taylthlem1 - Metamath Proof Explorer

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

Theorem taylthlem1 26321

Description: Lemma for taylth 26324. 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 26324 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 13471	. . . 4 ⊢ (𝑁 ∈ ℕ ↔ 𝑁 ∈ (1...𝑁))
3	1, 2	sylib 218	. . 3 ⊢ (𝜑 → 𝑁 ∈ (1...𝑁))
4		oveq2 7366	. . . . . . . . . . . 12 ⊢ (𝑚 = 1 → (𝑁 − 𝑚) = (𝑁 − 1))
5	4	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
6	5	fveq1d 6834	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
7	4	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑚 = 1 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)))
8	7	fveq1d 6834	. . . . . . . . . 10 ⊢ (𝑚 = 1 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
9	6, 8	oveq12d 7376	. . . . . . . . 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 7366	. . . . . . . . 9 ⊢ (𝑚 = 1 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑1))
11	9, 10	oveq12d 7376	. . . . . . . 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 5180	. . . . . . 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 7373	. . . . . 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 2823	. . . . 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 340	. . . 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 7366	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑛 → (𝑁 − 𝑚) = (𝑁 − 𝑛))
17	16	fveq2d 6836	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛)))
18	17	fveq1d 6834	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥))
19	16	fveq2d 6836	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑛 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛)))
20	19	fveq1d 6834	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑛 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥))
21	18, 20	oveq12d 7376	. . . . . . . . . 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 7366	. . . . . . . . . 10 ⊢ (𝑚 = 𝑛 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑛))
23	21, 22	oveq12d 7376	. . . . . . . . 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 5180	. . . . . . . 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 6832	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑛))‘𝑦))
26		fveq2 6832	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑛))‘𝑦))
27	25, 26	oveq12d 7376	. . . . . . . . . 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 7365	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝑥 − 𝐵) = (𝑦 − 𝐵))
29	28	oveq1d 7373	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝑥 − 𝐵)↑𝑛) = ((𝑦 − 𝐵)↑𝑛))
30	27, 29	oveq12d 7376	. . . . . . . . 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 5190	. . . . . . . 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 2788	. . . . . . 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 7373	. . . . . 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 2823	. . . . 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 340	. . . 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 7366	. . . . . . . . . . . 12 ⊢ (𝑚 = (𝑛 + 1) → (𝑁 − 𝑚) = (𝑁 − (𝑛 + 1)))
37	36	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1))))
38	37	fveq1d 6834	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − (𝑛 + 1)))‘𝑥))
39	36	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑚 = (𝑛 + 1) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1))))
40	39	fveq1d 6834	. . . . . . . . . 10 ⊢ (𝑚 = (𝑛 + 1) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑛 + 1)))‘𝑥))
41	38, 40	oveq12d 7376	. . . . . . . . 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 7366	. . . . . . . . 9 ⊢ (𝑚 = (𝑛 + 1) → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑(𝑛 + 1)))
43	41, 42	oveq12d 7376	. . . . . . . 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 5180	. . . . . . 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 7373	. . . . . 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 2823	. . . . 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 340	. . . 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 7366	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑁 → (𝑁 − 𝑚) = (𝑁 − 𝑁))
49	48	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚)) = ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)))
50	49	fveq1d 6834	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑚))‘𝑥) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥))
51	48	fveq2d 6836	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑁 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)))
52	51	fveq1d 6834	. . . . . . . . . 10 ⊢ (𝑚 = 𝑁 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑚))‘𝑥) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥))
