taylply2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > taylply2		Structured version Visualization version GIF version

Theorem taylply2 25887

Description: The Taylor polynomial is a polynomial of degree (at most) 𝑁. This version of taylply 25888 shows that the coefficients of 𝑇 are in a subring of the complex numbers. (Contributed by Mario Carneiro, 1-Jan-2017.)

**Hypotheses**
Ref	Expression
taylpfval.s	⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
taylpfval.f	⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
taylpfval.a	⊢ (𝜑 → 𝐴 ⊆ 𝑆)
taylpfval.n	⊢ (𝜑 → 𝑁 ∈ ℕ₀)
taylpfval.b	⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
taylpfval.t	⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
taylply2.1	⊢ (𝜑 → 𝐷 ∈ (SubRing‘ℂ_fld))
taylply2.2	⊢ (𝜑 → 𝐵 ∈ 𝐷)
taylply2.3	⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷)

**Assertion**
Ref	Expression
taylply2	⊢ (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁))

Distinct variable groups: 𝐵,𝑘 𝑘,𝐹 𝑘,𝑁 𝜑,𝑘 𝐷,𝑘 𝑆,𝑘 Allowed substitution hints: 𝐴(𝑘) 𝑇(𝑘)

Proof of Theorem taylply2 Dummy variables 𝑢 𝑣 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		taylpfval.s	. . . . 5 ⊢ (𝜑 → 𝑆 ∈ {ℝ, ℂ})
2		taylpfval.f	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ℂ)
3		taylpfval.a	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝑆)
4		taylpfval.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
5		taylpfval.b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
6		taylpfval.t	. . . . 5 ⊢ 𝑇 = (𝑁(𝑆 Tayl 𝐹)𝐵)
7	1, 2, 3, 4, 5, 6	taylpfval 25884	. . . 4 ⊢ (𝜑 → 𝑇 = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))
8		simpr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ)
9		cnex 11193	. . . . . . . . . . . . 13 ⊢ ℂ ∈ V
10	9	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → ℂ ∈ V)
11		elpm2r 8841	. . . . . . . . . . . 12 ⊢ (((ℂ ∈ V ∧ 𝑆 ∈ {ℝ, ℂ}) ∧ (𝐹:𝐴⟶ℂ ∧ 𝐴 ⊆ 𝑆)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
12	10, 1, 2, 3, 11	syl22anc 837	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐹 ∈ (ℂ ↑_pm 𝑆))
13		dvnbss 25452	. . . . . . . . . . 11 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑁 ∈ ℕ₀) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
14	1, 12, 4, 13	syl3anc 1371	. . . . . . . . . 10 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom 𝐹)
15	2, 14	fssdmd 6736	. . . . . . . . 9 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ 𝐴)
16		recnprss 25428	. . . . . . . . . . 11 ⊢ (𝑆 ∈ {ℝ, ℂ} → 𝑆 ⊆ ℂ)
17	1, 16	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ⊆ ℂ)
18	3, 17	sstrd 3992	. . . . . . . . 9 ⊢ (𝜑 → 𝐴 ⊆ ℂ)
19	15, 18	sstrd 3992	. . . . . . . 8 ⊢ (𝜑 → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ ℂ)
20	19, 5	sseldd 3983	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℂ)
21	20	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → 𝐵 ∈ ℂ)
22	8, 21	subcld 11573	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℂ) → (𝑥 − 𝐵) ∈ ℂ)
