Theorem eirraplem 12418

Description: Lemma for eirrap 12419. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Jim Kingdon, 5-Jan-2022.)

Hypotheses

Ref	Expression
eirr.1	⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))
eirr.2	⊢ (𝜑 → 𝑃 ∈ ℤ)
eirr.3	⊢ (𝜑 → 𝑄 ∈ ℕ)

Assertion

Ref	Expression
eirraplem	⊢ (𝜑 → e # (𝑃 / 𝑄))

Distinct variable group:   𝑄,𝑛

Allowed substitution hints:   𝜑(𝑛)   𝑃(𝑛)   𝐹(𝑛)

Proof of Theorem eirraplem

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		esum 12303	. . . . . . . . . . 11 ⊢ e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
2		faccl 11060	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ)
3	2	nnrecred 9249	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (1 / (!‘𝑘)) ∈ ℝ)
4		fveq2 5648	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (!‘𝑛) = (!‘𝑘))
5	4	oveq2d 6044	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (1 / (!‘𝑛)) = (1 / (!‘𝑘)))
6		eirr.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))
7	5, 6	fvmptg 5731	. . . . . . . . . . . . 13 ⊢ ((𝑘 ∈ ℕ0 ∧ (1 / (!‘𝑘)) ∈ ℝ) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
8	3, 7	mpdan 421	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = (1 / (!‘𝑘)))
9	8	sumeq2i 12004	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
10	1, 9	eqtr4i 2255	. . . . . . . . . 10 ⊢ e = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)
11		nn0uz 9852	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
12		eqid 2231	. . . . . . . . . . 11 ⊢ (ℤ≥‘(𝑄 + 1)) = (ℤ≥‘(𝑄 + 1))
13		eirr.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ℕ)
14	13	peano2nnd 9217	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ)
15	14	nnnn0d 9516	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ0)
16		eqidd 2232	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘))
17		ax-1cn 8185	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
18		nn0z 9560	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
19		1exp 10893	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
20	18, 19	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ0 → (1↑𝑛) = 1)
21	20	oveq1d 6043	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ0 → ((1↑𝑛) / (!‘𝑛)) = (1 / (!‘𝑛)))
22	21	mpteq2ia 4180	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ0 ↦ ((1↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))
23	6, 22	eqtr4i 2255	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((1↑𝑛) / (!‘𝑛)))
24	23	eftvalcn 12298	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
25	17, 24	mpan 424	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
26	25	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘)))
27	17	a1i 9	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ∈ ℂ)
28		eftcl 12295	. . . . . . . . . . . . 13 ⊢ ((1 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ)
29	27, 28	sylan 283	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ)
30	26, 29	eqeltrd 2308	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ)
31	23	efcllem 12300	. . . . . . . . . . . 12 ⊢ (1 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
32	27, 31	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
33	11, 12, 15, 16, 30, 32	isumsplit 12132	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
34	10, 33	eqtrid 2276	. . . . . . . . 9 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
35	13	nncnd 9216	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ℂ)
36		pncan 8444	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑄 + 1) − 1) = 𝑄)
37	35, 17, 36	sylancl 413	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄 + 1) − 1) = 𝑄)
38	37	oveq2d 6044	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑄 + 1) − 1)) = (0...𝑄))
39	38	sumeq1d 12006	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))
40	39	oveq1d 6043	. . . . . . . . 9 ⊢ (𝜑 → (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
41	34, 40	eqtrd 2264	. . . . . . . 8 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
42	41	oveq1d 6043	. . . . . . 7 ⊢ (𝜑 → (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = ((Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))
43		0zd 9552	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
44	13	nnzd 9662	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℤ)
45	43, 44	fzfigd 10756	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑄) ∈ Fin)
46		elfznn0 10411	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑄) → 𝑘 ∈ ℕ0)
47	46, 30	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) ∈ ℂ)
48	45, 47	fsumcl 12041	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) ∈ ℂ)
49	8	adantl 277	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
