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Theorem taylthlem2 25542

Description: Lemma for taylth 25543. (Contributed by Mario Carneiro, 1-Jan-2017.)

**Hypotheses**
Ref	Expression
taylth.f	⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
taylth.a	⊢ (𝜑 → 𝐴 ⊆ ℝ)
taylth.d	⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) = 𝐴)
taylth.n	⊢ (𝜑 → 𝑁 ∈ ℕ)
taylth.b	⊢ (𝜑 → 𝐵 ∈ 𝐴)
taylth.t	⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)
taylthlem2.m	⊢ (𝜑 → 𝑀 ∈ (1..^𝑁))
taylthlem2.i	⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀))) lim_ℂ 𝐵))

**Assertion**
Ref	Expression
taylthlem2	⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))) lim_ℂ 𝐵))

Distinct variable groups: 𝑥,𝐴 𝑥,𝐵 𝑥,𝐹 𝑥,𝑀 𝑥,𝑇 𝑥,𝑁 𝜑,𝑥

Proof of Theorem taylthlem2 Dummy variables 𝑦 𝑘 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylth.a	. . 3 ⊢ (𝜑 → 𝐴 ⊆ ℝ)
2		taylth.f	. . . . . . . 8 ⊢ (𝜑 → 𝐹:𝐴⟶ℝ)
3		fz1ssfz0 13361	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (0...𝑁)
4		taylthlem2.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (1..^𝑁))
5		fzofzp1 13493	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (1..^𝑁) → (𝑀 + 1) ∈ (1...𝑁))
6	4, 5	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁))
7	3, 6	sselid 3920	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ (0...𝑁))
8		fznn0sub2 13372	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (0...𝑁) → (𝑁 − (𝑀 + 1)) ∈ (0...𝑁))
9	7, 8	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (𝑀 + 1)) ∈ (0...𝑁))
10		elfznn0 13358	. . . . . . . . 9 ⊢ ((𝑁 − (𝑀 + 1)) ∈ (0...𝑁) → (𝑁 − (𝑀 + 1)) ∈ ℕ₀)
11	9, 10	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − (𝑀 + 1)) ∈ ℕ₀)
12		dvnfre 25125	. . . . . . . 8 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))⟶ℝ)
13	2, 1, 11, 12	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))⟶ℝ)
14		reelprrecn 10972	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
15	14	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
16		cnex 10961	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
18		reex 10971	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
20		ax-resscn 10937	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
21		fss 6626	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
22	2, 20, 21	sylancl 586	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
23		elpm2r 8642	. . . . . . . . . . . 12 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
24	17, 19, 22, 1, 23	syl22anc 836	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
25		dvnbss 25101	. . . . . . . . . . 11 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ⊆ dom 𝐹)
26	15, 24, 11, 25	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ⊆ dom 𝐹)
27	2, 26	fssdmd 6628	. . . . . . . . 9 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ⊆ 𝐴)
28		taylth.d	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) = 𝐴)
29		dvn2bss 25103	. . . . . . . . . . 11 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − (𝑀 + 1)) ∈ (0...𝑁)) → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))))
30	15, 24, 9, 29	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))))
31	28, 30	eqsstrrd 3961	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))))
32	27, 31	eqssd 3939	. . . . . . . 8 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) = 𝐴)
33	32	feq2d 6595	. . . . . . 7 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℝ))
34	13, 33	mpbid 231	. . . . . 6 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℝ)
35	34	ffvelrnda 6970	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℝ)
36	1	sselda 3922	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
37		fvres 6802	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ)‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))
38	37	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ)‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))
39		resubdrg 20822	. . . . . . . . . . . 12 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
40	39	simpli 484	. . . . . . . . . . 11 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
41	40	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (SubRing‘ℂ_fld))
42		taylth.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
43	42	nnnn0d 12302	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
44		taylth.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
45	44, 28	eleqtrrd 2843	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ dom ((ℝ D^𝑛 𝐹)‘𝑁))
46		taylth.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)
47	1, 44	sseldd 3923	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
48		elfznn0 13358	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
49		dvnfre 25125	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑘 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘𝑘):dom ((ℝ D^𝑛 𝐹)‘𝑘)⟶ℝ)
50	2, 1, 48, 49	syl2an3an 1421	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((ℝ D^𝑛 𝐹)‘𝑘):dom ((ℝ D^𝑛 𝐹)‘𝑘)⟶ℝ)
51		simpr 485	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
52		dvn2bss 25103	. . . . . . . . . . . . . . . 16 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑘 ∈ (0...𝑁)) → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘𝑘))
53	14, 24, 51, 52	mp3an2ani 1467	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘𝑘))
54	45	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((ℝ D^𝑛 𝐹)‘𝑁))
55	53, 54	sseldd 3923	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((ℝ D^𝑛 𝐹)‘𝑘))
56	50, 55	ffvelrnd 6971	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((ℝ D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℝ)
57	48	adantl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