23		df-idp 25710	. . . . . . . 8 ⊢ X_p = ( I ↾ ℂ)
24		mptresid 6050	. . . . . . . 8 ⊢ ( I ↾ ℂ) = (𝑥 ∈ ℂ ↦ 𝑥)
25	23, 24	eqtri 2760	. . . . . . 7 ⊢ X_p = (𝑥 ∈ ℂ ↦ 𝑥)
26	25	a1i 11	. . . . . 6 ⊢ (𝜑 → X_p = (𝑥 ∈ ℂ ↦ 𝑥))
27		fconstmpt 5738	. . . . . . 7 ⊢ (ℂ × {𝐵}) = (𝑥 ∈ ℂ ↦ 𝐵)
28	27	a1i 11	. . . . . 6 ⊢ (𝜑 → (ℂ × {𝐵}) = (𝑥 ∈ ℂ ↦ 𝐵))
29	10, 8, 21, 26, 28	offval2 7692	. . . . 5 ⊢ (𝜑 → (X_p ∘_f − (ℂ × {𝐵})) = (𝑥 ∈ ℂ ↦ (𝑥 − 𝐵)))
30		eqidd 2733	. . . . 5 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) = (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))))
31		oveq1 7418	. . . . . . 7 ⊢ (𝑦 = (𝑥 − 𝐵) → (𝑦↑𝑘) = ((𝑥 − 𝐵)↑𝑘))
32	31	oveq2d 7427	. . . . . 6 ⊢ (𝑦 = (𝑥 − 𝐵) → (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)) = (((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))
33	32	sumeq2sdv 15652	. . . . 5 ⊢ (𝑦 = (𝑥 − 𝐵) → Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)) = Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘)))
34	22, 29, 30, 33	fmptco 7129	. . . 4 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∘ (X_p ∘_f − (ℂ × {𝐵}))) = (𝑥 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · ((𝑥 − 𝐵)↑𝑘))))
35	7, 34	eqtr4d 2775	. . 3 ⊢ (𝜑 → 𝑇 = ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∘ (X_p ∘_f − (ℂ × {𝐵}))))
36		taylply2.1	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (SubRing‘ℂ_fld))
37		cnfldbas 20954	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
38	37	subrgss 20324	. . . . . 6 ⊢ (𝐷 ∈ (SubRing‘ℂ_fld) → 𝐷 ⊆ ℂ)
39	36, 38	syl 17	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ℂ)
40		taylply2.3	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ 𝐷)
41	39, 4, 40	elplyd 25723	. . . 4 ⊢ (𝜑 → (𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∈ (Poly‘𝐷))
42		cnfld1 20976	. . . . . . . 8 ⊢ 1 = (1_r‘ℂ_fld)
43	42	subrg1cl 20331	. . . . . . 7 ⊢ (𝐷 ∈ (SubRing‘ℂ_fld) → 1 ∈ 𝐷)
44	36, 43	syl 17	. . . . . 6 ⊢ (𝜑 → 1 ∈ 𝐷)
45		plyid 25730	. . . . . 6 ⊢ ((𝐷 ⊆ ℂ ∧ 1 ∈ 𝐷) → X_p ∈ (Poly‘𝐷))
46	39, 44, 45	syl2anc 584	. . . . 5 ⊢ (𝜑 → X_p ∈ (Poly‘𝐷))
47		taylply2.2	. . . . . 6 ⊢ (𝜑 → 𝐵 ∈ 𝐷)
48		plyconst 25727	. . . . . 6 ⊢ ((𝐷 ⊆ ℂ ∧ 𝐵 ∈ 𝐷) → (ℂ × {𝐵}) ∈ (Poly‘𝐷))
49	39, 47, 48	syl2anc 584	. . . . 5 ⊢ (𝜑 → (ℂ × {𝐵}) ∈ (Poly‘𝐷))
50		subrgsubg 20329	. . . . . . 7 ⊢ (𝐷 ∈ (SubRing‘ℂ_fld) → 𝐷 ∈ (SubGrp‘ℂ_fld))
51	36, 50	syl 17	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ (SubGrp‘ℂ_fld))
52		cnfldadd 20955	. . . . . . . 8 ⊢ + = (+_g‘ℂ_fld)
53	52	subgcl 19018	. . . . . . 7 ⊢ ((𝐷 ∈ (SubGrp‘ℂ_fld) ∧ 𝑢 ∈ 𝐷 ∧ 𝑣 ∈ 𝐷) → (𝑢 + 𝑣) ∈ 𝐷)
54	53	3expb 1120	. . . . . 6 ⊢ ((𝐷 ∈ (SubGrp‘ℂ_fld) ∧ (𝑢 ∈ 𝐷 ∧ 𝑣 ∈ 𝐷)) → (𝑢 + 𝑣) ∈ 𝐷)
55	51, 54	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑣 ∈ 𝐷)) → (𝑢 + 𝑣) ∈ 𝐷)
56		cnfldmul 20956	. . . . . . . 8 ⊢ · = (._r‘ℂ_fld)
57	56	subrgmcl 20335	. . . . . . 7 ⊢ ((𝐷 ∈ (SubRing‘ℂ_fld) ∧ 𝑢 ∈ 𝐷 ∧ 𝑣 ∈ 𝐷) → (𝑢 · 𝑣) ∈ 𝐷)
58	57	3expb 1120	. . . . . 6 ⊢ ((𝐷 ∈ (SubRing‘ℂ_fld) ∧ (𝑢 ∈ 𝐷 ∧ 𝑣 ∈ 𝐷)) → (𝑢 · 𝑣) ∈ 𝐷)