53	50, 52	oveq12d 7376	. . . . . . . . 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 7366	. . . . . . . . 9 ⊢ (𝑚 = 𝑁 → ((𝑥 − 𝐵)↑𝑚) = ((𝑥 − 𝐵)↑𝑁))
55	53, 54	oveq12d 7376	. . . . . . . 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 5180	. . . . . . 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 7373	. . . . . 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 2823	. . . . 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 340	. . . 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 6832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
62		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) = (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵))
63	61, 62	oveq12d 7376	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 7391	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵)) ∈ V
66	63, 64, 65	fvmpt 6939	. . . . . . . . . . . 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 12463	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
72		nn0uz 12790	. . . . . . . . . . . . . . 15 ⊢ ℕ₀ = (ℤ_≥‘0)
73	71, 72	eleqtrdi 2847	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘0))
74		eluzfz2b 13450	. . . . . . . . . . . . . 14 ⊢ (𝑁 ∈ (ℤ_≥‘0) ↔ 𝑁 ∈ (0...𝑁))
75	73, 74	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ (0...𝑁))
76		taylthlem1.d	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) = 𝐴)
77	60, 76	eleqtrrd 2840	. . . . . . . . . . . . 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 26320	. . . . . . . . . . . 12 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝐵) = (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵))
80	79	oveq2d 7374	. . . . . . . . . . 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 11108	. . . . . . . . . . . . . . . 16 ⊢ ℂ ∈ V
82	81	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ℂ ∈ V)
83		elpm2r 8783	. . . . . . . . . . . . . . 15 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
84	82, 68, 69, 70, 83	syl22anc 839	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
85		dvnf 25872	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):dom ((𝑆 D^𝑛 𝐹)‘𝑁)⟶ℂ)
87	86, 77	ffvelcdmd 7029	. . . . . . . . . . . 12 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) ∈ ℂ)
88	87	subidd 11481	. . . . . . . . . . 11 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵) − (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝐵)) = 0)
89	67, 80, 88	3eqtrd 2776	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))‘𝐵) = 0)
90		funmpt 6528	. . . . . . . . . . 11 ⊢ Fun (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
91		ovex 7391	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)) ∈ V
92	91, 64	dmmpti 6634	. . . . . . . . . . . 12 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))) = 𝐴
93	60, 92	eleqtrrdi 2848	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐵 ∈ dom (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦))))
94		funbrfvb 6885	. . . . . . . . . . 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 588	. . . . . . . . . 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 232	. . . . . . . . 9 ⊢ (𝜑 → 𝐵(𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) − (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))0)
97		nnm1nn0 12443	. . . . . . . . . . . . . . 15 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ ℕ₀)
98	1, 97	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (𝑁 − 1) ∈ ℕ₀)
99		dvnf 25872	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))⟶ℂ)
101		dvnbss 25873	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ dom 𝐹)
103	69, 102	fssdmd 6678	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ⊆ 𝐴)
104		fzo0end 13675	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑁 ∈ ℕ → (𝑁 − 1) ∈ (0..^𝑁))
105		elfzofz 13592	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 − 1) ∈ (0..^𝑁) → (𝑁 − 1) ∈ (0...𝑁))
106	1, 104, 105	3syl 18	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (𝑁 − 1) ∈ (0...𝑁))
107		dvn2bss 25875	. . . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
109	76, 108	eqsstrrd 3958	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐴 ⊆ dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))
110	103, 109	eqssd 3940	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = 𝐴)
111	110	feq2d 6644	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)):𝐴⟶ℂ)
113	112	ffvelcdmda 7028	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
114	76	feq2d 6644	. . . . . . . . . . . . 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 232	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁):𝐴⟶ℂ)
116	115	ffvelcdmda 7028	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦) ∈ ℂ)
117	1	nncnd 12162	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝑁 ∈ ℂ)
