Step | Hyp | Ref
| Expression |
1 | | esum 11559 |
. . . . . . . . . . 11
⊢ e =
Σ𝑘 ∈
ℕ0 (1 / (!‘𝑘)) |
2 | | faccl 10609 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ∈
ℕ) |
3 | 2 | nnrecred 8880 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ (1 / (!‘𝑘))
∈ ℝ) |
4 | | fveq2 5468 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → (!‘𝑛) = (!‘𝑘)) |
5 | 4 | oveq2d 5840 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → (1 / (!‘𝑛)) = (1 / (!‘𝑘))) |
6 | | eirr.1 |
. . . . . . . . . . . . . 14
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦ (1 /
(!‘𝑛))) |
7 | 5, 6 | fvmptg 5544 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ0
∧ (1 / (!‘𝑘))
∈ ℝ) → (𝐹‘𝑘) = (1 / (!‘𝑘))) |
8 | 3, 7 | mpdan 418 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ (𝐹‘𝑘) = (1 / (!‘𝑘))) |
9 | 8 | sumeq2i 11261 |
. . . . . . . . . . 11
⊢
Σ𝑘 ∈
ℕ0 (𝐹‘𝑘) = Σ𝑘 ∈ ℕ0 (1 /
(!‘𝑘)) |
10 | 1, 9 | eqtr4i 2181 |
. . . . . . . . . 10
⊢ e =
Σ𝑘 ∈
ℕ0 (𝐹‘𝑘) |
11 | | nn0uz 9473 |
. . . . . . . . . . 11
⊢
ℕ0 = (ℤ≥‘0) |
12 | | eqid 2157 |
. . . . . . . . . . 11
⊢
(ℤ≥‘(𝑄 + 1)) =
(ℤ≥‘(𝑄 + 1)) |
13 | | eirr.3 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑄 ∈ ℕ) |
14 | 13 | peano2nnd 8848 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄 + 1) ∈ ℕ) |
15 | 14 | nnnn0d 9143 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑄 + 1) ∈
ℕ0) |
16 | | eqidd 2158 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (𝐹‘𝑘)) |
17 | | ax-1cn 7825 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℂ |
18 | | nn0z 9187 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℤ) |
19 | | 1exp 10448 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 ∈ ℤ →
(1↑𝑛) =
1) |
20 | 18, 19 | syl 14 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ ℕ0
→ (1↑𝑛) =
1) |
21 | 20 | oveq1d 5839 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ ℕ0
→ ((1↑𝑛) /
(!‘𝑛)) = (1 /
(!‘𝑛))) |
22 | 21 | mpteq2ia 4050 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ ℕ0
↦ ((1↑𝑛) /
(!‘𝑛))) = (𝑛 ∈ ℕ0
↦ (1 / (!‘𝑛))) |
23 | 6, 22 | eqtr4i 2181 |
. . . . . . . . . . . . . . 15
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦
((1↑𝑛) /
(!‘𝑛))) |
24 | 23 | eftvalcn 11554 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℂ ∧ 𝑘
∈ ℕ0) → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘))) |
25 | 17, 24 | mpan 421 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘))) |
26 | 25 | adantl 275 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = ((1↑𝑘) / (!‘𝑘))) |
27 | 17 | a1i 9 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ∈
ℂ) |
28 | | eftcl 11551 |
. . . . . . . . . . . . 13
⊢ ((1
∈ ℂ ∧ 𝑘
∈ ℕ0) → ((1↑𝑘) / (!‘𝑘)) ∈ ℂ) |
29 | 27, 28 | sylan 281 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
((1↑𝑘) /
(!‘𝑘)) ∈
ℂ) |
30 | 26, 29 | eqeltrd 2234 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈ ℂ) |
31 | 23 | efcllem 11556 |
. . . . . . . . . . . 12
⊢ (1 ∈
ℂ → seq0( + , 𝐹)
∈ dom ⇝ ) |
32 | 27, 31 | syl 14 |
. . . . . . . . . . 11
⊢ (𝜑 → seq0( + , 𝐹) ∈ dom ⇝
) |
33 | 11, 12, 15, 16, 30, 32 | isumsplit 11388 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐹‘𝑘) = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))) |
34 | 10, 33 | syl5eq 2202 |
. . . . . . . . 9
⊢ (𝜑 → e = (Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))) |
35 | 13 | nncnd 8847 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝑄 ∈ ℂ) |
36 | | pncan 8081 |
. . . . . . . . . . . . 13
⊢ ((𝑄 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑄 + 1)
− 1) = 𝑄) |
37 | 35, 17, 36 | sylancl 410 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((𝑄 + 1) − 1) = 𝑄) |
38 | 37 | oveq2d 5840 |
. . . . . . . . . . 11
⊢ (𝜑 → (0...((𝑄 + 1) − 1)) = (0...𝑄)) |
39 | 38 | sumeq1d 11263 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (0...((𝑄 + 1) − 1))(𝐹‘𝑘) = Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) |