58	57	faccld 14007	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
59	56, 58	nndivred 12036	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((ℝ D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℝ)
60	15, 22, 1, 43, 45, 46, 41, 47, 59	taylply2 25536	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℝ) ∧ (deg‘𝑇) ≤ 𝑁))
61	60	simpld 495	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (Poly‘ℝ))
62		dvnply2 25456	. . . . . . . . . 10 ⊢ ((ℝ ∈ (SubRing‘ℂ_fld) ∧ 𝑇 ∈ (Poly‘ℝ) ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ))
63	41, 61, 11, 62	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ))
64		plyreres 25452	. . . . . . . . 9 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ):ℝ⟶ℝ)
65	63, 64	syl 17	. . . . . . . 8 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ):ℝ⟶ℝ)
66	65	ffvelrnda 6970	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ)‘𝑦) ∈ ℝ)
67	38, 66	eqeltrrd 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℝ)
68	36, 67	syldan 591	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℝ)
69	35, 68	resubcld 11412	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)) ∈ ℝ)
70	69	fmpttd 6998	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))):𝐴⟶ℝ)
71	47	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ ℝ)
72	36, 71	resubcld 11412	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 − 𝐵) ∈ ℝ)
73		elfzouz 13400	. . . . . . . . . 10 ⊢ (𝑀 ∈ (1..^𝑁) → 𝑀 ∈ (ℤ_≥‘1))
74	4, 73	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
75		nnuz 12630	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
76	74, 75	eleqtrrdi 2851	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
77	76	nnnn0d 12302	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
78	77	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑀 ∈ ℕ₀)
79		1nn0 12258	. . . . . . 7 ⊢ 1 ∈ ℕ₀
80	79	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 1 ∈ ℕ₀)
81	78, 80	nn0addcld 12306	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑀 + 1) ∈ ℕ₀)
82	72, 81	reexpcld 13890	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℝ)
83	82	fmpttd 6998	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))):𝐴⟶ℝ)
84		retop 23934	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
85		uniretop 23935	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
86	85	ntrss2 22217	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴)
87	84, 1, 86	sylancr 587	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴)
88	42	nncnd 11998	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℂ)
89	76	nncnd 11998	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
90		1cnd 10979	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
91	88, 89, 90	nppcan2d 11367	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 − (𝑀 + 1)) + 1) = (𝑁 − 𝑀))
92	91	fveq2d 6787	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘((𝑁 − (𝑀 + 1)) + 1)) = ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
93	20	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
94		dvnp1 25098	. . . . . . . . . 10 ⊢ ((ℝ ⊆ ℂ ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘((𝑁 − (𝑀 + 1)) + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
95	93, 24, 11, 94	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘((𝑁 − (𝑀 + 1)) + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
96	92, 95	eqtr3d 2781	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
97	96	dmeqd 5817	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
98		fzonnsub 13421	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (1..^𝑁) → (𝑁 − 𝑀) ∈ ℕ)
99	4, 98	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ)
100	99	nnnn0d 12302	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
101		dvnbss 25101	. . . . . . . . . 10 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − 𝑀) ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) ⊆ dom 𝐹)
102	15, 24, 100, 101	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) ⊆ dom 𝐹)
103	2, 102	fssdmd 6628	. . . . . . . 8 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) ⊆ 𝐴)
104		elfzofz 13412	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1..^𝑁) → 𝑀 ∈ (1...𝑁))
105	4, 104	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
106	3, 105	sselid 3920	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))
107		fznn0sub2 13372	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ (0...𝑁))
108	106, 107	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ (0...𝑁))
109		dvn2bss 25103	. . . . . . . . . 10 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − 𝑀) ∈ (0...𝑁)) → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
110	15, 24, 108, 109	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
111	28, 110	eqsstrrd 3961	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
112	103, 111	eqssd 3939	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = 𝐴)
113	97, 112	eqtr3d 2781	. . . . . 6 ⊢ (𝜑 → dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) = 𝐴)
114		fss 6626	. . . . . . . 8 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℂ)
115	34, 20, 114	sylancl 586	. . . . . . 7 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℂ)
116		eqid 2739	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
117	116	tgioo2 23975	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
118	93, 115, 1, 117, 116	dvbssntr 25073	. . . . . 6 ⊢ (𝜑 → dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) ⊆ ((int‘(topGen‘ran (,)))‘𝐴))
119	113, 118	eqsstrrd 3961	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ((int‘(topGen‘ran (,)))‘𝐴))
120	87, 119	eqssd 3939	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝐴) = 𝐴)
121	85	isopn3 22226	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ⊆ ℝ) → (𝐴 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝐴) = 𝐴))
122	84, 1, 121	sylancr 587	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝐴) = 𝐴))
123	120, 122	mpbird 256	. . 3 ⊢ (𝜑 → 𝐴 ∈ (topGen‘ran (,)))
124		eqid 2739	. . 3 ⊢ (𝐴 ∖ {𝐵}) = (𝐴 ∖ {𝐵})
125		difss 4067	. . . 4 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
126	35	recnd 11012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
127		dvnf 25100	. . . . . . . . . 10 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − 𝑀) ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℂ)
128	15, 24, 100, 127	syl3anc 1370	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℂ)
129	112	feq2d 6595	. . . . . . . . 9 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℂ))
130	128, 129	mpbid 231	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℂ)
131	130	ffvelrnda 6970	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) ∈ ℂ)
132		dvnfre 25125	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ (𝑁 − 𝑀) ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℝ)
133	2, 1, 100, 132	syl3anc 1370	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℝ)
134	112	feq2d 6595	. . . . . . . . . 10 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℝ))
135	133, 134	mpbid 231	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℝ)
136	135	feqmptd 6846	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦)))
137	34	feqmptd 6846	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) = (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦)))
138	137	oveq2d 7300	. . . . . . . 8 ⊢ (𝜑 → (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) = (ℝ D (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦))))
139	96, 136, 138	3eqtr3rd 2788	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦)))
140	68	recnd 11012	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
141		fvexd 6798	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) ∈ V)
142	67	recnd 11012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
143		recn 10970	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
144		dvnply2 25456	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ (SubRing‘ℂ_fld) ∧ 𝑇 ∈ (Poly‘ℝ) ∧ (𝑁 − 𝑀) ∈ ℕ₀) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) ∈ (Poly‘ℝ))
145	41, 61, 100, 144	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) ∈ (Poly‘ℝ))
146		plyf 25368	. . . . . . . . . . 11 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) ∈ (Poly‘ℝ) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)):ℂ⟶ℂ)
147	145, 146	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)):ℂ⟶ℂ)
148	147	ffvelrnda 6970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) ∈ ℂ)
149	143, 148	sylan2 593	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) ∈ ℂ)
150	116	cnfldtopon 23955	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
151		toponmax 22084	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
152	150, 151	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
153		df-ss 3905	. . . . . . . . . 10 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
154	93, 153	sylib 217	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∩ ℂ) = ℝ)
155		plyf 25368	. . . . . . . . . . 11 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ) → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))):ℂ⟶ℂ)
156	63, 155	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))):ℂ⟶ℂ)
157	156	ffvelrnda 6970	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
158	91	fveq2d 6787	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − (𝑀 + 1)) + 1)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)))
159		ssid 3944	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
160	159	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ⊆ ℂ)
161		mapsspm 8673	. . . . . . . . . . . . 13 ⊢ (ℂ ↑_m ℂ) ⊆ (ℂ ↑_pm ℂ)
162		plyf 25368	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (Poly‘ℝ) → 𝑇:ℂ⟶ℂ)
163	61, 162	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
164	16, 16	elmap 8668	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ (ℂ ↑_m ℂ) ↔ 𝑇:ℂ⟶ℂ)
165	163, 164	sylibr 233	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_m ℂ))
166	161, 165	sselid 3920	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
167		dvnp1 25098	. . . . . . . . . . . 12 ⊢ ((ℂ ⊆ ℂ ∧ 𝑇 ∈ (ℂ ↑_pm ℂ) ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → ((ℂ D^𝑛 𝑇)‘((𝑁 − (𝑀 + 1)) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))))
168	160, 166, 11, 167	syl3anc 1370	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − (𝑀 + 1)) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))))
169	158, 168	eqtr3d 2781	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))))
170	147	feqmptd 6846	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
171	156	feqmptd 6846	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))
172	171	oveq2d 7300	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))
173	169, 170, 172	3eqtr3rd 2788	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
174	116, 15, 152, 154, 157, 148, 173	dvmptres3 25129	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ ℝ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
175	15, 142, 149, 174, 1, 117, 116, 123	dvmptres 25136	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
176	15, 126, 131, 139, 140, 141, 175	dvmptsub 25140	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))) = (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦))))
177	176	dmeqd 5817	. . . . 5 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))) = dom (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦))))
