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Theorem taylthlem2 24128

Description: Lemma for taylth 24129. (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		1eluzge0 11732	. . . . . . . . . . . 12 ⊢ 1 ∈ (ℤ_≥‘0)
4		fzss1 12380	. . . . . . . . . . . 12 ⊢ (1 ∈ (ℤ_≥‘0) → (1...𝑁) ⊆ (0...𝑁))
5	3, 4	ax-mp 5	. . . . . . . . . . 11 ⊢ (1...𝑁) ⊆ (0...𝑁)
6		taylthlem2.m	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (1..^𝑁))
7		fzofzp1 12565	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (1..^𝑁) → (𝑀 + 1) ∈ (1...𝑁))
8	6, 7	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑀 + 1) ∈ (1...𝑁))
9	5, 8	sseldi 3601	. . . . . . . . . 10 ⊢ (𝜑 → (𝑀 + 1) ∈ (0...𝑁))
10		fznn0sub2 12446	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (0...𝑁) → (𝑁 − (𝑀 + 1)) ∈ (0...𝑁))
11	9, 10	syl 17	. . . . . . . . 9 ⊢ (𝜑 → (𝑁 − (𝑀 + 1)) ∈ (0...𝑁))
12		elfznn0 12433	. . . . . . . . 9 ⊢ ((𝑁 − (𝑀 + 1)) ∈ (0...𝑁) → (𝑁 − (𝑀 + 1)) ∈ ℕ₀)
13	11, 12	syl 17	. . . . . . . 8 ⊢ (𝜑 → (𝑁 − (𝑀 + 1)) ∈ ℕ₀)
14		dvnfre 23715	. . . . . . . 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)
15	2, 1, 13, 14	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))⟶ℝ)
16		reelprrecn 10028	. . . . . . . . . . . 12 ⊢ ℝ ∈ {ℝ, ℂ}
17	16	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → ℝ ∈ {ℝ, ℂ})
18		cnex 10017	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
19	18	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
20		reex 10027	. . . . . . . . . . . . 13 ⊢ ℝ ∈ V
21	20	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℝ ∈ V)
22		ax-resscn 9993	. . . . . . . . . . . . 13 ⊢ ℝ ⊆ ℂ
23		fss 6056	. . . . . . . . . . . . 13 ⊢ ((𝐹:𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → 𝐹:𝐴⟶ℂ)
24	2, 22, 23	sylancl 694	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
25		elpm2r 7875	. . . . . . . . . . . 12 ⊢ (((ℂ ∈ V ∧ ℝ ∈ V) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ ℝ)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
26	19, 21, 24, 1, 25	syl22anc 1327	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm ℝ))
27		dvnbss 23691	. . . . . . . . . . 11 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ⊆ dom 𝐹)
28	17, 26, 13, 27	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ⊆ dom 𝐹)
29		fdm 6051	. . . . . . . . . . 11 ⊢ (𝐹:𝐴⟶ℝ → dom 𝐹 = 𝐴)
30	2, 29	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → dom 𝐹 = 𝐴)
31	28, 30	sseqtrd 3641	. . . . . . . . 9 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ⊆ 𝐴)
32		taylth.d	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) = 𝐴)
33		dvn2bss 23693	. . . . . . . . . . 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))))
34	17, 26, 11, 33	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))))
35	32, 34	eqsstr3d 3640	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))))
36	31, 35	eqssd 3620	. . . . . . . 8 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) = 𝐴)
37	36	feq2d 6031	. . . . . . 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))
38	15, 37	mpbid 222	. . . . . 6 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℝ)
39	38	ffvelrnda 6359	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℝ)
40	1	sselda 3603	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℝ)
41		fvres 6207	. . . . . . . 8 ⊢ (𝑦 ∈ ℝ → ((((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ)‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))
42	41	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ)‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))
43		resubdrg 19954	. . . . . . . . . . . 12 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
44	43	simpli 474	. . . . . . . . . . 11 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
45	44	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ∈ (SubRing‘ℂ_fld))
46		taylth.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑁 ∈ ℕ)
47	46	nnnn0d 11351	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
48		taylth.b	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ 𝐴)
49	48, 32	eleqtrrd 2704	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ dom ((ℝ D^𝑛 𝐹)‘𝑁))
50		taylth.t	. . . . . . . . . . . 12 ⊢ 𝑇 = (𝑁(ℝ Tayl 𝐹)𝐵)
51	1, 48	sseldd 3604	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐵 ∈ ℝ)
52	2	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹:𝐴⟶ℝ)
53	1	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐴 ⊆ ℝ)
54		elfznn0 12433	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
55	54	adantl 482	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
56		dvnfre 23715	. . . . . . . . . . . . . . 15 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ 𝑘 ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘𝑘):dom ((ℝ D^𝑛 𝐹)‘𝑘)⟶ℝ)
57	52, 53, 55, 56	syl3anc 1326	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((ℝ D^𝑛 𝐹)‘𝑘):dom ((ℝ D^𝑛 𝐹)‘𝑘)⟶ℝ)
58	16	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ℝ ∈ {ℝ, ℂ})
59	26	adantr 481	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm ℝ))
60		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
61		dvn2bss 23693	. . . . . . . . . . . . . . . 16 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ 𝑘 ∈ (0...𝑁)) → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘𝑘))
62	58, 59, 60, 61	syl3anc 1326	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘𝑘))