118		1cnd 11128	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℂ)
119	117, 118	npcand 11497	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑁 − 1) + 1) = 𝑁)
120	119	fveq2d 6836	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
121		recnprss 25849	. . . . . . . . . . . . . . 15 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
122	68, 121	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
123		dvnp1 25870	. . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘((𝑁 − 1) + 1)) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
125	120, 124	eqtr3d 2774	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
126	115	feqmptd 6900	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘𝑁)‘𝑦)))
127	112	feqmptd 6900	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) = (𝑦 ∈ 𝐴 ↦ (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦)))
128	127	oveq2d 7374	. . . . . . . . . . . 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 2781	. . . . . . . . . . 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 3933	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
131	130	sselda 3922	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
132		1nn0 12418	. . . . . . . . . . . . . . . 16 ⊢ 1 ∈ ℕ₀
133	132	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 1 ∈ ℕ₀)
134		elpm2r 8783	. . . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)) ∈ (ℂ ↑_pm 𝑆))
136		dvn1 25871	. . . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))))
138	124, 120	eqtr3d 2774	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
139	137, 138	eqtrd 2772	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = ((𝑆 D^𝑛 𝐹)‘𝑁))
140	139	dmeqd 5852	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
141	77, 140	eleqtrrd 2840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))‘1))
142		eqid 2737	. . . . . . . . . . . . . . 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 26313	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵):ℂ⟶ℂ)
144	118, 117	pncan3d 11496	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (1 + (𝑁 − 1)) = 𝑁)
145	144	oveq1d 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
146	78, 145	eqtr4id 2791	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇 = ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))
147	146	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 𝑇) = (ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵)))
148	147	fveq1d 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = ((ℂ D^𝑛 ((1 + (𝑁 − 1))(𝑆 Tayl 𝐹)𝐵))‘(𝑁 − 1)))
149	144	fveq2d 6836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = ((𝑆 D^𝑛 𝐹)‘𝑁))
150	149	dmeqd 5852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
151	77, 150	eleqtrrd 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(1 + (𝑁 − 1))))
152	68, 69, 70, 98, 133, 151	dvntaylp 26319	. . . . . . . . . . . . . . . 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 2772	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (1(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1)))𝐵))
154	153	feq1d 6642	. . . . . . . . . . . . . 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 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)):ℂ⟶ℂ)
156	155	ffvelcdmda 7028	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
157	131, 156	syldan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
158		0nn0 12417	. . . . . . . . . . . . . . . 16 ⊢ 0 ∈ ℕ₀
159	158	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 0 ∈ ℕ₀)
160		elpm2r 8783	. . . . . . . . . . . . . . . . . . 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 839	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘𝑁) ∈ (ℂ ↑_pm 𝑆))
162		dvn0 25869	. . . . . . . . . . . . . . . . . 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 585	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = ((𝑆 D^𝑛 𝐹)‘𝑁))
164	163	dmeqd 5852	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
165	77, 164	eleqtrrd 2840	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 ((𝑆 D^𝑛 𝐹)‘𝑁))‘0))
166		eqid 2737	. . . . . . . . . . . . . . 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 26313	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ)
168	117	addlidd 11335	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → (0 + 𝑁) = 𝑁)
169	168	oveq1d 7373	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = (𝑁(𝑆 Tayl 𝐹)𝐵))
170	169, 78	eqtr4di 2790	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵) = 𝑇)
171	170	oveq2d 7374	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → (ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵)) = (ℂ D^𝑛 𝑇))
172	171	fveq1d 6834	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ((ℂ D^𝑛 ((0 + 𝑁)(𝑆 Tayl 𝐹)𝐵))‘𝑁) = ((ℂ D^𝑛 𝑇)‘𝑁))