40 | 39 | oveq1d 5839 |
. . . . . . . . 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 2190 |
. . . . . . . 8
⊢ (𝜑 → e = (Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) + Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))) |
42 | 41 | oveq1d 5839 |
. . . . . . 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 9179 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℤ) |
44 | 13 | nnzd 9285 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ∈ ℤ) |
45 | 43, 44 | fzfigd 10330 |
. . . . . . . . 9
⊢ (𝜑 → (0...𝑄) ∈ Fin) |
46 | | elfznn0 10016 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (0...𝑄) → 𝑘 ∈ ℕ0) |
47 | 46, 30 | sylan2 284 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) ∈ ℂ) |
48 | 45, 47 | fsumcl 11297 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘) ∈ ℂ) |
49 | 8 | adantl 275 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) = (1 / (!‘𝑘))) |
50 | 2 | adantl 275 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
(!‘𝑘) ∈
ℕ) |
51 | 50 | nnrpd 9601 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) →
(!‘𝑘) ∈
ℝ+) |
52 | 51 | rpreccld 9614 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (1 /
(!‘𝑘)) ∈
ℝ+) |
53 | 49, 52 | eqeltrd 2234 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐹‘𝑘) ∈
ℝ+) |
54 | 11, 12, 15, 16, 53, 32 | isumrpcl 11391 |
. . . . . . . . . 10
⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈
ℝ+) |
55 | 54 | rpred 9603 |
. . . . . . . . 9
⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ) |
56 | 55 | recnd 7906 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℂ) |
57 | 48, 56 | pncan2d 8188 |
. . . . . . 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 2190 |
. . . . . 6
⊢ (𝜑 → (e − Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) |
59 | 58 | oveq2d 5840 |
. . . . 5
⊢ (𝜑 → ((!‘𝑄) · (e −
Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = ((!‘𝑄) · Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))) |
60 | 13 | nnnn0d 9143 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 ∈
ℕ0) |
61 | 60 | faccld 10610 |
. . . . . . 7
⊢ (𝜑 → (!‘𝑄) ∈ ℕ) |
62 | 61 | nncnd 8847 |
. . . . . 6
⊢ (𝜑 → (!‘𝑄) ∈ ℂ) |
63 | | ere 11567 |
. . . . . . . 8
⊢ e ∈
ℝ |
64 | 63 | recni 7890 |
. . . . . . 7
⊢ e ∈
ℂ |
65 | 64 | a1i 9 |
. . . . . 6
⊢ (𝜑 → e ∈
ℂ) |
66 | 62, 65, 48 | subdid 8289 |
. . . . 5
⊢ (𝜑 → ((!‘𝑄) · (e −
Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))) |
67 | 59, 66 | eqtr3d 2192 |
. . . 4
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) = (((!‘𝑄) · e) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))) |
68 | 61 | nnrpd 9601 |
. . . . . . 7
⊢ (𝜑 → (!‘𝑄) ∈
ℝ+) |
69 | 68, 54 | rpmulcld 9620 |
. . . . . 6
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈
ℝ+) |
70 | 69 | rpred 9603 |
. . . . 5
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ∈ ℝ) |
71 | | eirr.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑃 ∈ ℤ) |
72 | 71 | zcnd 9287 |
. . . . . . . 8
⊢ (𝜑 → 𝑃 ∈ ℂ) |
73 | 13 | nnap0d 8879 |
. . . . . . . 8
⊢ (𝜑 → 𝑄 # 0) |
74 | 62, 72, 35, 73 | div12apd 8700 |
. . . . . . 7
⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) = (𝑃 · ((!‘𝑄) / 𝑄))) |
75 | 13 | nnred 8846 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑄 ∈ ℝ) |
76 | 75 | leidd 8389 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑄 ≤ 𝑄) |
77 | | facdiv 10612 |
. . . . . . . . . 10
⊢ ((𝑄 ∈ ℕ0
∧ 𝑄 ∈ ℕ
∧ 𝑄 ≤ 𝑄) → ((!‘𝑄) / 𝑄) ∈ ℕ) |
78 | 60, 13, 76, 77 | syl3anc 1220 |
. . . . . . . . 9
⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℕ) |
79 | 78 | nnzd 9285 |
. . . . . . . 8
⊢ (𝜑 → ((!‘𝑄) / 𝑄) ∈ ℤ) |
80 | 71, 79 | zmulcld 9292 |
. . . . . . 7
⊢ (𝜑 → (𝑃 · ((!‘𝑄) / 𝑄)) ∈ ℤ) |
81 | 74, 80 | eqeltrd 2234 |
. . . . . 6
⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℤ) |