63	49	adantr 481	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((ℝ D^𝑛 𝐹)‘𝑁))
64	62, 63	sseldd 3604	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((ℝ D^𝑛 𝐹)‘𝑘))
65	57, 64	ffvelrnd 6360	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((ℝ D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℝ)
66		faccl 13070	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ₀ → (!‘𝑘) ∈ ℕ)
67	55, 66	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
68	65, 67	nndivred 11069	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((ℝ D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℝ)
69	17, 24, 1, 47, 49, 50, 45, 51, 68	taylply2 24122	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑇 ∈ (Poly‘ℝ) ∧ (deg‘𝑇) ≤ 𝑁))
70	69	simpld 475	. . . . . . . . . 10 ⊢ (𝜑 → 𝑇 ∈ (Poly‘ℝ))
71		dvnply2 24042	. . . . . . . . . 10 ⊢ ((ℝ ∈ (SubRing‘ℂ_fld) ∧ 𝑇 ∈ (Poly‘ℝ) ∧ (𝑁 − (𝑀 + 1)) ∈ ℕ₀) → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ))
72	45, 70, 13, 71	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ))
73		plyreres 24038	. . . . . . . . 9 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ):ℝ⟶ℝ)
74	72, 73	syl 17	. . . . . . . 8 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ):ℝ⟶ℝ)
75	74	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ↾ ℝ)‘𝑦) ∈ ℝ)
76	42, 75	eqeltrrd 2702	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℝ)
77	40, 76	syldan 487	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℝ)
78	39, 77	resubcld 10458	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)) ∈ ℝ)
79		eqid 2622	. . . 4 ⊢ (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)))
80	78, 79	fmptd 6385	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))):𝐴⟶ℝ)
81	51	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝐵 ∈ ℝ)
82	40, 81	resubcld 10458	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑦 − 𝐵) ∈ ℝ)
83		elfzouz 12474	. . . . . . . . . 10 ⊢ (𝑀 ∈ (1..^𝑁) → 𝑀 ∈ (ℤ_≥‘1))
84	6, 83	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
85		nnuz 11723	. . . . . . . . 9 ⊢ ℕ = (ℤ_≥‘1)
86	84, 85	syl6eleqr 2712	. . . . . . . 8 ⊢ (𝜑 → 𝑀 ∈ ℕ)
87	86	nnnn0d 11351	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ₀)
88	87	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑀 ∈ ℕ₀)
89		1nn0 11308	. . . . . . 7 ⊢ 1 ∈ ℕ₀
90	89	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 1 ∈ ℕ₀)
91	88, 90	nn0addcld 11355	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (𝑀 + 1) ∈ ℕ₀)
92	82, 91	reexpcld 13025	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℝ)
93		eqid 2622	. . . 4 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) = (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))
94	92, 93	fmptd 6385	. . 3 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))):𝐴⟶ℝ)
95		retop 22565	. . . . . 6 ⊢ (topGen‘ran (,)) ∈ Top
96		uniretop 22566	. . . . . . 7 ⊢ ℝ = ∪ (topGen‘ran (,))
97	96	ntrss2 20861	. . . . . 6 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ⊆ ℝ) → ((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴)
98	95, 1, 97	sylancr 695	. . . . 5 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝐴) ⊆ 𝐴)
99	46	nncnd 11036	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑁 ∈ ℂ)
100	86	nncnd 11036	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℂ)
101		1cnd 10056	. . . . . . . . . . 11 ⊢ (𝜑 → 1 ∈ ℂ)
102	99, 100, 101	nppcan2d 10418	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑁 − (𝑀 + 1)) + 1) = (𝑁 − 𝑀))
103	102	fveq2d 6195	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘((𝑁 − (𝑀 + 1)) + 1)) = ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
104	22	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℝ ⊆ ℂ)
105		dvnp1 23688	. . . . . . . . . 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)))))
106	104, 26, 13, 105	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘((𝑁 − (𝑀 + 1)) + 1)) = (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
107	103, 106	eqtr3d 2658	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
108	107	dmeqd 5326	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))))
109		fzonnsub 12493	. . . . . . . . . . . 12 ⊢ (𝑀 ∈ (1..^𝑁) → (𝑁 − 𝑀) ∈ ℕ)
110	6, 109	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ)
111	110	nnnn0d 11351	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ ℕ₀)
112		dvnbss 23691	. . . . . . . . . 10 ⊢ ((ℝ ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm ℝ) ∧ (𝑁 − 𝑀) ∈ ℕ₀) → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) ⊆ dom 𝐹)
113	17, 26, 111, 112	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) ⊆ dom 𝐹)
114	113, 30	sseqtrd 3641	. . . . . . . 8 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) ⊆ 𝐴)
115		elfzofz 12485	. . . . . . . . . . . . 13 ⊢ (𝑀 ∈ (1..^𝑁) → 𝑀 ∈ (1...𝑁))
116	6, 115	syl 17	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
117	5, 116	sseldi 3601	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ (0...𝑁))
118		fznn0sub2 12446	. . . . . . . . . . 11 ⊢ (𝑀 ∈ (0...𝑁) → (𝑁 − 𝑀) ∈ (0...𝑁))
119	117, 118	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (𝑁 − 𝑀) ∈ (0...𝑁))
120		dvn2bss 23693	. . . . . . . . . 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)))
121	17, 26, 119, 120	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘𝑁) ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