173	168	fveq2d 6836	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = ((𝑆 D^𝑛 𝐹)‘𝑁))
174	173	dmeqd 5852	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
175	77, 174	eleqtrrd 2840	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘(0 + 𝑁)))
176	68, 69, 70, 71, 159, 175	dvntaylp 26319	. . . . . . . . . . . . . . . 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 2774	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵))
178	177	feq1d 6642	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ ↔ (0(𝑆 Tayl ((𝑆 D^𝑛 𝐹)‘𝑁))𝐵):ℂ⟶ℂ))
179	167, 178	mpbird 257	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁):ℂ⟶ℂ)
180	179	ffvelcdmda 7028	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
181	131, 180	syldan 592	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
182	122	sselda 3922	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → 𝑦 ∈ ℂ)
183	182, 156	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) ∈ ℂ)
184	182, 180	syldan 592	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝑆) → (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦) ∈ ℂ)
185		eqid 2737	. . . . . . . . . . . . 13 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
186	185	cnfldtopon 24725	. . . . . . . . . . . . . 14 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
187		toponmax 22869	. . . . . . . . . . . . . 14 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
188	186, 187	mp1i 13	. . . . . . . . . . . . 13 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
189		dfss2 3908	. . . . . . . . . . . . . 14 ⊢ (𝑆 ⊆ ℂ ↔ (𝑆 ∩ ℂ) = 𝑆)
190	122, 189	sylib 218	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑆 ∩ ℂ) = 𝑆)
191		ssid 3945	. . . . . . . . . . . . . . . . 17 ⊢ ℂ ⊆ ℂ
192	191	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → ℂ ⊆ ℂ)
193		mapsspm 8815	. . . . . . . . . . . . . . . . 17 ⊢ (ℂ ↑_m ℂ) ⊆ (ℂ ↑_pm ℂ)
194	68, 69, 70, 71, 77, 78	taylpf 26313	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
195	81, 81	elmap 8810	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑇 ∈ (ℂ ↑_m ℂ) ↔ 𝑇:ℂ⟶ℂ)
196	194, 195	sylibr 234	. . . . . . . . . . . . . . . . 17 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_m ℂ))
197	193, 196	sselid 3920	. . . . . . . . . . . . . . . 16 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
198		dvnp1 25870	. . . . . . . . . . . . . . . 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 1374	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))))
200	119	fveq2d 6836	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − 1) + 1)) = ((ℂ D^𝑛 𝑇)‘𝑁))
201	199, 200	eqtr3d 2774	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = ((ℂ D^𝑛 𝑇)‘𝑁))
202	155	feqmptd 6900	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 1)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))
203	202	oveq2d 7374	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − 1))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))
204	179	feqmptd 6900	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘𝑁) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
205	201, 203, 204	3eqtr3d 2780	. . . . . . . . . . . . 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 25901	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑆 D (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))) = (𝑦 ∈ 𝑆 ↦ (((ℂ D^𝑛 𝑇)‘𝑁)‘𝑦)))
207		eqid 2737	. . . . . . . . . . . 12 ⊢ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ((TopOpen‘ℂ_fld) ↾_t 𝑆)
208		resttopon 23104	. . . . . . . . . . . . . . . 16 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ 𝑆 ⊆ ℂ) → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
209	186, 122, 208	sylancr 588	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → ((TopOpen‘ℂ_fld) ↾_t 𝑆) ∈ (TopOn‘𝑆))
210		topontop 22856	. . . . . . . . . . . . . . 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 22857	. . . . . . . . . . . . . . . 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 3959	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ⊆ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆))
215		eqid 2737	. . . . . . . . . . . . . . 15 ⊢ ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆) = ∪ ((TopOpen‘ℂ_fld) ↾_t 𝑆)
216	215	ntrss2 23000	. . . . . . . . . . . . . 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 585	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) ⊆ 𝐴)
218	138	dmeqd 5852	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = dom ((𝑆 D^𝑛 𝐹)‘𝑁))
219	218, 76	eqtrd 2772	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) = 𝐴)
220	122, 112, 70, 207, 185	dvbssntr 25845	. . . . . . . . . . . . . 14 ⊢ (𝜑 → dom (𝑆 D ((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))) ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