178		ovex 7317	. . . . . 6 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)) ∈ V
179		eqid 2739	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
180	178, 179	dmmpti 6586	. . . . 5 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦))) = 𝐴
181	177, 180	eqtrdi 2795	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))) = 𝐴)
182	125, 181	sseqtrrid 3975	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))))
183		simpr 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
184	47	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐵 ∈ ℝ)
185	184	recnd 11012	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐵 ∈ ℂ)
186	183, 185	subcld 11341	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦 − 𝐵) ∈ ℂ)
187	77	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℕ₀)
188	79	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 1 ∈ ℕ₀)
189	187, 188	nn0addcld 12306	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑀 + 1) ∈ ℕ₀)
190	186, 189	expcld 13873	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℂ)
191	143, 190	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℂ)
192	89	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ)
193		1cnd 10979	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
194	192, 193	addcld 11003	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑀 + 1) ∈ ℂ)
195	186, 187	expcld 13873	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑦 − 𝐵)↑𝑀) ∈ ℂ)
196	194, 195	mulcld 11004	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ∈ ℂ)
197	143, 196	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ∈ ℂ)
198	16	prid2 4700	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
199	198	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
200		simpr 485	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
201		elfznn 13294	. . . . . . . . . . . . . 14 ⊢ ((𝑀 + 1) ∈ (1...𝑁) → (𝑀 + 1) ∈ ℕ)
202	6, 201	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
203	202	nnnn0d 12302	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
204	203	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑀 + 1) ∈ ℕ₀)
205	200, 204	expcld 13873	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑀 + 1)) ∈ ℂ)
206		ovexd 7319	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝑀 + 1) · (𝑥↑𝑀)) ∈ V)
207	199	dvmptid 25130	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
208		0cnd 10977	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
209	47	recnd 11012	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
210	199, 209	dvmptc 25131	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ 𝐵)) = (𝑦 ∈ ℂ ↦ 0))
211	199, 183, 193, 207, 185, 208, 210	dvmptsub 25140	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦 − 𝐵))) = (𝑦 ∈ ℂ ↦ (1 − 0)))
212		1m0e1 12103	. . . . . . . . . . . 12 ⊢ (1 − 0) = 1
213	212	mpteq2i 5180	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ (1 − 0)) = (𝑦 ∈ ℂ ↦ 1)
214	211, 213	eqtrdi 2795	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦 − 𝐵))) = (𝑦 ∈ ℂ ↦ 1))
215		dvexp 25126	. . . . . . . . . . . 12 ⊢ ((𝑀 + 1) ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1)))))
216	202, 215	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1)))))
217	89, 90	pncand 11342	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
218	217	oveq2d 7300	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥↑((𝑀 + 1) − 1)) = (𝑥↑𝑀))
219	218	oveq2d 7300	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1))) = ((𝑀 + 1) · (𝑥↑𝑀)))
220	219	mpteq2dv 5177	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑𝑀))))
221	216, 220	eqtrd 2779	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑𝑀))))
222		oveq1 7291	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐵) → (𝑥↑(𝑀 + 1)) = ((𝑦 − 𝐵)↑(𝑀 + 1)))
223		oveq1 7291	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 − 𝐵) → (𝑥↑𝑀) = ((𝑦 − 𝐵)↑𝑀))
224	223	oveq2d 7300	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐵) → ((𝑀 + 1) · (𝑥↑𝑀)) = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
225	199, 199, 186, 193, 205, 206, 214, 221, 222, 224	dvmptco 25145	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ ℂ ↦ (((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) · 1)))
226	196	mulid1d 11001	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) · 1) = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
227	226	mpteq2dva 5175	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) · 1)) = (𝑦 ∈ ℂ ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
228	225, 227	eqtrd 2779	. . . . . . . 8 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ ℂ ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
229	116, 15, 152, 154, 190, 196, 228	dvmptres3 25129	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ ℝ ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
230	15, 191, 197, 229, 1, 117, 116, 123	dvmptres 25136	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
231	230	dmeqd 5817	. . . . 5 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = dom (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
232		ovex 7317	. . . . . 6 ⊢ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ∈ V
233		eqid 2739	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) = (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
234	232, 233	dmmpti 6586	. . . . 5 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) = 𝐴
235	231, 234	eqtrdi 2795	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = 𝐴)
236	125, 235	sseqtrrid 3975	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))))