59	36, 58	sylan 580	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐷 ∧ 𝑣 ∈ 𝐷)) → (𝑢 · 𝑣) ∈ 𝐷)
60		ax-1cn 11170	. . . . . . 7 ⊢ 1 ∈ ℂ
61		cnfldneg 20977	. . . . . . 7 ⊢ (1 ∈ ℂ → ((inv_g‘ℂ_fld)‘1) = -1)
62	60, 61	ax-mp 5	. . . . . 6 ⊢ ((inv_g‘ℂ_fld)‘1) = -1
63		eqid 2732	. . . . . . . 8 ⊢ (inv_g‘ℂ_fld) = (inv_g‘ℂ_fld)
64	63	subginvcl 19017	. . . . . . 7 ⊢ ((𝐷 ∈ (SubGrp‘ℂ_fld) ∧ 1 ∈ 𝐷) → ((inv_g‘ℂ_fld)‘1) ∈ 𝐷)
65	51, 44, 64	syl2anc 584	. . . . . 6 ⊢ (𝜑 → ((inv_g‘ℂ_fld)‘1) ∈ 𝐷)
66	62, 65	eqeltrrid 2838	. . . . 5 ⊢ (𝜑 → -1 ∈ 𝐷)
67	46, 49, 55, 59, 66	plysub 25740	. . . 4 ⊢ (𝜑 → (X_p ∘_f − (ℂ × {𝐵})) ∈ (Poly‘𝐷))
68	41, 67, 55, 59	plyco 25762	. . 3 ⊢ (𝜑 → ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∘ (X_p ∘_f − (ℂ × {𝐵}))) ∈ (Poly‘𝐷))
69	35, 68	eqeltrd 2833	. 2 ⊢ (𝜑 → 𝑇 ∈ (Poly‘𝐷))
70	35	fveq2d 6895	. . . 4 ⊢ (𝜑 → (deg‘𝑇) = (deg‘((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∘ (X_p ∘_f − (ℂ × {𝐵})))))
71		eqid 2732	. . . . 5 ⊢ (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) = (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))))
72		eqid 2732	. . . . 5 ⊢ (deg‘(X_p ∘_f − (ℂ × {𝐵}))) = (deg‘(X_p ∘_f − (ℂ × {𝐵})))
73	71, 72, 41, 67	dgrco 25796	. . . 4 ⊢ (𝜑 → (deg‘((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∘ (X_p ∘_f − (ℂ × {𝐵})))) = ((deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) · (deg‘(X_p ∘_f − (ℂ × {𝐵})))))
74		eqid 2732	. . . . . . . . 9 ⊢ (X_p ∘_f − (ℂ × {𝐵})) = (X_p ∘_f − (ℂ × {𝐵}))
75	74	plyremlem 25824	. . . . . . . 8 ⊢ (𝐵 ∈ ℂ → ((X_p ∘_f − (ℂ × {𝐵})) ∈ (Poly‘ℂ) ∧ (deg‘(X_p ∘_f − (ℂ × {𝐵}))) = 1 ∧ (^◡(X_p ∘_f − (ℂ × {𝐵})) “ {0}) = {𝐵}))
76	20, 75	syl 17	. . . . . . 7 ⊢ (𝜑 → ((X_p ∘_f − (ℂ × {𝐵})) ∈ (Poly‘ℂ) ∧ (deg‘(X_p ∘_f − (ℂ × {𝐵}))) = 1 ∧ (^◡(X_p ∘_f − (ℂ × {𝐵})) “ {0}) = {𝐵}))
77	76	simp2d 1143	. . . . . 6 ⊢ (𝜑 → (deg‘(X_p ∘_f − (ℂ × {𝐵}))) = 1)
78	77	oveq2d 7427	. . . . 5 ⊢ (𝜑 → ((deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) · (deg‘(X_p ∘_f − (ℂ × {𝐵})))) = ((deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) · 1))
79		dgrcl 25754	. . . . . . . 8 ⊢ ((𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘))) ∈ (Poly‘𝐷) → (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) ∈ ℕ₀)
80	41, 79	syl 17	. . . . . . 7 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) ∈ ℕ₀)
81	80	nn0cnd 12536	. . . . . 6 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) ∈ ℂ)
82	81	mulridd 11233	. . . . 5 ⊢ (𝜑 → ((deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) · 1) = (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))))
83	78, 82	eqtrd 2772	. . . 4 ⊢ (𝜑 → ((deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) · (deg‘(X_p ∘_f − (ℂ × {𝐵})))) = (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))))
84	70, 73, 83	3eqtrd 2776	. . 3 ⊢ (𝜑 → (deg‘𝑇) = (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))))
85	1	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑆 ∈ {ℝ, ℂ})