50	2	adantl 277	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ)
51	50	nnrpd 9990	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ+)
52	51	rpreccld 10003	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (!‘𝑘)) ∈ ℝ+)
53	49, 52	eqeltrd 2308	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ+)
54	11, 12, 15, 16, 53, 32	isumrpcl 12135	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ+)
55	54	rpred 9992	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ)
56	55	recnd 8267	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℂ)
57	48, 56	pncan2d 8551	. . . . . . 7 ⊢ (𝜑 → ((Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))
58	42, 57	eqtrd 2264	. . . . . 6 ⊢ (𝜑 → (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))
59	58	oveq2d 6044	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
60	13	nnnn0d 9516	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℕ0)
61	60	faccld 11061	. . . . . . 7 ⊢ (𝜑 → (!‘𝑄) ∈ ℕ)
62	61	nncnd 9216	. . . . . 6 ⊢ (𝜑 → (!‘𝑄) ∈ ℂ)
63		ere 12311	. . . . . . . 8 ⊢ e ∈ ℝ
64	63	recni 8251	. . . . . . 7 ⊢ e ∈ ℂ
65	64	a1i 9	. . . . . 6 ⊢ (𝜑 → e ∈ ℂ)
66	62, 65, 48	subdid 8652	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
67	59, 66	eqtr3d 2266	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
68	61	nnrpd 9990	. . . . . . 7 ⊢ (𝜑 → (!‘𝑄) ∈ ℝ+)
69	68, 54	rpmulcld 10009	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ+)
70	69	rpred 9992	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ)
71		eirr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
72	71	zcnd 9664	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
73	13	nnap0d 9248	. . . . . . . 8 ⊢ (𝜑 → 𝑄 # 0)
74	62, 72, 35, 73	div12apd 9066	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) = (𝑃 · ((!‘𝑄) / 𝑄)))
75	13	nnred 9215	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℝ)
76	75	leidd 8753	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ 𝑄)
77		facdiv 11063	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℕ0 ∧ 𝑄 ∈ ℕ ∧ 𝑄 ≤ 𝑄) → ((!‘𝑄) / 𝑄) ∈ ℕ)
78	60, 13, 76, 77	syl3anc 1274	. . . . . . . . 9 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℕ)
79	78	nnzd 9662	. . . . . . . 8 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℤ)
80	71, 79	zmulcld 9669	. . . . . . 7 ⊢ (𝜑 → (𝑃 · ((!‘𝑄) / 𝑄)) ∈ ℤ)
81	74, 80	eqeltrd 2308	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℤ)
82	45, 62, 47	fsummulc2 12089	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)))
83	46	adantl 277	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → 𝑘 ∈ ℕ0)
84	83, 8	syl 14	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) = (1 / (!‘𝑘)))
85	84	oveq2d 6044	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
86	62	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑄) ∈ ℂ)
87	46, 50	sylan2 286	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℕ)
88	87	nncnd 9216	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℂ)
89	87	nnap0d 9248	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) # 0)
90	86, 88, 89	divrecapd 9032	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
91	85, 90	eqtr4d 2267	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) / (!‘𝑘)))
92		permnn 11096	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑄) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
93	92	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
94	91, 93	eqeltrd 2308	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℕ)
95	94	nnzd 9662	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
96	45, 95	fsumzcl 12043	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
97	82, 96	eqeltrd 2308	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℤ)
98	81, 97	zsubcld 9668	. . . . 5 ⊢ (𝜑 → (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) ∈ ℤ)
99	69	rpgt0d 9995	. . . . 5 ⊢ (𝜑 → 0 < ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
100	14	peano2nnd 9217	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℕ)
101	100	nnred 9215	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℝ)
102	15	faccld 11061	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℕ)
103	102, 14	nnmulcld 9251	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℕ)
104	101, 103	nndivred 9252	. . . . . . . 8 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℝ)
105	61	nnrecred 9249	. . . . . . . 8 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℝ)
106		abs1 11712	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
107	106	oveq1i 6038	. . . . . . . . . . . . 13 ⊢ ((abs‘1)↑𝑛) = (1↑𝑛)
108	107	oveq1i 6038	. . . . . . . . . . . 12 ⊢ (((abs‘1)↑𝑛) / (!‘𝑛)) = ((1↑𝑛) / (!‘𝑛))