122	32, 121	eqsstr3d 3640	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)))
123	114, 122	eqssd 3620	. . . . . . 7 ⊢ (𝜑 → dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = 𝐴)
124	108, 123	eqtr3d 2658	. . . . . 6 ⊢ (𝜑 → dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) = 𝐴)
125		fss 6056	. . . . . . . 8 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℝ ∧ ℝ ⊆ ℂ) → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℂ)
126	38, 22, 125	sylancl 694	. . . . . . 7 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))):𝐴⟶ℂ)
127		eqid 2622	. . . . . . . 8 ⊢ (TopOpen‘ℂ_fld) = (TopOpen‘ℂ_fld)
128	127	tgioo2 22606	. . . . . . 7 ⊢ (topGen‘ran (,)) = ((TopOpen‘ℂ_fld) ↾_t ℝ)
129	104, 126, 1, 128, 127	dvbssntr 23664	. . . . . 6 ⊢ (𝜑 → dom (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) ⊆ ((int‘(topGen‘ran (,)))‘𝐴))
130	124, 129	eqsstr3d 3640	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ((int‘(topGen‘ran (,)))‘𝐴))
131	98, 130	eqssd 3620	. . . 4 ⊢ (𝜑 → ((int‘(topGen‘ran (,)))‘𝐴) = 𝐴)
132	96	isopn3 20870	. . . . 5 ⊢ (((topGen‘ran (,)) ∈ Top ∧ 𝐴 ⊆ ℝ) → (𝐴 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝐴) = 𝐴))
133	95, 1, 132	sylancr 695	. . . 4 ⊢ (𝜑 → (𝐴 ∈ (topGen‘ran (,)) ↔ ((int‘(topGen‘ran (,)))‘𝐴) = 𝐴))
134	131, 133	mpbird 247	. . 3 ⊢ (𝜑 → 𝐴 ∈ (topGen‘ran (,)))
135		eqid 2622	. . 3 ⊢ (𝐴 ∖ {𝐵}) = (𝐴 ∖ {𝐵})
136		difss 3737	. . . 4 ⊢ (𝐴 ∖ {𝐵}) ⊆ 𝐴
137	39	recnd 10068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
138		dvnf 23690	. . . . . . . . . 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)
139	17, 26, 111, 138	syl3anc 1326	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℂ)
140	123	feq2d 6031	. . . . . . . . 9 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℂ ↔ ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℂ))
141	139, 140	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℂ)
142	141	ffvelrnda 6359	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) ∈ ℂ)
143		dvnfre 23715	. . . . . . . . . . 11 ⊢ ((𝐹:𝐴⟶ℝ ∧ 𝐴 ⊆ ℝ ∧ (𝑁 − 𝑀) ∈ ℕ₀) → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℝ)
144	2, 1, 111, 143	syl3anc 1326	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℝ)
145	123	feq2d 6031	. . . . . . . . . 10 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):dom ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))⟶ℝ ↔ ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℝ))
146	144, 145	mpbid 222	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)):𝐴⟶ℝ)
147	146	feqmptd 6249	. . . . . . . 8 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀)) = (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦)))
148	38	feqmptd 6249	. . . . . . . . 9 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) = (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦)))
149	148	oveq2d 6666	. . . . . . . 8 ⊢ (𝜑 → (ℝ D ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))) = (ℝ D (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦))))
150	107, 147, 149	3eqtr3rd 2665	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦)))
151	77	recnd 10068	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
152		fvexd 6203	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) ∈ V)
153	76	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
154		recn 10026	. . . . . . . . 9 ⊢ (𝑦 ∈ ℝ → 𝑦 ∈ ℂ)
155		dvnply2 24042	. . . . . . . . . . . 12 ⊢ ((ℝ ∈ (SubRing‘ℂ_fld) ∧ 𝑇 ∈ (Poly‘ℝ) ∧ (𝑁 − 𝑀) ∈ ℕ₀) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) ∈ (Poly‘ℝ))
156	45, 70, 111, 155	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) ∈ (Poly‘ℝ))
157		plyf 23954	. . . . . . . . . . 11 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) ∈ (Poly‘ℝ) → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)):ℂ⟶ℂ)
158	156, 157	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)):ℂ⟶ℂ)
159	158	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) ∈ ℂ)
160	154, 159	sylan2 491	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) ∈ ℂ)
161	127	cnfldtopon 22586	. . . . . . . . . 10 ⊢ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)
162		toponmax 20730	. . . . . . . . . 10 ⊢ ((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) → ℂ ∈ (TopOpen‘ℂ_fld))
163	161, 162	mp1i 13	. . . . . . . . 9 ⊢ (𝜑 → ℂ ∈ (TopOpen‘ℂ_fld))
164		df-ss 3588	. . . . . . . . . 10 ⊢ (ℝ ⊆ ℂ ↔ (ℝ ∩ ℂ) = ℝ)
165	104, 164	sylib 208	. . . . . . . . 9 ⊢ (𝜑 → (ℝ ∩ ℂ) = ℝ)
166		plyf 23954	. . . . . . . . . . 11 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ) → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))):ℂ⟶ℂ)
167	72, 166	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))):ℂ⟶ℂ)
168	167	ffvelrnda 6359	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) ∈ ℂ)
169	102	fveq2d 6195	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − (𝑀 + 1)) + 1)) = ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)))
170		ssid 3624	. . . . . . . . . . . . 13 ⊢ ℂ ⊆ ℂ
171	170	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ⊆ ℂ)
172		mapsspm 7891	. . . . . . . . . . . . 13 ⊢ (ℂ ↑_𝑚 ℂ) ⊆ (ℂ ↑_pm ℂ)
173		plyf 23954	. . . . . . . . . . . . . . 15 ⊢ (𝑇 ∈ (Poly‘ℝ) → 𝑇:ℂ⟶ℂ)
174	70, 173	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑇:ℂ⟶ℂ)
175	18, 18	elmap 7886	. . . . . . . . . . . . . 14 ⊢ (𝑇 ∈ (ℂ ↑_𝑚 ℂ) ↔ 𝑇:ℂ⟶ℂ)