237	15, 22, 1, 9, 45, 46	dvntaylp0 25540	. . . . . 6 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵) = (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵))
238	237	oveq2d 7300	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵)) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵)))
239	115, 44	ffvelrnd 6971	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) ∈ ℂ)
240	239	subidd 11329	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵)) = 0)
241	238, 240	eqtrd 2779	. . . 4 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵)) = 0)
242	116	subcn 24038	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
243	242	a1i 11	. . . . . 6 ⊢ (𝜑 → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
244		dvcn 25094	. . . . . . . 8 ⊢ (((ℝ ⊆ ℂ ∧ ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) = 𝐴) → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ∈ (𝐴–cn→ℂ))
245	93, 115, 1, 113, 244	syl31anc 1372	. . . . . . 7 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ∈ (𝐴–cn→ℂ))
246	137, 245	eqeltrrd 2841	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦)) ∈ (𝐴–cn→ℂ))
247		plycn 25431	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ) → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (ℂ–cn→ℂ))
248	63, 247	syl 17	. . . . . . 7 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (ℂ–cn→ℂ))
249	1, 20	sstrdi 3934	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
250		cncfmptid 24085	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑦 ∈ 𝐴 ↦ 𝑦) ∈ (𝐴–cn→ℂ))
251	249, 159, 250	sylancl 586	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ 𝑦) ∈ (𝐴–cn→ℂ))
252	248, 251	cncfmpt1f 24086	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)) ∈ (𝐴–cn→ℂ))
253	116, 243, 246, 252	cncfmpt2f 24087	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) ∈ (𝐴–cn→ℂ))
254		fveq2 6783	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵))
255		fveq2 6783	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵))
256	254, 255	oveq12d 7302	. . . . 5 ⊢ (𝑦 = 𝐵 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵)))
257	253, 44, 256	cnmptlimc 25063	. . . 4 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵)) ∈ ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) lim_ℂ 𝐵))
258	241, 257	eqeltrrd 2841	. . 3 ⊢ (𝜑 → 0 ∈ ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) lim_ℂ 𝐵))
259	209	subidd 11329	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐵) = 0)
260	259	oveq1d 7299	. . . . 5 ⊢ (𝜑 → ((𝐵 − 𝐵)↑(𝑀 + 1)) = (0↑(𝑀 + 1)))
261	202	0expd 13866	. . . . 5 ⊢ (𝜑 → (0↑(𝑀 + 1)) = 0)
262	260, 261	eqtrd 2779	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝐵)↑(𝑀 + 1)) = 0)
263	249	sselda 3922	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
264	263, 190	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℂ)
265	264	fmpttd 6998	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))):𝐴⟶ℂ)
266		dvcn 25094	. . . . . 6 ⊢ (((ℝ ⊆ ℂ ∧ (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))):𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ) ∧ dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = 𝐴) → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ∈ (𝐴–cn→ℂ))
267	93, 265, 1, 235, 266	syl31anc 1372	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ∈ (𝐴–cn→ℂ))
268		oveq1 7291	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 − 𝐵) = (𝐵 − 𝐵))
269	268	oveq1d 7299	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 − 𝐵)↑(𝑀 + 1)) = ((𝐵 − 𝐵)↑(𝑀 + 1)))
270	267, 44, 269	cnmptlimc 25063	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝐵)↑(𝑀 + 1)) ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) lim_ℂ 𝐵))
271	262, 270	eqeltrrd 2841	. . 3 ⊢ (𝜑 → 0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) lim_ℂ 𝐵))
272	249	ssdifssd 4078	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
273	272	sselda 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℂ)
274	209	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
275	273, 274	subcld 11341	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑦 − 𝐵) ∈ ℂ)
276		eldifsni 4724	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝐵}) → 𝑦 ≠ 𝐵)
277	276	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ≠ 𝐵)
278	273, 274, 277	subne0d 11350	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑦 − 𝐵) ≠ 0)
279	202	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℕ)
280	279	nnzd 12434	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℤ)
281	275, 278, 280	expne0d 13879	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ≠ 0)
282	281	necomd 3000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 0 ≠ ((𝑦 − 𝐵)↑(𝑀 + 1)))
283	282	neneqd 2949	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ¬ 0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
284	283	nrexdv 3199	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
285		df-ima 5603	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})) = ran ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵}))
286	285	eleq2i 2831	. . . . 5 ⊢ (0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})) ↔ 0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})))
287		resmpt 5948	. . . . . . 7 ⊢ ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))
288	125, 287	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))
289		ovex 7317	. . . . . 6 ⊢ ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ V
290	288, 289	elrnmpti 5872	. . . . 5 ⊢ (0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
291	286, 290	bitri 274	. . . 4 ⊢ (0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
292	284, 291	sylnibr 329	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})))