221	219, 220	eqsstrrd 3958	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ⊆ ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴))
222	217, 221	eqssd 3940	. . . . . . . . . . . 12 ⊢ (𝜑 → ((int‘((TopOpen‘ℂ_fld) ↾_t 𝑆))‘𝐴) = 𝐴)
223	68, 183, 184, 206, 70, 207, 185, 222	dvmptres2 25907	. . . . . . . . . . 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 25912	. . . . . . . . . 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 5097	. . . . . . . . 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 257	. . . . . . . 8 ⊢ (𝜑 → 𝐵(𝑆 D (𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦))))0)
227		eqid 2737	. . . . . . . . 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 11493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)) ∈ ℂ)
229	228	fmpttd 7059	. . . . . . . . 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 25843	. . . . . . . 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 232	. . . . . . 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 495	. . . . . 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 4072	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
234		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥))
235		fveq2 6832	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥))
236	234, 235	oveq12d 7376	. . . . . . . . . . . . 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 2737	. . . . . . . . . . . . 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 7391	. . . . . . . . . . . . 13 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ V
239	236, 237, 238	fvmpt 6939	. . . . . . . . . . . 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 6832	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
241		fveq2 6832	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵))
242	240, 241	oveq12d 7376	. . . . . . . . . . . . . . 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 7391	. . . . . . . . . . . . . . 15 ⊢ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵)) ∈ V
244	242, 237, 243	fvmpt 6939	. . . . . . . . . . . . . 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 26320	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝐵) = (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵))
247	246	oveq2d 7374	. . . . . . . . . . . . 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 7029	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) ∈ ℂ)
249	248	subidd 11481	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵) − (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝐵)) = 0)
250	245, 247, 249	3eqtrd 2776	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑦 ∈ 𝐴 ↦ ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑦)))‘𝐵) = 0)
251	239, 250	oveqan12rd 7378	. . . . . . . . . . 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 7028	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
253	130	sselda 3922	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ ℂ)
254	155	ffvelcdmda 7028	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
255	253, 254	syldan 592	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥) ∈ ℂ)
256	252, 255	subcld 11493	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 1))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 1))‘𝑥)) ∈ ℂ)
257	256	subid1d 11482	. . . . . . . . . . 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 2773	. . . . . . . . . 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 594	. . . . . . . . 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 4088	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
261	260	sselda 3922	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
262	130, 60	sseldd 3923	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℂ)
263	262	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
264	261, 263	subcld 11493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
265	264	exp1d 14065	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑1) = (𝑥 − 𝐵))
266	259, 265	oveq12d 7376	. . . . . . . 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 5179	. . . . . . 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 7373	. . . . . 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 2840	. . . . 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 456	. . . . . 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 413	. . . . 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 13705	. . 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 11481	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑁) = 0)
278	277	fveq2d 6836	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = ((𝑆 D^𝑛 𝐹)‘0))
279		dvn0 25869	. . . . . . . . . 10 ⊢ ((𝑆 ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆)) → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