176	174, 175	sylibr 224	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_𝑚 ℂ))
177	172, 176	sseldi 3601	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑇 ∈ (ℂ ↑_pm ℂ))
178		dvnp1 23688	. . . . . . . . . . . 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)))))
179	171, 177, 13, 178	syl3anc 1326	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘((𝑁 − (𝑀 + 1)) + 1)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))))
180	169, 179	eqtr3d 2658	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) = (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))))
181	158	feqmptd 6249	. . . . . . . . . 10 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀)) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
182	167	feqmptd 6249	. . . . . . . . . . 11 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))
183	182	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))) = (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))))
184	180, 181, 183	3eqtr3rd 2665	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ ℂ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
185	127, 17, 163, 165, 168, 159, 184	dvmptres3 23719	. . . . . . . 8 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ ℝ ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
186	17, 153, 160, 185, 1, 128, 127, 134	dvmptres 23726	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) = (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)))
187	17, 137, 142, 150, 151, 152, 186	dvmptsub 23730	. . . . . 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))))
188	187	dmeqd 5326	. . . . 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))))
189		ovex 6678	. . . . . 6 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦)) ∈ V
190		eqid 2622	. . . . . 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)))
191	189, 190	dmmpti 6023	. . . . 5 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦))) = 𝐴
192	188, 191	syl6eq 2672	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)))) = 𝐴)
193	136, 192	syl5sseqr 3654	. . 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)))))
194		simpr 477	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑦 ∈ ℂ)
195	51	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐵 ∈ ℝ)
196	195	recnd 10068	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝐵 ∈ ℂ)
197	194, 196	subcld 10392	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑦 − 𝐵) ∈ ℂ)
198	87	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℕ₀)
199	89	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 1 ∈ ℕ₀)
200	198, 199	nn0addcld 11355	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑀 + 1) ∈ ℕ₀)
201	197, 200	expcld 13008	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℂ)
202	154, 201	sylan2 491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℂ)
203	100	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 𝑀 ∈ ℂ)
204		1cnd 10056	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 1 ∈ ℂ)
205	203, 204	addcld 10059	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (𝑀 + 1) ∈ ℂ)
206	197, 198	expcld 13008	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑦 − 𝐵)↑𝑀) ∈ ℂ)
207	205, 206	mulcld 10060	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ∈ ℂ)
208	154, 207	sylan2 491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ∈ ℂ)
209	18	prid2 4298	. . . . . . . . . . 11 ⊢ ℂ ∈ {ℝ, ℂ}
210	209	a1i 11	. . . . . . . . . 10 ⊢ (𝜑 → ℂ ∈ {ℝ, ℂ})
211		simpr 477	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
212		elfznn 12370	. . . . . . . . . . . . . 14 ⊢ ((𝑀 + 1) ∈ (1...𝑁) → (𝑀 + 1) ∈ ℕ)
213	8, 212	syl 17	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ)
214	213	nnnn0d 11351	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑀 + 1) ∈ ℕ₀)
215	214	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑀 + 1) ∈ ℕ₀)
216	211, 215	expcld 13008	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥↑(𝑀 + 1)) ∈ ℂ)
217		ovexd 6680	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → ((𝑀 + 1) · (𝑥↑𝑀)) ∈ V)
218	210	dvmptid 23720	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ 𝑦)) = (𝑦 ∈ ℂ ↦ 1))
219		0cnd 10033	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → 0 ∈ ℂ)
220	51	recnd 10068	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐵 ∈ ℂ)
221	210, 220	dvmptc 23721	. . . . . . . . . . . 12 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ 𝐵)) = (𝑦 ∈ ℂ ↦ 0))
222	210, 194, 204, 218, 196, 219, 221	dvmptsub 23730	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦 − 𝐵))) = (𝑦 ∈ ℂ ↦ (1 − 0)))
223		1m0e1 11131	. . . . . . . . . . . 12 ⊢ (1 − 0) = 1
224	223	mpteq2i 4741	. . . . . . . . . . 11 ⊢ (𝑦 ∈ ℂ ↦ (1 − 0)) = (𝑦 ∈ ℂ ↦ 1)
225	222, 224	syl6eq 2672	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ (𝑦 − 𝐵))) = (𝑦 ∈ ℂ ↦ 1))
226		dvexp 23716	. . . . . . . . . . . 12 ⊢ ((𝑀 + 1) ∈ ℕ → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1)))))
227	213, 226	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1)))))
228	100, 101	pncand 10393	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ((𝑀 + 1) − 1) = 𝑀)
229	228	oveq2d 6666	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑥↑((𝑀 + 1) − 1)) = (𝑥↑𝑀))
230	229	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1))) = ((𝑀 + 1) · (𝑥↑𝑀)))
