Theorem eirraplem 11923

Description: Lemma for eirrap 11924. (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 11808	. . . . . . . . . . 11 ⊢ e = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
2		faccl 10809	. . . . . . . . . . . . . 14 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ)
3	2	nnrecred 9031	. . . . . . . . . . . . 13 ⊢ (𝑘 ∈ ℕ0 → (1 / (!‘𝑘)) ∈ ℝ)
4		fveq2 5555	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑘 → (!‘𝑛) = (!‘𝑘))
5	4	oveq2d 5935	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑘 → (1 / (!‘𝑛)) = (1 / (!‘𝑘)))
6		eirr.1	. . . . . . . . . . . . . 14 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))
7	5, 6	fvmptg 5634	. . . . . . . . . . . . 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 11510	. . . . . . . . . . 11 ⊢ Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ ℕ0 (1 / (!‘𝑘))
10	1, 9	eqtr4i 2217	. . . . . . . . . 10 ⊢ e = Σ𝑘 ∈ ℕ0 (𝐹‘𝑘)
11		nn0uz 9630	. . . . . . . . . . 11 ⊢ ℕ0 = (ℤ≥‘0)
12		eqid 2193	. . . . . . . . . . 11 ⊢ (ℤ≥‘(𝑄 + 1)) = (ℤ≥‘(𝑄 + 1))
13		eirr.3	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ℕ)
14	13	peano2nnd 8999	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ)
15	14	nnnn0d 9296	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑄 + 1) ∈ ℕ0)
16		eqidd 2194	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘))
17		ax-1cn 7967	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℂ
18		nn0z 9340	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℕ0 → 𝑛 ∈ ℤ)
19		1exp 10642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑛 ∈ ℤ → (1↑𝑛) = 1)
20	18, 19	syl 14	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑛 ∈ ℕ0 → (1↑𝑛) = 1)
21	20	oveq1d 5934	. . . . . . . . . . . . . . . . 17 ⊢ (𝑛 ∈ ℕ0 → ((1↑𝑛) / (!‘𝑛)) = (1 / (!‘𝑛)))
22	21	mpteq2ia 4116	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ ℕ0 ↦ ((1↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ (1 / (!‘𝑛)))
23	6, 22	eqtr4i 2217	. . . . . . . . . . . . . . 15 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ ((1↑𝑛) / (!‘𝑛)))
24	23	eftvalcn 11803	. . . . . . . . . . . . . 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 11800	. . . . . . . . . . . . 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 2270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ)
31	23	efcllem 11805	. . . . . . . . . . . 12 ⊢ (1 ∈ ℂ → seq0( + , 𝐹) ∈ dom ⇝ )
32	27, 31	syl 14	. . . . . . . . . . 11 ⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝ )
33	11, 12, 15, 16, 30, 32	isumsplit 11637	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
34	10, 33	eqtrid 2238	. . . . . . . . 9 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
35	13	nncnd 8998	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑄 ∈ ℂ)
36		pncan 8227	. . . . . . . . . . . . 13 ⊢ ((𝑄 ∈ ℂ ∧ 1 ∈ ℂ) → ((𝑄 + 1) − 1) = 𝑄)
37	35, 17, 36	sylancl 413	. . . . . . . . . . . 12 ⊢ (𝜑 → ((𝑄 + 1) − 1) = 𝑄)
38	37	oveq2d 5935	. . . . . . . . . . 11 ⊢ (𝜑 → (0...((𝑄 + 1) − 1)) = (0...𝑄))
39	38	sumeq1d 11512	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))
40	39	oveq1d 5934	. . . . . . . . 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 2226	. . . . . . . 8 ⊢ (𝜑 → e = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
42	41	oveq1d 5934	. . . . . . 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 9332	. . . . . . . . . 10 ⊢ (𝜑 → 0 ∈ ℤ)
44	13	nnzd 9441	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ∈ ℤ)
45	43, 44	fzfigd 10505	. . . . . . . . 9 ⊢ (𝜑 → (0...𝑄) ∈ Fin)
46		elfznn0 10183	. . . . . . . . . 10 ⊢ (𝑘 ∈ (0...𝑄) → 𝑘 ∈ ℕ0)
47	46, 30	sylan2 286	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) ∈ ℂ)
48	45, 47	fsumcl 11546	. . . . . . . 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 9763	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ+)