109	108	mpteq2i 4181	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 ↦ (((abs‘1)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((1↑𝑛) / (!‘𝑛)))
110	23, 109	eqtr4i 2255	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((abs‘1)↑𝑛) / (!‘𝑛)))
111		eqid 2231	. . . . . . . . . 10 ⊢ (𝑛 ∈ ℕ0 ↦ ((((abs‘1)↑(𝑄 + 1)) / (!‘(𝑄 + 1))) · ((1 / ((𝑄 + 1) + 1))↑𝑛))) = (𝑛 ∈ ℕ0 ↦ ((((abs‘1)↑(𝑄 + 1)) / (!‘(𝑄 + 1))) · ((1 / ((𝑄 + 1) + 1))↑𝑛)))
112		1le1 8811	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
113	106, 112	eqbrtri 4114	. . . . . . . . . . 11 ⊢ (abs‘1) ≤ 1
114	113	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘1) ≤ 1)
115	23, 110, 111, 14, 27, 114	eftlub 12331	. . . . . . . . 9 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ≤ (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
116	54	rprege0d 10000	. . . . . . . . . 10 ⊢ (𝜑 → (Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
117		absid 11711	. . . . . . . . . 10 ⊢ ((Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))
118	116, 117	syl 14	. . . . . . . . 9 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))
119	106	oveq1i 6038	. . . . . . . . . . . 12 ⊢ ((abs‘1)↑(𝑄 + 1)) = (1↑(𝑄 + 1))
120	14	nnzd 9662	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 1) ∈ ℤ)
121		1exp 10893	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 1) ∈ ℤ → (1↑(𝑄 + 1)) = 1)
122	120, 121	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (1↑(𝑄 + 1)) = 1)
123	119, 122	eqtrid 2276	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘1)↑(𝑄 + 1)) = 1)
124	123	oveq1d 6043	. . . . . . . . . 10 ⊢ (𝜑 → (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
125	104	recnd 8267	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℂ)
126	125	mullidd 8257	. . . . . . . . . 10 ⊢ (𝜑 → (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
127	124, 126	eqtrd 2264	. . . . . . . . 9 ⊢ (𝜑 → (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
128	115, 118, 127	3brtr3d 4124	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ≤ (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
129	14	nnred 9215	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 1) ∈ ℝ)
130	129, 129	readdcld 8268	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ∈ ℝ)
131	129, 129	remulcld 8269	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) ∈ ℝ)
132		1red 8254	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
133	13	nnge1d 9245	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ≤ 𝑄)
134		1nn 9213	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
135		nnleltp1 9600	. . . . . . . . . . . . . 14 ⊢ ((1 ∈ ℕ ∧ 𝑄 ∈ ℕ) → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1)))
136	134, 13, 135	sylancr 414	. . . . . . . . . . . . 13 ⊢ (𝜑 → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1)))
137	133, 136	mpbid 147	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 < (𝑄 + 1))
138	132, 129, 129, 137	ltadd2dd 8661	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) + (𝑄 + 1)))
139	14	nncnd 9216	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 1) ∈ ℂ)
140	139	2timesd 9446	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · (𝑄 + 1)) = ((𝑄 + 1) + (𝑄 + 1)))
141		df-2 9261	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
142	132, 75, 132, 133	leadd1dd 8798	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 + 1) ≤ (𝑄 + 1))
143	141, 142	eqbrtrid 4128	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≤ (𝑄 + 1))
144		2re 9272	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
145	144	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
146	14	nngt0d 9246	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑄 + 1))
147		lemul1 8832	. . . . . . . . . . . . . 14 ⊢ ((2 ∈ ℝ ∧ (𝑄 + 1) ∈ ℝ ∧ ((𝑄 + 1) ∈ ℝ ∧ 0 < (𝑄 + 1))) → (2 ≤ (𝑄 + 1) ↔ (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1))))
148	145, 129, 129, 146, 147	syl112anc 1278	. . . . . . . . . . . . 13 ⊢ (𝜑 → (2 ≤ (𝑄 + 1) ↔ (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1))))
149	143, 148	mpbid 147	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))
150	140, 149	eqbrtrrd 4117	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))
151	101, 130, 131, 138, 150	ltletrd 8662	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) · (𝑄 + 1)))
152		facp1 11055	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ ℕ0 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
153	60, 152	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
154	153	oveq1d 6043	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)))
155	102	nncnd 9216	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℂ)
156	61	nnap0d 9248	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘𝑄) # 0)