86	12	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐹 ∈ (ℂ ↑_pm 𝑆))
87		elfznn0 13596	. . . . . . . 8 ⊢ (𝑘 ∈ (0...𝑁) → 𝑘 ∈ ℕ₀)
88	87	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ ℕ₀)
89		dvnf 25451	. . . . . . 7 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑘 ∈ ℕ₀) → ((𝑆 D^𝑛 𝐹)‘𝑘):dom ((𝑆 D^𝑛 𝐹)‘𝑘)⟶ℂ)
90	85, 86, 88, 89	syl3anc 1371	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((𝑆 D^𝑛 𝐹)‘𝑘):dom ((𝑆 D^𝑛 𝐹)‘𝑘)⟶ℂ)
91		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝑘 ∈ (0...𝑁))
92		dvn2bss 25454	. . . . . . . 8 ⊢ ((𝑆 ∈ {ℝ, ℂ} ∧ 𝐹 ∈ (ℂ ↑_pm 𝑆) ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
93	85, 86, 91, 92	syl3anc 1371	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → dom ((𝑆 D^𝑛 𝐹)‘𝑁) ⊆ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
94	5	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑁))
95	93, 94	sseldd 3983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → 𝐵 ∈ dom ((𝑆 D^𝑛 𝐹)‘𝑘))
96	90, 95	ffvelcdmd 7087	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) ∈ ℂ)
97	88	faccld 14246	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℕ)
98	97	nncnd 12230	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ∈ ℂ)
99	97	nnne0d 12264	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → (!‘𝑘) ≠ 0)
100	96, 98, 99	divcld 11992	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑁)) → ((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) ∈ ℂ)
101	41, 4, 100, 30	dgrle 25764	. . 3 ⊢ (𝜑 → (deg‘(𝑦 ∈ ℂ ↦ Σ𝑘 ∈ (0...𝑁)(((((𝑆 D^𝑛 𝐹)‘𝑘)‘𝐵) / (!‘𝑘)) · (𝑦↑𝑘)))) ≤ 𝑁)
102	84, 101	eqbrtrd 5170	. 2 ⊢ (𝜑 → (deg‘𝑇) ≤ 𝑁)
103	69, 102	jca 512	1 ⊢ (𝜑 → (𝑇 ∈ (Poly‘𝐷) ∧ (deg‘𝑇) ≤ 𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 {csn 4628 {cpr 4630 class class class wbr 5148 ↦ cmpt 5231 I cid 5573 × cxp 5674 ^◡ccnv 5675 dom cdm 5676 ↾ cres 5678 “ cima 5679 ∘ ccom 5680 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ∘_f cof 7670 ↑_pm cpm 8823 ℂcc 11110 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 ≤ cle 11251 − cmin 11446 -cneg 11447 / cdiv 11873 ℕ₀cn0 12474 ...cfz 13486 ↑cexp 14029 !cfa 14235 Σcsu 15634 inv_gcminusg 18822 SubGrpcsubg 19002 SubRingcsubrg 20319 ℂ_fldccnfld 20950 D^𝑛 cdvn 25388 Polycply 25705 X_pcidp 25706 degcdgr 25708 Tayl ctayl 25872

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-seq 13969 df-exp 14030 df-fac 14236 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-clim 15434 df-rlim 15435 df-sum 15635 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-grp 18824 df-minusg 18825 df-subg 19005 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-subrg 20321 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cnp 22739 df-haus 22826 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-tsms 23638 df-xms 23833 df-ms 23834 df-0p 25194 df-limc 25390 df-dv 25391 df-dvn 25392 df-ply 25709 df-idp 25710 df-coe 25711 df-dgr 25712 df-tayl 25874

This theorem is referenced by: taylply 25888 taylthlem2 25893

W3C validator