52	51	rpreccld 9776	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 / (!‘𝑘)) ∈ ℝ+)
53	49, 52	eqeltrd 2270	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℝ+)
54	11, 12, 15, 16, 53, 32	isumrpcl 11640	. . . . . . . . . 10 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ+)
55	54	rpred 9765	. . . . . . . . 9 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ)
56	55	recnd 8050	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℂ)
57	48, 56	pncan2d 8334	. . . . . . 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 2226	. . . . . 6 ⊢ (𝜑 → (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))
59	58	oveq2d 5935	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
60	13	nnnn0d 9296	. . . . . . . 8 ⊢ (𝜑 → 𝑄 ∈ ℕ0)
61	60	faccld 10810	. . . . . . 7 ⊢ (𝜑 → (!‘𝑄) ∈ ℕ)
62	61	nncnd 8998	. . . . . 6 ⊢ (𝜑 → (!‘𝑄) ∈ ℂ)
63		ere 11816	. . . . . . . 8 ⊢ e ∈ ℝ
64	63	recni 8033	. . . . . . 7 ⊢ e ∈ ℂ
65	64	a1i 9	. . . . . 6 ⊢ (𝜑 → e ∈ ℂ)
66	62, 65, 48	subdid 8435	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
67	59, 66	eqtr3d 2228	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
68	61	nnrpd 9763	. . . . . . 7 ⊢ (𝜑 → (!‘𝑄) ∈ ℝ+)
69	68, 54	rpmulcld 9782	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ+)
70	69	rpred 9765	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ)
71		eirr.2	. . . . . . . . 9 ⊢ (𝜑 → 𝑃 ∈ ℤ)
72	71	zcnd 9443	. . . . . . . 8 ⊢ (𝜑 → 𝑃 ∈ ℂ)
73	13	nnap0d 9030	. . . . . . . 8 ⊢ (𝜑 → 𝑄 # 0)
74	62, 72, 35, 73	div12apd 8848	. . . . . . 7 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) = (𝑃 · ((!‘𝑄) / 𝑄)))
75	13	nnred 8997	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑄 ∈ ℝ)
76	75	leidd 8535	. . . . . . . . . 10 ⊢ (𝜑 → 𝑄 ≤ 𝑄)
77		facdiv 10812	. . . . . . . . . 10 ⊢ ((𝑄 ∈ ℕ0 ∧ 𝑄 ∈ ℕ ∧ 𝑄 ≤ 𝑄) → ((!‘𝑄) / 𝑄) ∈ ℕ)
78	60, 13, 76, 77	syl3anc 1249	. . . . . . . . 9 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℕ)
79	78	nnzd 9441	. . . . . . . 8 ⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℤ)
80	71, 79	zmulcld 9448	. . . . . . 7 ⊢ (𝜑 → (𝑃 · ((!‘𝑄) / 𝑄)) ∈ ℤ)
81	74, 80	eqeltrd 2270	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℤ)
82	45, 62, 47	fsummulc2 11594	. . . . . . 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 5935	. . . . . . . . . . 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 8998	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℂ)
89	87	nnap0d 9030	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) # 0)
90	86, 88, 89	divrecapd 8814	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘))))
91	85, 90	eqtr4d 2229	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) / (!‘𝑘)))
92		permnn 10845	. . . . . . . . . . 11 ⊢ (𝑘 ∈ (0...𝑄) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
93	92	adantl 277	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ)
94	91, 93	eqeltrd 2270	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℕ)
95	94	nnzd 9441	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
96	45, 95	fsumzcl 11548	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ)
97	82, 96	eqeltrd 2270	. . . . . 6 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℤ)
98	81, 97	zsubcld 9447	. . . . 5 ⊢ (𝜑 → (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) ∈ ℤ)
99	69	rpgt0d 9768	. . . . 5 ⊢ (𝜑 → 0 < ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
100	14	peano2nnd 8999	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℕ)
101	100	nnred 8997	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℝ)
102	15	faccld 10810	. . . . . . . . . 10 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℕ)
103	102, 14	nnmulcld 9033	. . . . . . . . 9 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℕ)
104	101, 103	nndivred 9034	. . . . . . . 8 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℝ)
105	61	nnrecred 9031	. . . . . . . 8 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℝ)