280	122, 84, 279	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘0) = 𝐹)
281	278, 280	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → ((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁)) = 𝐹)
282	281	fveq1d 6834	. . . . . . 7 ⊢ (𝜑 → (((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) = (𝐹‘𝑥))
283	277	fveq2d 6836	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = ((ℂ D^𝑛 𝑇)‘0))
284		dvn0 25869	. . . . . . . . . 10 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ)) → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
285	191, 197, 284	sylancr 588	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘0) = 𝑇)
286	283, 285	eqtrd 2772	. . . . . . . 8 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁)) = 𝑇)
287	286	fveq1d 6834	. . . . . . 7 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥) = (𝑇‘𝑥))
288	282, 287	oveq12d 7376	. . . . . 6 ⊢ (𝜑 → ((((𝑆 D^𝑛 𝐹)‘(𝑁 − 𝑁))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑁))‘𝑥)) = ((𝐹‘𝑥) − (𝑇‘𝑥)))
289	288	oveq1d 7373	. . . . 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 5180	. . . 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 2790	. . 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 7373	. 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 2839	1 ⊢ (𝜑 → 0 ∈ (𝑅 lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3430 ∖ cdif 3887 ∩ cin 3889 ⊆ wss 3890 {csn 4568 {cpr 4570 ∪ cuni 4851 class class class wbr 5086 ↦ cmpt 5167 dom cdm 5622 Fun wfun 6484 ⟶wf 6486 ‘cfv 6490 (class class class)co 7358 ↑_m cmap 8764 ↑_pm cpm 8765 ℂcc 11025 ℝcr 11026 0cc0 11027 1c1 11028 + caddc 11030 − cmin 11365 / cdiv 11795 ℕcn 12146 ℕ₀cn0 12402 ℤ_≥cuz 12752 ...cfz 13424 ..^cfzo 13571 ↑cexp 13985 ↾_t crest 17341 TopOpenctopn 17342 ℂ_fldccnfld 21311 Topctop 22836 TopOnctopon 22853 intcnt 22960 lim_ℂ climc 25807 D cdv 25808 D^𝑛 cdvn 25809 Tayl ctayl 26300

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5368 ax-un 7680 ax-inf2 9551 ax-cnex 11083 ax-resscn 11084 ax-1cn 11085 ax-icn 11086 ax-addcl 11087 ax-addrcl 11088 ax-mulcl 11089 ax-mulrcl 11090 ax-mulcom 11091 ax-addass 11092 ax-mulass 11093 ax-distr 11094 ax-i2m1 11095 ax-1ne0 11096 ax-1rid 11097 ax-rnegex 11098 ax-rrecex 11099 ax-cnre 11100 ax-pre-lttri 11101 ax-pre-lttrn 11102 ax-pre-ltadd 11103 ax-pre-mulgt0 11104 ax-pre-sup 11105 ax-addf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3343 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-tp 4573 df-op 4575 df-uni 4852 df-int 4891 df-iun 4936 df-iin 4937 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5517 df-eprel 5522 df-po 5530 df-so 5531 df-fr 5575 df-se 5576 df-we 5577 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-pred 6257 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-isom 6499 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8102 df-frecs 8222 df-wrecs 8253 df-recs 8302 df-rdg 8340 df-1o 8396 df-2o 8397 df-er 8634 df-map 8766 df-pm 8767 df-ixp 8837 df-en 8885 df-dom 8886 df-sdom 8887 df-fin 8888 df-fsupp 9266 df-fi 9315 df-sup 9346 df-inf 9347 df-oi 9416 df-card 9852 df-pnf 11169 df-mnf 11170 df-xr 11171 df-ltxr 11172 df-le 11173 df-sub 11367 df-neg 11368 df-div 11796 df-nn 12147 df-2 12209 df-3 12210 df-4 12211 df-5 12212 df-6 12213 df-7 12214 df-8 12215 df-9 12216 df-n0 12403 df-xnn0 12476 df-z 12490 df-dec 12609 df-uz 12753 df-q 12863 df-rp 12907 df-xneg 13027 df-xadd 13028 df-xmul 13029 df-icc 13269 df-fz 13425 df-fzo 13572 df-seq 13926 df-exp 13986 df-fac 14198 df-hash 14255 df-cj 15023 df-re 15024 df-im 15025 df-sqrt 15159 df-abs 15160 df-clim 15412 df-sum 15611 df-struct 17075 df-sets 17092 df-slot 17110 df-ndx 17122 df-base 17138 df-ress 17159 df-plusg 17191 df-mulr 17192 df-starv 17193 df-sca 17194 df-vsca 17195 df-ip 17196 df-tset 17197 df-ple 17198 df-ds 17200 df-unif 17201 df-hom 17202 df-cco 17203 df-rest 17343 df-topn 17344 df-0g 17362 df-gsum 17363 df-topgen 17364 df-pt 17365 df-prds 17368 df-xrs 17424 df-qtop 17429 df-imas 17430 df-xps 17432 df-mre 17506 df-mrc 17507 df-acs 17509 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18710 df-grp 18870 df-minusg 18871 df-mulg 19002 df-cntz 19250 df-cmn 19715 df-abl 19716 df-mgp 20080 df-ur 20121 df-ring 20174 df-cring 20175 df-psmet 21303 df-xmet 21304 df-met 21305 df-bl 21306 df-mopn 21307 df-fbas 21308 df-fg 21309 df-cnfld 21312 df-top 22837 df-topon 22854 df-topsp 22876 df-bases 22889 df-cld 22962 df-ntr 22963 df-cls 22964 df-nei 23041 df-lp 23079 df-perf 23080 df-cn 23170 df-cnp 23171 df-haus 23258 df-tx 23505 df-hmeo 23698 df-fil 23789 df-fm 23881 df-flim 23882 df-flf 23883 df-tsms 24070 df-xms 24263 df-ms 24264 df-tms 24265 df-cncf 24823 df-limc 25811 df-dv 25812 df-dvn 25813 df-tayl 26302

This theorem is referenced by: taylth 26324

W3C validator