157	155, 62, 156	divrecapd 9032	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
158	139, 62, 156	divcanap3d 9034	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)) = (𝑄 + 1))
159	154, 157, 158	3eqtr3rd 2273	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 1) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
160	159	oveq1d 6043	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)))
161	105	recnd 8267	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℂ)
162	155, 161, 139	mul32d 8391	. . . . . . . . . . 11 ⊢ (𝜑 → (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
163	160, 162	eqtrd 2264	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
164	151, 163	breqtrd 4119	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
165	103	nnred 9215	. . . . . . . . . 10 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℝ)
166	103	nngt0d 9246	. . . . . . . . . 10 ⊢ (𝜑 → 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1)))
167		ltdivmul 9115	. . . . . . . . . 10 ⊢ ((((𝑄 + 1) + 1) ∈ ℝ ∧ (1 / (!‘𝑄)) ∈ ℝ ∧ (((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℝ ∧ 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1)))) → ((((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)) ↔ ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄)))))
168	101, 105, 165, 166, 167	syl112anc 1278	. . . . . . . . 9 ⊢ (𝜑 → ((((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)) ↔ ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄)))))
169	164, 168	mpbird 167	. . . . . . . 8 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)))
170	55, 104, 105, 128, 169	lelttrd 8363	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄)))
171	55, 132, 68	ltmuldiv2d 10041	. . . . . . 7 ⊢ (𝜑 → (((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < 1 ↔ Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄))))
172	170, 171	mpbird 167	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < 1)
173		0p1e1 9316	. . . . . 6 ⊢ (0 + 1) = 1
174	172, 173	breqtrrdi 4135	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < (0 + 1))
175	43, 70, 98, 99, 174	btwnapz 9671	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
176	67, 175	eqbrtrrd 4117	. . 3 ⊢ (𝜑 → (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
177	62, 65	mulcld 8259	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · e) ∈ ℂ)
178	81	zcnd 9664	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℂ)
179	62, 48	mulcld 8259	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℂ)
180		apsub1 8881	. . . 4 ⊢ ((((!‘𝑄) · e) ∈ ℂ ∧ ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℂ ∧ ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℂ) → (((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄)) ↔ (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))))
181	177, 178, 179, 180	syl3anc 1274	. . 3 ⊢ (𝜑 → (((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄)) ↔ (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))))
182	176, 181	mpbird 167	. 2 ⊢ (𝜑 → ((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄)))
183		znq 9919	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℚ)
184	71, 13, 183	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑃 / 𝑄) ∈ ℚ)
185		qcn 9929	. . . 4 ⊢ ((𝑃 / 𝑄) ∈ ℚ → (𝑃 / 𝑄) ∈ ℂ)
186	184, 185	syl 14	. . 3 ⊢ (𝜑 → (𝑃 / 𝑄) ∈ ℂ)
187		apmul2 9028	. . 3 ⊢ ((e ∈ ℂ ∧ (𝑃 / 𝑄) ∈ ℂ ∧ ((!‘𝑄) ∈ ℂ ∧ (!‘𝑄) # 0)) → (e # (𝑃 / 𝑄) ↔ ((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄))))
188	65, 186, 62, 156, 187	syl112anc 1278	. 2 ⊢ (𝜑 → (e # (𝑃 / 𝑄) ↔ ((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄))))
189	182, 188	mpbird 167	1 ⊢ (𝜑 → e # (𝑃 / 𝑄))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202   class class class wbr 4093   ↦ cmpt 4155  dom cdm 4731  ‘cfv 5333  (class class class)co 6028  ℂcc 8090  ℝcr 8091  0cc0 8092  1c1 8093   + caddc 8095   · cmul 8097   < clt 8273   ≤ cle 8274   − cmin 8409   # cap 8820   / cdiv 8911  ℕcn 9202  2c2 9253  ℕ₀cn0 9461  ℤcz 9540  ℤ_≥cuz 9816  ℚcq 9914  ℝ⁺crp 9949  ...cfz 10305  seqcseq 10772  ↑cexp 10863  !cfa 11050  abscabs 11637   ⇝ cli 11918  Σcsu 11993  eceu 12284

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-irdg 6579  df-frec 6600  df-1o 6625  df-oadd 6629  df-er 6745  df-en 6953  df-dom 6954  df-fin 6955  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-q 9915  df-rp 9950  df-ico 10190  df-fz 10306  df-fzo 10440  df-seqfrec 10773  df-exp 10864  df-fac 11051  df-bc 11073  df-ihash 11101  df-shft 11455  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639  df-clim 11919  df-sumdc 11994  df-ef 12289  df-e 12290

This theorem is referenced by:  eirrap  12419