231	230	mpteq2dv 4745	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑((𝑀 + 1) − 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑𝑀))))
232	227, 231	eqtrd 2656	. . . . . . . . . 10 ⊢ (𝜑 → (ℂ D (𝑥 ∈ ℂ ↦ (𝑥↑(𝑀 + 1)))) = (𝑥 ∈ ℂ ↦ ((𝑀 + 1) · (𝑥↑𝑀))))
233		oveq1 6657	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐵) → (𝑥↑(𝑀 + 1)) = ((𝑦 − 𝐵)↑(𝑀 + 1)))
234		oveq1 6657	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑦 − 𝐵) → (𝑥↑𝑀) = ((𝑦 − 𝐵)↑𝑀))
235	234	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑥 = (𝑦 − 𝐵) → ((𝑀 + 1) · (𝑥↑𝑀)) = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
236	210, 210, 197, 204, 216, 217, 225, 232, 233, 235	dvmptco 23735	. . . . . . . . 9 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ ℂ ↦ (((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) · 1)))
237	207	mulid1d 10057	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ ℂ) → (((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) · 1) = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
238	237	mpteq2dva 4744	. . . . . . . . 9 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ (((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) · 1)) = (𝑦 ∈ ℂ ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
239	236, 238	eqtrd 2656	. . . . . . . 8 ⊢ (𝜑 → (ℂ D (𝑦 ∈ ℂ ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ ℂ ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
240	127, 17, 163, 165, 201, 207, 239	dvmptres3 23719	. . . . . . 7 ⊢ (𝜑 → (ℝ D (𝑦 ∈ ℝ ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ ℝ ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
241	17, 202, 208, 240, 1, 128, 127, 134	dvmptres 23726	. . . . . 6 ⊢ (𝜑 → (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
242	241	dmeqd 5326	. . . . 5 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = dom (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
243		ovex 6678	. . . . . 6 ⊢ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ∈ V
244		eqid 2622	. . . . . 6 ⊢ (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) = (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
245	243, 244	dmmpti 6023	. . . . 5 ⊢ dom (𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) = 𝐴
246	242, 245	syl6eq 2672	. . . 4 ⊢ (𝜑 → dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) = 𝐴)
247	136, 246	syl5sseqr 3654	. . 3 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ dom (ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))))
248	17, 24, 1, 11, 49, 50	dvntaylp0 24126	. . . . . 6 ⊢ (𝜑 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵) = (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵))
249	248	oveq2d 6666	. . . . 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)))
250	126, 48	ffvelrnd 6360	. . . . . 6 ⊢ (𝜑 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) ∈ ℂ)
251	250	subidd 10380	. . . . 5 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵)) = 0)
252	249, 251	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵)) = 0)
253	127	subcn 22669	. . . . . . 7 ⊢ − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
254	253	a1i 11	. . . . . 6 ⊢ (𝜑 → − ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld)))
255		dvcn 23684	. . . . . . . 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))
256	104, 126, 1, 124, 255	syl31anc 1329	. . . . . . 7 ⊢ (𝜑 → ((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1))) ∈ (𝐴–cn→ℂ))
257	148, 256	eqeltrrd 2702	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦)) ∈ (𝐴–cn→ℂ))
258		plycn 24017	. . . . . . . 8 ⊢ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (Poly‘ℝ) → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (ℂ–cn→ℂ))
259	72, 258	syl 17	. . . . . . 7 ⊢ (𝜑 → ((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1))) ∈ (ℂ–cn→ℂ))
260	1, 22	syl6ss 3615	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
261		cncfmptid 22715	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℂ ∧ ℂ ⊆ ℂ) → (𝑦 ∈ 𝐴 ↦ 𝑦) ∈ (𝐴–cn→ℂ))
262	260, 170, 261	sylancl 694	. . . . . . 7 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ 𝑦) ∈ (𝐴–cn→ℂ))
263	259, 262	cncfmpt1f 22716	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦)) ∈ (𝐴–cn→ℂ))
264	127, 254, 257, 263	cncfmpt2f 22717	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) ∈ (𝐴–cn→ℂ))
265		fveq2 6191	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝐵))
266		fveq2 6191	. . . . . 6 ⊢ (𝑦 = 𝐵 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝐵))
267	265, 266	oveq12d 6668	. . . . 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)))
268	264, 48, 267	cnmptlimc 23654	. . . 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))
269	252, 268	eqeltrrd 2702	. . 3 ⊢ (𝜑 → 0 ∈ ((𝑦 ∈ 𝐴 ↦ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦))) lim_ℂ 𝐵))
270	220	subidd 10380	. . . . . 6 ⊢ (𝜑 → (𝐵 − 𝐵) = 0)
271	270	oveq1d 6665	. . . . 5 ⊢ (𝜑 → ((𝐵 − 𝐵)↑(𝑀 + 1)) = (0↑(𝑀 + 1)))
272	213	0expd 13024	. . . . 5 ⊢ (𝜑 → (0↑(𝑀 + 1)) = 0)
273	271, 272	eqtrd 2656	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝐵)↑(𝑀 + 1)) = 0)
274	260	sselda 3603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ ℂ)
275	274, 201	syldan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ ℂ)
276	275, 93	fmptd 6385	. . . . . 6 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))):𝐴⟶ℂ)