293	89	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℂ)
294		1cnd 10979	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 1 ∈ ℂ)
295	293, 294	addcld 11003	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℂ)
296	273, 195	syldan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 − 𝐵)↑𝑀) ∈ ℂ)
297	279	nnne0d 12032	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ≠ 0)
298	76	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℕ)
299	298	nnzd 12434	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℤ)
300	275, 278, 299	expne0d 13879	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 − 𝐵)↑𝑀) ≠ 0)
301	295, 296, 297, 300	mulne0d 11636	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ≠ 0)
302	301	necomd 3000	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 0 ≠ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
303	302	neneqd 2949	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ¬ 0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
304	303	nrexdv 3199	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
305	230	imaeq1d 5971	. . . . . . 7 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) = ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) “ (𝐴 ∖ {𝐵})))
306		df-ima 5603	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) “ (𝐴 ∖ {𝐵})) = ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵}))
307	305, 306	eqtrdi 2795	. . . . . 6 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) = ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})))
308	307	eleq2d 2825	. . . . 5 ⊢ (𝜑 → (0 ∈ ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) ↔ 0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵}))))
309		resmpt 5948	. . . . . . 7 ⊢ ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
310	125, 309	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
311	310, 232	elrnmpti 5872	. . . . 5 ⊢ (0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
312	308, 311	bitrdi 287	. . . 4 ⊢ (𝜑 → (0 ∈ ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
313	304, 312	mtbird 325	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})))
314		eldifi 4062	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
315	130	ffvelrnda 6970	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
316	314, 315	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
317	1	ssdifssd 4078	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℝ)
318	317	sselda 3922	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ)
319	318	recnd 11012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
320	147	ffvelrnda 6970	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
321	319, 320	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
322	316, 321	subcld 11341	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) ∈ ℂ)
323	47	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℝ)
324	318, 323	resubcld 11412	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℝ)
325	77	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℕ₀)
326	324, 325	reexpcld 13890	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑𝑀) ∈ ℝ)
327	326	recnd 11012	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑𝑀) ∈ ℂ)
328	323	recnd 11012	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
329	319, 328	subcld 11341	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
330		eldifsni 4724	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ≠ 𝐵)
331	330	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ≠ 𝐵)
332	319, 328, 331	subne0d 11350	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ≠ 0)
333	325	nn0zd 12433	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℤ)
334	329, 332, 333	expne0d 13879	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑𝑀) ≠ 0)
335	322, 327, 334	divcld 11760	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) ∈ ℂ)
336	202	nnrecred 12033	. . . . . . 7 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ℝ)
337	336	recnd 11012	. . . . . 6 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ℂ)
338	337	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (1 / (𝑀 + 1)) ∈ ℂ)
339		txtopon 22751	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ∈ (TopOn‘(ℂ × ℂ)))
340	150, 150, 339	mp2an 689	. . . . . 6 ⊢ ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ∈ (TopOn‘(ℂ × ℂ))
341	340	toponrestid 22079	. . . . 5 ⊢ ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) = (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ↾_t (ℂ × ℂ))
342		taylthlem2.i	. . . . 5 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀))) lim_ℂ 𝐵))
343		limcresi 25058	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵) ⊆ (((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) lim_ℂ 𝐵)
344		resmpt 5948	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))))
345	125, 344	ax-mp 5	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1)))
346	345	oveq1i 7294	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵)
347	343, 346	sseqtri 3958	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵) ⊆ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵)
348		cncfmptc 24084	. . . . . . . 8 ⊢ (((1 / (𝑀 + 1)) ∈ ℝ ∧ 𝐴 ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ∈ (𝐴–cn→ℝ))
349	336, 249, 93, 348	syl3anc 1370	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ∈ (𝐴–cn→ℝ))
350		eqidd 2740	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (1 / (𝑀 + 1)) = (1 / (𝑀 + 1)))
351	349, 44, 350	cnmptlimc 25063	. . . . . 6 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵))
352	347, 351	sselid 3920	. . . . 5 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵))
353	116	mulcn 24039	. . . . . 6 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
