 7409	. . . . . . . . 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 7399	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑁))
55	53, 54	oveq12d 7409	. . . . . . . 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 5191	. . . . . . 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 7406	. . . . . 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 342	. . . 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 6862	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
62		fveq2 6862	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) = (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵))
63	61, 62	oveq12d 7409	. . . . . . . . . . . . 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 7424	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)) ∈ V
66	63, 64, 65	fvmpt 6970	. . . . . . . . . . . 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 12536	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
72		nn0uz 12871	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
73	71, 72	eleqtrdi 2871	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
74		eluzfz2b 13532	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ 𝑁 ∈ (0...𝑁))
75	73, 74	sylib 220	. . . . . . . . . . . . 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 26423	. . . . . . . . . . . 12 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
80	79	oveq2d 7407	. . . . . . . . . . 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 11148	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℂ ∈ V)
83		elpm2r 8820	. . . . . . . . . . . . . . 15 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
84	82, 68, 69, 70, 83	syl22anc 849	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
85		dvnf 25977	. . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)
87	86, 77	ffvelcdmd 7061	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) ∈ ℂ)
88	87	subidd 11524	. . . . . . . . . . 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 6554	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
91		ovex 7424	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)) ∈ V
92	91, 64	dmmpti 6660	. . . . . . . . . . . 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 6915	. . . . . . . . . . 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 596	. . . . . . . . . 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 234	. . . . . . . . 9 ⊢ (𝜑 → 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0)
97		nnm1nn0 12516	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
98	1, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
99		dvnf 25977	. . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ)
101		dvnbss 25978	. . . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ dom 𝐹)
103	69, 102	fssdmd 6705	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ 𝐴)
104		fzo0end 13758	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
105		elfzofz 13675	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ (0..^𝑁) → (𝑁 − 1) ∈ (0...𝑁))
106	1, 104, 105	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...𝑁))
107		dvn2bss 25980	. . . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
109	76, 108	eqsstrrd 3969	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
110	103, 109	eqssd 3951	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = 𝐴)
111	110	feq2d 6670	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ)
113	112	ffvelcdmda 7060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
114	76	feq2d 6670	. . . . . . . . . . . . 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 234	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ)
116	115	ffvelcdmda 7060	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) ∈ ℂ)
117	1	nncnd 12220	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
118		1cnd 11169	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
119	117, 118	npcand 11540	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
120	119	fveq2d 6866	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
121		recnprss 25954	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
122	68, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
123		dvnp1 25975	. . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . 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 6930	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
127	112	feqmptd 6930	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦)))
128	127	oveq2d 7407	. . . . . . . . . . . 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 3944	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
131	130	sselda 3934	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
132		1nn0 12491	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℕ₀)
134		elpm2r 8820	. . . . . . . . . . . . . . . . . . . 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 849	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆))
136		dvn1 25976	. . . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . . 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 5877	. . . . . . . . . . . . . . . 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 26417	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ)
144	118, 117	pncan3d 11539	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
145	144	oveq1d 7406	. . . . . . . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 𝑇) = (ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵)))
148	147	fveq1d 6864	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)))
149	144	fveq2d 6866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
150	149	dmeqd 5877	. . . . . . . . . . . . . . . . . 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 26422	. . . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . 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 259	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)):ℂ⟶ℂ)
156	155	ffvelcdmda 7060	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
157	131, 156	syldan 600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
158		0nn0 12490	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
159	158	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℕ₀)
160		elpm2r 8820	. . . . . . . . . . . . . . . . . . 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 849	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
162		dvn0 25974	. . . . . . . . . . . . . . . . . 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 593	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑁))
164	163	dmeqd 5877	. . . . . . . . . . . . . . . 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 26417	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ)
168	117	addlidd 11378	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
169	168	oveq1d 7406	. . . . . . . . . . . . . . . . . . 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 7407	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
172	171	fveq1d 6864	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = ((ℂ D^𝑛 𝑇)‘𝑁))
173	168	fveq2d 6866	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
174	173	dmeqd 5877	. . . . . . . . . . . . . . . . . 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 26422	. . . . . . . . . . . . . . . 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 6668	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ ↔ (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ))
179	167, 178	mpbird 259	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ)
180	179	ffvelcdmda 7060	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
181	131, 180	syldan 600	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
182	122	sselda 3934	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
183	182, 156	syldan 600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
184	182, 180	syldan 600	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
185		eqid 2761	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
186	185	cnfldtopon 24830	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
187		toponmax 22974	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
188	186, 187	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
189		dfss2 3920	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆)
190	122, 189	sylib 220	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆)
191		ssid 3956	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ⊆ ℂ
192	191	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℂ ⊆ ℂ)
193		mapsspm 8852	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ↑_m ℂ) ⊆ (ℂ ↑_pm ℂ)
194	68, 69, 70, 71, 77, 78	taylpf 26417	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
195	81, 81	elmap 8847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ (ℂ ↑_m ℂ) ↔ 𝑇:ℂ⟶ℂ)
196	194, 195	sylibr 236	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_m ℂ))
197	193, 196	sselid 3932	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
198		dvnp1 25975	. . . . . . . . . . . . . . . 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 1389	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))))
200	119	fveq2d 6866	. . . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))
203	202	oveq2d 7407	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))
204	179	feqmptd 6930	. . . . . . . . . . . . . 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 26006	. . . . . . . . . . . 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 23209	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
209	186, 122, 208	sylancr 596	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
210		topontop 22961	. . . . . . . . . . . . . . 15 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
211	209, 210	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ Top)
212		toponuni 22962	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆) → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
213	209, 212	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑆 = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
214	70, 213	sseqtrd 3970	. . . . . . . . . . . . . 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 23105	. . . . . . . . . . . . . 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 593	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
218	138	dmeqd 5877	. . . . . . . . . . . . . . 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 25950	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
221	219, 220	eqsstrrd 3969	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
222	217, 221	eqssd 3951	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) = 𝐴)
223	68, 183, 184, 206, 70, 207, 185, 222	dvmptres2 26012	. . . . . . . . . . 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 26017	. . . . . . . . . 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 5108	. . . . . . . . 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 259	. . . . . . . 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 11536	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) ∈ ℂ)
229	228	fmpttd 7091	. . . . . . . . 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 25948	. . . . . . . 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 234	. . . . . . 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 499	. . . . . 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 4082	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
234		fveq2 6862	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
235		fveq2 6862	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
236	234, 235	oveq12d 7409	. . . . . . . . . . . . 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 7424	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ V
239	236, 237, 238	fvmpt 6970	. . . . . . . . . . . 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 6862	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
241		fveq2 6862	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵))
242	240, 241	oveq12d 7409	. . . . . . . . . . . . . . 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 7424	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)) ∈ V
244	242, 237, 243	fvmpt 6970	. . . . . . . . . . . . . 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 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)))
246	68, 69, 70, 106, 77, 78	dvntaylp0 26423	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
247	246	oveq2d 7407	. . . . . . . . . . . . 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 7061	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) ∈ ℂ)
249	248	subidd 11524	. . . . . . . . . . . . 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 7411	. . . . . . . . . . 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 7060	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
253	130	sselda 3934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
254	155	ffvelcdmda 7060	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
255	253, 254	syldan 600	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
256	252, 255	subcld 11536	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ ℂ)
257	256	subid1d 11525	. . . . . . . . . . 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 602	. . . . . . . . 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 4098	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
261	260	sselda 3934	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
262	130, 60	sseldd 3935	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℂ)
263	262	adantr 484	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
264	261, 263	subcld 11536	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
265	264	exp1d 14148	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑1) = (𝑥 − 𝐵))
266	259, 265	oveq12d 7409	. . . . . . . 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 5190	. . . . . . 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 7406	. . . . . 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 460	. . . . . 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 417	. . . . 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 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))))
275	15, 35, 47, 59, 270, 274	fzind2 13788	. . 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 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))
277	117	subidd 11524	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
278	277	fveq2d 6866	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = ((𝑆 D^𝑛 𝐹)‘0))
279		dvn0 25974	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
280	122, 84, 279	syl2anc 593	. . . . . . . . 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 6864	. . . . . . 7 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) = (𝐹‘𝑥))
283	277	fveq2d 6866	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = ((ℂ D^𝑛 𝑇)‘0))
284		dvn0 25974	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
285	191, 197, 284	sylancr 596	. . . . . . . . 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 6864	. . . . . . 7 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥) = (𝑇‘𝑥))
288	282, 287	oveq12d 7409	. . . . . 6 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) = ((𝐹‘𝑥) − (𝑇‘𝑥)))
289	288	oveq1d 7406	. . . . 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 5191	. . . 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 7406	. 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