277		dvcn 23684	. . . . . 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))
278	104, 276, 1, 246, 277	syl31anc 1329	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ∈ (𝐴–cn→ℂ))
279		oveq1 6657	. . . . . 6 ⊢ (𝑦 = 𝐵 → (𝑦 − 𝐵) = (𝐵 − 𝐵))
280	279	oveq1d 6665	. . . . 5 ⊢ (𝑦 = 𝐵 → ((𝑦 − 𝐵)↑(𝑀 + 1)) = ((𝐵 − 𝐵)↑(𝑀 + 1)))
281	278, 48, 280	cnmptlimc 23654	. . . 4 ⊢ (𝜑 → ((𝐵 − 𝐵)↑(𝑀 + 1)) ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) lim_ℂ 𝐵))
282	273, 281	eqeltrrd 2702	. . 3 ⊢ (𝜑 → 0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) lim_ℂ 𝐵))
283	260	ssdifssd 3748	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℂ)
284	283	sselda 3603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ∈ ℂ)
285	220	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
286	284, 285	subcld 10392	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑦 − 𝐵) ∈ ℂ)
287		eldifsni 4320	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝐴 ∖ {𝐵}) → 𝑦 ≠ 𝐵)
288	287	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑦 ≠ 𝐵)
289	284, 285, 288	subne0d 10401	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑦 − 𝐵) ≠ 0)
290	213	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℕ)
291	290	nnzd 11481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℤ)
292	286, 289, 291	expne0d 13014	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 − 𝐵)↑(𝑀 + 1)) ≠ 0)
293	292	necomd 2849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 0 ≠ ((𝑦 − 𝐵)↑(𝑀 + 1)))
294	293	neneqd 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ¬ 0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
295	294	nrexdv 3001	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
296		df-ima 5127	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})) = ran ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵}))
297	296	eleq2i 2693	. . . . 5 ⊢ (0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})) ↔ 0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})))
298		resmpt 5449	. . . . . . 7 ⊢ ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))
299	136, 298	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))
300		ovex 6678	. . . . . 6 ⊢ ((𝑦 − 𝐵)↑(𝑀 + 1)) ∈ V
301	299, 300	elrnmpti 5376	. . . . 5 ⊢ (0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
302	297, 301	bitri 264	. . . 4 ⊢ (0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑦 − 𝐵)↑(𝑀 + 1)))
303	295, 302	sylnibr 319	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))) “ (𝐴 ∖ {𝐵})))
304	100	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℂ)
305		1cnd 10056	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 1 ∈ ℂ)
306	304, 305	addcld 10059	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℂ)
307	284, 206	syldan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 − 𝐵)↑𝑀) ∈ ℂ)
308	290	nnne0d 11065	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ≠ 0)
309	86	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℕ)
310	309	nnzd 11481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℤ)
311	286, 289, 310	expne0d 13014	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 − 𝐵)↑𝑀) ≠ 0)
312	306, 307, 308, 311	mulne0d 10679	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) ≠ 0)
313	312	necomd 2849	. . . . . 6 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → 0 ≠ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
314	313	neneqd 2799	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐴 ∖ {𝐵})) → ¬ 0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
315	314	nrexdv 3001	. . . 4 ⊢ (𝜑 → ¬ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
316	241	imaeq1d 5465	. . . . . . 7 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) = ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) “ (𝐴 ∖ {𝐵})))
317		df-ima 5127	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) “ (𝐴 ∖ {𝐵})) = ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵}))
318	316, 317	syl6eq 2672	. . . . . 6 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) = ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})))
319	318	eleq2d 2687	. . . . 5 ⊢ (𝜑 → (0 ∈ ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) ↔ 0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵}))))
320		resmpt 5449	. . . . . . 7 ⊢ ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
321	136, 320	ax-mp 5	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})) = (𝑦 ∈ (𝐴 ∖ {𝐵}) ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
322	321, 243	elrnmpti 5376	. . . . 5 ⊢ (0 ∈ ran ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))) ↾ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))
323	319, 322	syl6bb 276	. . . 4 ⊢ (𝜑 → (0 ∈ ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})) ↔ ∃𝑦 ∈ (𝐴 ∖ {𝐵})0 = ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀))))
324	315, 323	mtbird 315	. . 3 ⊢ (𝜑 → ¬ 0 ∈ ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))) “ (𝐴 ∖ {𝐵})))
325		eldifi 3732	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ∈ 𝐴)
326	141	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
327	325, 326	sylan2 491	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
328	1	ssdifssd 3748	. . . . . . . . . 10 ⊢ (𝜑 → (𝐴 ∖ {𝐵}) ⊆ ℝ)
329	328	sselda 3603	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℝ)