354		0cn 10976	. . . . . . 7 ⊢ 0 ∈ ℂ
355		opelxpi 5627	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ (1 / (𝑀 + 1)) ∈ ℂ) → ⟨0, (1 / (𝑀 + 1))⟩ ∈ (ℂ × ℂ))
356	354, 337, 355	sylancr 587	. . . . . 6 ⊢ (𝜑 → ⟨0, (1 / (𝑀 + 1))⟩ ∈ (ℂ × ℂ))
357	340	toponunii 22074	. . . . . . 7 ⊢ (ℂ × ℂ) = ∪ ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld))
358	357	cncnpi 22438	. . . . . 6 ⊢ (( · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)) ∧ ⟨0, (1 / (𝑀 + 1))⟩ ∈ (ℂ × ℂ)) → · ∈ ((((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) CnP (TopOpen‘ℂ_fld))‘⟨0, (1 / (𝑀 + 1))⟩))
359	353, 356, 358	sylancr 587	. . . . 5 ⊢ (𝜑 → · ∈ ((((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) CnP (TopOpen‘ℂ_fld))‘⟨0, (1 / (𝑀 + 1))⟩))
360	335, 338, 160, 160, 116, 341, 342, 352, 359	limccnp2 25065	. . . 4 ⊢ (𝜑 → (0 · (1 / (𝑀 + 1))) ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) · (1 / (𝑀 + 1)))) lim_ℂ 𝐵))
361	337	mul02d 11182	. . . 4 ⊢ (𝜑 → (0 · (1 / (𝑀 + 1))) = 0)
362	176	fveq1d 6785	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) = ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))‘𝑥))
363		fveq2 6783	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥))
364		fveq2 6783	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥))
365	363, 364	oveq12d 7302	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)))
366		ovex 7317	. . . . . . . . . . 11 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) ∈ V
367	365, 179, 366	fvmpt 6884	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))‘𝑥) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)))
368	314, 367	syl 17	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))‘𝑥) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)))
369	362, 368	sylan9eq 2799	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)))
370	230	fveq1d 6785	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥) = ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))‘𝑥))
371		oveq1 7291	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵))
372	371	oveq1d 7299	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑀) = ((𝑥 − 𝐵)↑𝑀))
373	372	oveq2d 7300	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
374		ovex 7317	. . . . . . . . . . . 12 ⊢ ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)) ∈ V
375	373, 233, 374	fvmpt 6884	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))‘𝑥) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
376	314, 375	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))‘𝑥) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
377	370, 376	sylan9eq 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
378	202	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℕ)
379	378	nncnd 11998	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℂ)
380	379, 327	mulcomd 11005	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)) = (((𝑥 − 𝐵)↑𝑀) · (𝑀 + 1)))
381	377, 380	eqtrd 2779	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥) = (((𝑥 − 𝐵)↑𝑀) · (𝑀 + 1)))
382	369, 381	oveq12d 7302	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) / ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥)) = (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / (((𝑥 − 𝐵)↑𝑀) · (𝑀 + 1))))
383	378	nnne0d 12032	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ≠ 0)
384	322, 327, 379, 334, 383	divdiv1d 11791	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) / (𝑀 + 1)) = (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / (((𝑥 − 𝐵)↑𝑀) · (𝑀 + 1))))
385	335, 379, 383	divrecd 11763	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) / (𝑀 + 1)) = ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) · (1 / (𝑀 + 1))))
386	382, 384, 385	3eqtr2rd 2786	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) · (1 / (𝑀 + 1))) = (((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) / ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥)))
387	386	mpteq2dva 5175	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) · (1 / (𝑀 + 1)))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) / ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥))))
388	387	oveq1d 7299	. . . 4 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ ((((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) · (1 / (𝑀 + 1)))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) / ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥))) lim_ℂ 𝐵))
389	360, 361, 388	3eltr3d 2854	. . 3 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))‘𝑥) / ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥))) lim_ℂ 𝐵))
390	1, 70, 83, 123, 44, 124, 182, 236, 258, 271, 292, 313, 389	lhop 25189	. 2 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))‘𝑥) / ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥))) lim_ℂ 𝐵))
391	314	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ 𝐴)
392		fveq2 6783	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥))
393		fveq2 6783	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥))
394	392, 393	oveq12d 7302	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)))
395		eqid 2739	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))
396		ovex 7317	. . . . . . 7 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) ∈ V
397	394, 395, 396	fvmpt 6884	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))‘𝑥) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)))
398	391, 397	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))‘𝑥) = ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)))
399	371	oveq1d 7299	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑(𝑀 + 1)) = ((𝑥 − 𝐵)↑(𝑀 + 1)))