106		abs1 11219	. . . . . . . . . . . . . 14 ⊢ (abs‘1) = 1
107	106	oveq1i 5929	. . . . . . . . . . . . 13 ⊢ ((abs‘1)↑𝑛) = (1↑𝑛)
108	107	oveq1i 5929	. . . . . . . . . . . 12 ⊢ (((abs‘1)↑𝑛) / (!‘𝑛)) = ((1↑𝑛) / (!‘𝑛))
109	108	mpteq2i 4117	. . . . . . . . . . 11 ⊢ (𝑛 ∈ ℕ0 ↦ (((abs‘1)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((1↑𝑛) / (!‘𝑛)))
110	23, 109	eqtr4i 2217	. . . . . . . . . 10 ⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (((abs‘1)↑𝑛) / (!‘𝑛)))
111		eqid 2193	. . . . . . . . . 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 8593	. . . . . . . . . . . 12 ⊢ 1 ≤ 1
113	106, 112	eqbrtri 4051	. . . . . . . . . . 11 ⊢ (abs‘1) ≤ 1
114	113	a1i 9	. . . . . . . . . 10 ⊢ (𝜑 → (abs‘1) ≤ 1)
115	23, 110, 111, 14, 27, 114	eftlub 11836	. . . . . . . . 9 ⊢ (𝜑 → (abs‘Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ≤ (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
116	54	rprege0d 9773	. . . . . . . . . 10 ⊢ (𝜑 → (Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)))
117		absid 11218	. . . . . . . . . 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 5929	. . . . . . . . . . . 12 ⊢ ((abs‘1)↑(𝑄 + 1)) = (1↑(𝑄 + 1))
120	14	nnzd 9441	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 1) ∈ ℤ)
121		1exp 10642	. . . . . . . . . . . . 13 ⊢ ((𝑄 + 1) ∈ ℤ → (1↑(𝑄 + 1)) = 1)
122	120, 121	syl 14	. . . . . . . . . . . 12 ⊢ (𝜑 → (1↑(𝑄 + 1)) = 1)
123	119, 122	eqtrid 2238	. . . . . . . . . . 11 ⊢ (𝜑 → ((abs‘1)↑(𝑄 + 1)) = 1)
124	123	oveq1d 5934	. . . . . . . . . 10 ⊢ (𝜑 → (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))))
125	104	recnd 8050	. . . . . . . . . . 11 ⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℂ)
126	125	mulid2d 8040	. . . . . . . . . 10 ⊢ (𝜑 → (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
127	124, 126	eqtrd 2226	. . . . . . . . 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 4061	. . . . . . . 8 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ≤ (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))
129	14	nnred 8997	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 1) ∈ ℝ)
130	129, 129	readdcld 8051	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ∈ ℝ)
131	129, 129	remulcld 8052	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) ∈ ℝ)
132		1red 8036	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
133	13	nnge1d 9027	. . . . . . . . . . . . 13 ⊢ (𝜑 → 1 ≤ 𝑄)
134		1nn 8995	. . . . . . . . . . . . . 14 ⊢ 1 ∈ ℕ
135		nnleltp1 9379	. . . . . . . . . . . . . 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 8443	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) + (𝑄 + 1)))
139	14	nncnd 8998	. . . . . . . . . . . . 13 ⊢ (𝜑 → (𝑄 + 1) ∈ ℂ)
140	139	2timesd 9228	. . . . . . . . . . . 12 ⊢ (𝜑 → (2 · (𝑄 + 1)) = ((𝑄 + 1) + (𝑄 + 1)))
141		df-2 9043	. . . . . . . . . . . . . 14 ⊢ 2 = (1 + 1)
142	132, 75, 132, 133	leadd1dd 8580	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (1 + 1) ≤ (𝑄 + 1))
143	141, 142	eqbrtrid 4065	. . . . . . . . . . . . 13 ⊢ (𝜑 → 2 ≤ (𝑄 + 1))
144		2re 9054	. . . . . . . . . . . . . . 15 ⊢ 2 ∈ ℝ
145	144	a1i 9	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 2 ∈ ℝ)
146	14	nngt0d 9028	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 0 < (𝑄 + 1))
147		lemul1 8614	. . . . . . . . . . . . . 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 1253	. . . . . . . . . . . . 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 4054	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))
151	101, 130, 131, 138, 150	ltletrd 8444	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) · (𝑄 + 1)))
152		facp1 10804	. . . . . . . . . . . . . . 15 ⊢ (𝑄 ∈ ℕ0 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
153	60, 152	syl 14	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1)))
154	153	oveq1d 5934	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)))