330	329	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ ℂ)
331	158	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
332	330, 331	syldan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥) ∈ ℂ)
333	327, 332	subcld 10392	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) ∈ ℂ)
334	51	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℝ)
335	329, 334	resubcld 10458	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℝ)
336	87	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℕ₀)
337	335, 336	reexpcld 13025	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑𝑀) ∈ ℝ)
338	337	recnd 10068	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑𝑀) ∈ ℂ)
339	334	recnd 10068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝐵 ∈ ℂ)
340	330, 339	subcld 10392	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ∈ ℂ)
341		eldifsni 4320	. . . . . . . . 9 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → 𝑥 ≠ 𝐵)
342	341	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ≠ 𝐵)
343	330, 339, 342	subne0d 10401	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑥 − 𝐵) ≠ 0)
344	336	nn0zd 11480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑀 ∈ ℤ)
345	340, 343, 344	expne0d 13014	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑥 − 𝐵)↑𝑀) ≠ 0)
346	333, 338, 345	divcld 10801	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) / ((𝑥 − 𝐵)↑𝑀)) ∈ ℂ)
347	213	nnrecred 11066	. . . . . . 7 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ℝ)
348	347	recnd 10068	. . . . . 6 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ℂ)
349	348	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (1 / (𝑀 + 1)) ∈ ℂ)
350		txtopon 21394	. . . . . . . 8 ⊢ (((TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ) ∧ (TopOpen‘ℂ_fld) ∈ (TopOn‘ℂ)) → ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ∈ (TopOn‘(ℂ × ℂ)))
351	161, 161, 350	mp2an 708	. . . . . . 7 ⊢ ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ∈ (TopOn‘(ℂ × ℂ))
352	351	toponunii 20721	. . . . . . . 8 ⊢ (ℂ × ℂ) = ∪ ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld))
353	352	restid 16094	. . . . . . 7 ⊢ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ∈ (TopOn‘(ℂ × ℂ)) → (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ↾_t (ℂ × ℂ)) = ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)))
354	351, 353	ax-mp 5	. . . . . 6 ⊢ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ↾_t (ℂ × ℂ)) = ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld))
355	354	eqcomi 2631	. . . . 5 ⊢ ((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) = (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) ↾_t (ℂ × ℂ))
356		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))
357		limcresi 23649	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵) ⊆ (((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) lim_ℂ 𝐵)
358		resmpt 5449	. . . . . . . . 9 ⊢ ((𝐴 ∖ {𝐵}) ⊆ 𝐴 → ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))))
359	136, 358	ax-mp 5	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) = (𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1)))
360	359	oveq1i 6660	. . . . . . 7 ⊢ (((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ↾ (𝐴 ∖ {𝐵})) lim_ℂ 𝐵) = ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵)
361	357, 360	sseqtri 3637	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵) ⊆ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵)
362		cncfmptc 22714	. . . . . . . 8 ⊢ (((1 / (𝑀 + 1)) ∈ ℝ ∧ 𝐴 ⊆ ℂ ∧ ℝ ⊆ ℂ) → (𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ∈ (𝐴–cn→ℝ))
363	347, 260, 104, 362	syl3anc 1326	. . . . . . 7 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) ∈ (𝐴–cn→ℝ))
364		eqidd 2623	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (1 / (𝑀 + 1)) = (1 / (𝑀 + 1)))
365	363, 48, 364	cnmptlimc 23654	. . . . . 6 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ((𝑥 ∈ 𝐴 ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵))
366	361, 365	sseldi 3601	. . . . 5 ⊢ (𝜑 → (1 / (𝑀 + 1)) ∈ ((𝑥 ∈ (𝐴 ∖ {𝐵}) ↦ (1 / (𝑀 + 1))) lim_ℂ 𝐵))
367	127	mulcn 22670	. . . . . 6 ⊢ · ∈ (((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) Cn (TopOpen‘ℂ_fld))
368		0cn 10032	. . . . . . 7 ⊢ 0 ∈ ℂ
369		opelxpi 5148	. . . . . . 7 ⊢ ((0 ∈ ℂ ∧ (1 / (𝑀 + 1)) ∈ ℂ) → ⟨0, (1 / (𝑀 + 1))⟩ ∈ (ℂ × ℂ))
370	368, 348, 369	sylancr 695	. . . . . 6 ⊢ (𝜑 → ⟨0, (1 / (𝑀 + 1))⟩ ∈ (ℂ × ℂ))
371	352	cncnpi 21082	. . . . . 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))⟩))
372	367, 370, 371	sylancr 695	. . . . 5 ⊢ (𝜑 → · ∈ ((((TopOpen‘ℂ_fld) ×_t (TopOpen‘ℂ_fld)) CnP (TopOpen‘ℂ_fld))‘⟨0, (1 / (𝑀 + 1))⟩))
373	346, 349, 171, 171, 127, 355, 356, 366, 372	limccnp2 23656	. . . 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))
374	348	mul02d 10234	. . . 4 ⊢ (𝜑 → (0 · (1 / (𝑀 + 1))) = 0)
375	187	fveq1d 6193	. . . . . . . . 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))
376		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥))
377		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥))
378	376, 377	oveq12d 6668	. . . . . . . . . . 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)))
379		ovex 6678	. . . . . . . . . . 11 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − 𝑀))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − 𝑀))‘𝑥)) ∈ V