400		eqid 2739	. . . . . . 7 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) = (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))
401		ovex 7317	. . . . . . 7 ⊢ ((𝑥 − 𝐵)↑(𝑀 + 1)) ∈ V
402	399, 400, 401	fvmpt 6884	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥) = ((𝑥 − 𝐵)↑(𝑀 + 1)))
403	391, 402	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥) = ((𝑥 − 𝐵)↑(𝑀 + 1)))
404	398, 403	oveq12d 7302	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))‘𝑥) / ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥)) = (((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1))))
405	404	mpteq2dva 5175	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))‘𝑥) / ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥))) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))))
406	405	oveq1d 7299	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))‘𝑥) / ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥))) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))) lim_ℂ 𝐵))
407	390, 406	eleqtrd 2842	1 ⊢ (𝜑 → 0 ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) / ((𝑥 − 𝐵)↑(𝑀 + 1)))) lim_ℂ 𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2107 ≠ wne 2944 ∃wrex 3066 Vcvv 3433 ∖ cdif 3885 ∩ cin 3887 ⊆ wss 3888 {csn 4562 {cpr 4564 ⟨cop 4568 class class class wbr 5075 ↦ cmpt 5158 × cxp 5588 dom cdm 5590 ran crn 5591 ↾ cres 5592 “ cima 5593 ⟶wf 6433 ‘cfv 6437 (class class class)co 7284 ↑_m cmap 8624 ↑_pm cpm 8625 ℂcc 10878 ℝcr 10879 0cc0 10880 1c1 10881 + caddc 10883 · cmul 10885 ≤ cle 11019 − cmin 11214 / cdiv 11641 ℕcn 11982 ℕ₀cn0 12242 ℤ_≥cuz 12591 (,)cioo 13088 ...cfz 13248 ..^cfzo 13391 ↑cexp 13791 !cfa 13996 TopOpenctopn 17141 topGenctg 17157 DivRingcdr 20000 SubRingcsubrg 20029 ℂ_fldccnfld 20606 ℝ_fldcrefld 20818 Topctop 22051 TopOnctopon 22068 intcnt 22177 Cn ccn 22384 CnP ccnp 22385 ×_t ctx 22720 –cn→ccncf 24048 lim_ℂ climc 25035 D cdv 25036 D^𝑛 cdvn 25037 Polycply 25354 degcdgr 25357 Tayl ctayl 25521

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2710 ax-rep 5210 ax-sep 5224 ax-nul 5231 ax-pow 5289 ax-pr 5353 ax-un 7597 ax-inf2 9408 ax-cnex 10936 ax-resscn 10937 ax-1cn 10938 ax-icn 10939 ax-addcl 10940 ax-addrcl 10941 ax-mulcl 10942 ax-mulrcl 10943 ax-mulcom 10944 ax-addass 10945 ax-mulass 10946 ax-distr 10947 ax-i2m1 10948 ax-1ne0 10949 ax-1rid 10950 ax-rnegex 10951 ax-rrecex 10952 ax-cnre 10953 ax-pre-lttri 10954 ax-pre-lttrn 10955 ax-pre-ltadd 10956 ax-pre-mulgt0 10957 ax-pre-sup 10958 ax-addf 10959 ax-mulf 10960

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2541 df-eu 2570 df-clab 2717 df-cleq 2731 df-clel 2817 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3070 df-rex 3071 df-rmo 3072 df-reu 3073 df-rab 3074 df-v 3435 df-sbc 3718 df-csb 3834 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-pss 3907 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-tp 4567 df-op 4569 df-uni 4841 df-int 4881 df-iun 4927 df-iin 4928 df-br 5076 df-opab 5138 df-mpt 5159 df-tr 5193 df-id 5490 df-eprel 5496 df-po 5504 df-so 5505 df-fr 5545 df-se 5546 df-we 5547 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-pred 6206 df-ord 6273 df-on 6274 df-lim 6275 df-suc 6276 df-iota 6395 df-fun 6439 df-fn 6440 df-f 6441 df-f1 6442 df-fo 6443 df-f1o 6444 df-fv 6445 df-isom 6446 df-riota 7241 df-ov 7287 df-oprab 7288 df-mpo 7289 df-of 7542 df-om 7722 df-1st 7840 df-2nd 7841 df-supp 7987 df-tpos 8051 df-frecs 8106 df-wrecs 8137 df-recs 8211 df-rdg 8250 df-1o 8306 df-2o 8307 df-er 8507 df-map 8626 df-pm 8627 df-ixp 8695 df-en 8743 df-dom 8744 df-sdom 8745 df-fin 8746 df-fsupp 9138 df-fi 9179 df-sup 9210 df-inf 9211 df-oi 9278 df-card 9706 df-pnf 11020 df-mnf 11021 df-xr 11022 df-ltxr 11023 df-le 11024 df-sub 11216 df-neg 11217 df-div 11642 df-nn 11983 df-2 12045 df-3 12046 df-4 12047 df-5 12048 df-6 12049 df-7 12050 df-8 12051 df-9 12052 df-n0 12243 df-xnn0 12315 df-z 12329 df-dec 12447 df-uz 12592 df-q 12698 df-rp 12740 df-xneg 12857 df-xadd 12858 df-xmul 12859 df-ioo 13092 df-ioc 13093 df-ico 13094 df-icc 13095 df-fz 13249 df-fzo 13392 df-fl 13521 df-seq 13731 df-exp 13792 df-fac 13997 df-hash 14054 df-cj 14819 df-re 14820 df-im 14821 df-sqrt 14955 df-abs 14956 df-clim 15206 df-rlim 15207 df-sum 15407 df-struct 16857 df-sets 16874 df-slot 16892 df-ndx 16904 df-base 16922 df-ress 16951 df-plusg 16984 df-mulr 16985 df-starv 16986 df-sca 16987 df-vsca 16988 df-ip 16989 df-tset 16990 df-ple 16991 df-ds 16993 df-unif 16994 df-hom 16995 df-cco 16996 df-rest 17142 df-topn 17143 df-0g 17161 df-gsum 17162 df-topgen 17163 df-pt 17164 df-prds 17167 df-xrs 17222 df-qtop 17227 df-imas 17228 df-xps 17230 df-mre 17304 df-mrc 17305 df-acs 17307 df-mgm 18335 df-sgrp 18384 df-mnd 18395 df-submnd 18440 df-grp 18589 df-minusg 18590 df-mulg 18710 df-subg 18761 df-cntz 18932 df-cmn 19397 df-abl 19398 df-mgp 19730 df-ur 19747 df-ring 19794 df-cring 19795 df-oppr 19871 df-dvdsr 19892 df-unit 19893 df-invr 19923 df-dvr 19934 df-drng 20002 df-subrg 20031 df-psmet 20598 df-xmet 20599 df-met 20600 df-bl 20601 df-mopn 20602 df-fbas 20603 df-fg 20604 df-cnfld 20607 df-refld 20819 df-top 22052 df-topon 22069 df-topsp 22091 df-bases 22105 df-cld 22179 df-ntr 22180 df-cls 22181 df-nei 22258 df-lp 22296 df-perf 22297 df-cn 22387 df-cnp 22388 df-haus 22475 df-cmp 22547 df-tx 22722 df-hmeo 22915 df-fil 23006 df-fm 23098 df-flim 23099 df-flf 23100 df-tsms 23287 df-xms 23482 df-ms 23483 df-tms 23484 df-cncf 24050 df-0p 24843 df-limc 25039 df-dv 25040 df-dvn 25041 df-ply 25358 df-idp 25359 df-coe 25360 df-dgr 25361 df-tayl 25523

This theorem is referenced by: taylth 25543

W3C validator