155	102	nncnd 8998	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘(𝑄 + 1)) ∈ ℂ)
156	61	nnap0d 9030	. . . . . . . . . . . . . 14 ⊢ (𝜑 → (!‘𝑄) # 0)
157	155, 62, 156	divrecapd 8814	. . . . . . . . . . . . 13 ⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
158	139, 62, 156	divcanap3d 8816	. . . . . . . . . . . . 13 ⊢ (𝜑 → (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)) = (𝑄 + 1))
159	154, 157, 158	3eqtr3rd 2235	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝑄 + 1) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄))))
160	159	oveq1d 5934	. . . . . . . . . . 11 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)))
161	105	recnd 8050	. . . . . . . . . . . 12 ⊢ (𝜑 → (1 / (!‘𝑄)) ∈ ℂ)
162	155, 161, 139	mul32d 8174	. . . . . . . . . . 11 ⊢ (𝜑 → (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
163	160, 162	eqtrd 2226	. . . . . . . . . 10 ⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
164	151, 163	breqtrd 4056	. . . . . . . . 9 ⊢ (𝜑 → ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄))))
165	103	nnred 8997	. . . . . . . . . 10 ⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈ ℝ)
166	103	nngt0d 9028	. . . . . . . . . 10 ⊢ (𝜑 → 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1)))
167		ltdivmul 8897	. . . . . . . . . 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 1253	. . . . . . . . 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 8146	. . . . . . 7 ⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄)))
171	55, 132, 68	ltmuldiv2d 9814	. . . . . . 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 9098	. . . . . 6 ⊢ (0 + 1) = 1
174	172, 173	breqtrrdi 4072	. . . . 5 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < (0 + 1))
175	43, 70, 98, 99, 174	btwnapz 9450	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
176	67, 175	eqbrtrrd 4054	. . 3 ⊢ (𝜑 → (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))))
177	62, 65	mulcld 8042	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · e) ∈ ℂ)
178	81	zcnd 9443	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℂ)
179	62, 48	mulcld 8042	. . . 4 ⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℂ)
180		apsub1 8663	. . . 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 1249	. . 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 9692	. . . . 5 ⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℚ)
184	71, 13, 183	syl2anc 411	. . . 4 ⊢ (𝜑 → (𝑃 / 𝑄) ∈ ℚ)
185		qcn 9702	. . . 4 ⊢ ((𝑃 / 𝑄) ∈ ℚ → (𝑃 / 𝑄) ∈ ℂ)
186	184, 185	syl 14	. . 3 ⊢ (𝜑 → (𝑃 / 𝑄) ∈ ℂ)
187		apmul2 8810	. . 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 1253	. 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 1364   ∈ wcel 2164   class class class wbr 4030   ↦ cmpt 4091  dom cdm 4660  ‘cfv 5255  (class class class)co 5919  ℂcc 7872  ℝcr 7873  0cc0 7874  1c1 7875   + caddc 7877   · cmul 7879   < clt 8056   ≤ cle 8057   − cmin 8192   # cap 8602   / cdiv 8693  ℕcn 8984  2c2 9035  ℕ₀cn0 9243  ℤcz 9320  ℤ_≥cuz 9595  ℚcq 9687  ℝ⁺crp 9722  ...cfz 10077  seqcseq 10521  ↑cexp 10612  !cfa 10799  abscabs 11144   ⇝ cli 11424  Σcsu 11499  eceu 11789

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992  ax-arch 7993  ax-caucvg 7994

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-irdg 6425  df-frec 6446  df-1o 6471  df-oadd 6475  df-er 6589  df-en 6797  df-dom 6798  df-fin 6799  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-2 9043  df-3 9044  df-4 9045  df-n0 9244  df-z 9321  df-uz 9596  df-q 9688  df-rp 9723  df-ico 9963  df-fz 10078  df-fzo 10212  df-seqfrec 10522  df-exp 10613  df-fac 10800  df-bc 10822  df-ihash 10850  df-shft 10962  df-cj 10989  df-re 10990  df-im 10991  df-rsqrt 11145  df-abs 11146  df-clim 11425  df-sumdc 11500  df-ef 11794  df-e 11795

This theorem is referenced by:  eirrap  11924