380	378, 190, 379	fvmpt 6282	. . . . . . . . . 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)))
381	325, 380	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)))
382	375, 381	sylan9eq 2676	. . . . . . . 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)))
383	241	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝜑 → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥) = ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))‘𝑥))
384		oveq1 6657	. . . . . . . . . . . . . 14 ⊢ (𝑦 = 𝑥 → (𝑦 − 𝐵) = (𝑥 − 𝐵))
385	384	oveq1d 6665	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑𝑀) = ((𝑥 − 𝐵)↑𝑀))
386	385	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝑥 → ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
387		ovex 6678	. . . . . . . . . . . 12 ⊢ ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)) ∈ V
388	386, 244, 387	fvmpt 6282	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))‘𝑥) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
389	325, 388	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝐴 ∖ {𝐵}) → ((𝑦 ∈ 𝐴 ↦ ((𝑀 + 1) · ((𝑦 − 𝐵)↑𝑀)))‘𝑥) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
390	383, 389	sylan9eq 2676	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥) = ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)))
391	213	adantr 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℕ)
392	391	nncnd 11036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ∈ ℂ)
393	392, 338	mulcomd 10061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑀 + 1) · ((𝑥 − 𝐵)↑𝑀)) = (((𝑥 − 𝐵)↑𝑀) · (𝑀 + 1)))
394	390, 393	eqtrd 2656	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((ℝ D (𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1))))‘𝑥) = (((𝑥 − 𝐵)↑𝑀) · (𝑀 + 1)))
395	382, 394	oveq12d 6668	. . . . . . 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))))
396	391	nnne0d 11065	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → (𝑀 + 1) ≠ 0)
397	333, 338, 392, 345, 396	divdiv1d 10832	. . . . . . 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))))
398	346, 392, 396	divrecd 10804	. . . . . . 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))))
399	395, 397, 398	3eqtr2rd 2663	. . . . . 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)))
400	399	mpteq2dva 4744	. . . . 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))))
401	400	oveq1d 6665	. . . 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))
402	373, 374, 401	3eltr3d 2715	. . 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))
403	1, 80, 94, 134, 48, 135, 193, 247, 269, 282, 303, 324, 402	lhop 23779	. 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))
404	325	adantl 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → 𝑥 ∈ 𝐴)
405		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥))
406		fveq2 6191	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑦) = (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥))
407	405, 406	oveq12d 6668	. . . . . . 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)))
408		ovex 6678	. . . . . . 7 ⊢ ((((ℝ D^𝑛 𝐹)‘(𝑁 − (𝑀 + 1)))‘𝑥) − (((ℂ D^𝑛 𝑇)‘(𝑁 − (𝑀 + 1)))‘𝑥)) ∈ V
409	407, 79, 408	fvmpt 6282	. . . . . 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)))
410	404, 409	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)))
411	384	oveq1d 6665	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑦 − 𝐵)↑(𝑀 + 1)) = ((𝑥 − 𝐵)↑(𝑀 + 1)))
412		ovex 6678	. . . . . . 7 ⊢ ((𝑥 − 𝐵)↑(𝑀 + 1)) ∈ V
413	411, 93, 412	fvmpt 6282	. . . . . 6 ⊢ (𝑥 ∈ 𝐴 → ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥) = ((𝑥 − 𝐵)↑(𝑀 + 1)))
414	404, 413	syl 17	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝐵})) → ((𝑦 ∈ 𝐴 ↦ ((𝑦 − 𝐵)↑(𝑀 + 1)))‘𝑥) = ((𝑥 − 𝐵)↑(𝑀 + 1)))
415	410, 414	oveq12d 6668	. . . 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))))
416	415	mpteq2dva 4744	. . 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)))))
417	416	oveq1d 6665	. 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))
418	403, 417	eleqtrd 2703	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 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊆ wss 3574 {csn 4177 {cpr 4179 ⟨cop 4183 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ran crn 5115 ↾ cres 5116 “ cima 5117 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ↑_𝑚 cmap 7857 ↑_pm cpm 7858 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 ℕ₀cn0 11292 ℤ_≥cuz 11687 (,)cioo 12175 ...cfz 12326 ..^cfzo 12465 ↑cexp 12860 !cfa 13060 ↾_t crest 16081 TopOpenctopn 16082 topGenctg 16098 DivRingcdr 18747 SubRingcsubrg 18776 ℂ_fldccnfld 19746 ℝ_fldcrefld 19950 Topctop 20698 TopOnctopon 20715 intcnt 20821 Cn ccn 21028 CnP ccnp 21029 ×_t ctx 21363 –cn→ccncf 22679 lim_ℂ climc 23626 D cdv 23627 D^𝑛 cdvn 23628 Polycply 23940 degcdgr 23943 Tayl ctayl 24107

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

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-rlim 14220 df-sum 14417 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tsms 21930 df-xms 22125 df-ms 22126 df-tms 22127 df-cncf 22681 df-0p 23437 df-limc 23630 df-dv 23631 df-dvn 23632 df-ply 23944 df-idp 23945 df-coe 23946 df-dgr 23947 df-tayl 24109

This theorem is referenced by: taylth 24129

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