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 Vcvv 3453 ∖ cdif 3899 ∩ cin 3901 ⊆ wss 3902 {csn 4579 {cpr 4581 ∪ cuni 4862 class class class wbr 5097 ↦ cmpt 5178 dom cdm 5643 Fun wfun 6510 ⟶wf 6512 ‘cfv 6516 (class class class)co 7391 ↑_m cmap 8802 ↑_pm cpm 8803 ℂcc 11065 ℝcr 11066 0cc0 11067 1c1 11068 + caddc 11070 − cmin 11408 / cdiv 11838 ℕcn 12204 ℕ₀cn0 12475 ℤ_≥cuz 12833 ...cfz 13506 ..^cfzo 13653 ↑cexp 14068 ↾_t crest 17440 TopOpenctopn 17441 ℂ_fldccnfld 21412 Topctop 22941 TopOnctopon 22958 intcnt 23065 lim_ℂ climc 25912 D cdv 25913 D^𝑛 cdvn 25914 Tayl ctayl 26404

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-inf2 9590 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-xnn0 12549 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-icc 13350 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-fac 14281 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-clim 15506 df-sum 15705 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18969 df-minusg 18970 df-mulg 19101 df-cntz 19348 df-cmn 19813 df-abl 19814 df-mgp 20178 df-ur 20219 df-ring 20272 df-cring 20273 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-cnfld 21413 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-lp 23184 df-perf 23185 df-cn 23275 df-cnp 23276 df-haus 23363 df-tx 23610 df-hmeo 23803 df-fil 23894 df-fm 23986 df-flim 23987 df-flf 23988 df-tsms 24175 df-xms 24368 df-ms 24369 df-tms 24370 df-cncf 24928 df-limc 25916 df-dv 25917 df-dvn 25918 df-tayl 26406

This theorem is referenced by: taylth 26426

W3C validator