82 | 45, 62, 47 | fsummulc2 11345 |
. . . . . . 7
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) = Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘))) |
83 | 46 | adantl 275 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → 𝑘 ∈ ℕ0) |
84 | 83, 8 | syl 14 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (𝐹‘𝑘) = (1 / (!‘𝑘))) |
85 | 84 | oveq2d 5840 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘)))) |
86 | 62 | adantr 274 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑄) ∈ ℂ) |
87 | 46, 50 | sylan2 284 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℕ) |
88 | 87 | nncnd 8847 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) ∈ ℂ) |
89 | 87 | nnap0d 8879 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → (!‘𝑘) # 0) |
90 | 86, 88, 89 | divrecapd 8666 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) = ((!‘𝑄) · (1 / (!‘𝑘)))) |
91 | 85, 90 | eqtr4d 2193 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) = ((!‘𝑄) / (!‘𝑘))) |
92 | | permnn 10645 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (0...𝑄) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ) |
93 | 92 | adantl 275 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) / (!‘𝑘)) ∈ ℕ) |
94 | 91, 93 | eqeltrd 2234 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℕ) |
95 | 94 | nnzd 9285 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ (0...𝑄)) → ((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ) |
96 | 45, 95 | fsumzcl 11299 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (0...𝑄)((!‘𝑄) · (𝐹‘𝑘)) ∈ ℤ) |
97 | 82, 96 | eqeltrd 2234 |
. . . . . 6
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℤ) |
98 | 81, 97 | zsubcld 9291 |
. . . . 5
⊢ (𝜑 → (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) ∈ ℤ) |
99 | 69 | rpgt0d 9606 |
. . . . 5
⊢ (𝜑 → 0 < ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))) |
100 | 14 | peano2nnd 8848 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℕ) |
101 | 100 | nnred 8846 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑄 + 1) + 1) ∈ ℝ) |
102 | 15 | faccld 10610 |
. . . . . . . . . 10
⊢ (𝜑 → (!‘(𝑄 + 1)) ∈
ℕ) |
103 | 102, 14 | nnmulcld 8882 |
. . . . . . . . 9
⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈
ℕ) |
104 | 101, 103 | nndivred 8883 |
. . . . . . . 8
⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℝ) |
105 | 61 | nnrecred 8880 |
. . . . . . . 8
⊢ (𝜑 → (1 / (!‘𝑄)) ∈
ℝ) |
106 | | abs1 10972 |
. . . . . . . . . . . . . 14
⊢
(abs‘1) = 1 |
107 | 106 | oveq1i 5834 |
. . . . . . . . . . . . 13
⊢
((abs‘1)↑𝑛) = (1↑𝑛) |
108 | 107 | oveq1i 5834 |
. . . . . . . . . . . 12
⊢
(((abs‘1)↑𝑛) / (!‘𝑛)) = ((1↑𝑛) / (!‘𝑛)) |
109 | 108 | mpteq2i 4051 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
↦ (((abs‘1)↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦
((1↑𝑛) /
(!‘𝑛))) |
110 | 23, 109 | eqtr4i 2181 |
. . . . . . . . . 10
⊢ 𝐹 = (𝑛 ∈ ℕ0 ↦
(((abs‘1)↑𝑛) /
(!‘𝑛))) |
111 | | eqid 2157 |
. . . . . . . . . 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 8447 |
. . . . . . . . . . . 12
⊢ 1 ≤
1 |
113 | 106, 112 | eqbrtri 3985 |
. . . . . . . . . . 11
⊢
(abs‘1) ≤ 1 |
114 | 113 | a1i 9 |
. . . . . . . . . 10
⊢ (𝜑 → (abs‘1) ≤
1) |
115 | 23, 110, 111, 14, 27, 114 | eftlub 11587 |
. . . . . . . . 9
⊢ (𝜑 → (abs‘Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) ≤ (((abs‘1)↑(𝑄 + 1)) · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))))) |
116 | 54 | rprege0d 9611 |
. . . . . . . . . 10
⊢ (𝜑 → (Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘))) |
117 | | absid 10971 |
. . . . . . . . . 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 5834 |
. . . . . . . . . . . 12
⊢
((abs‘1)↑(𝑄 + 1)) = (1↑(𝑄 + 1)) |
120 | 14 | nnzd 9285 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑄 + 1) ∈ ℤ) |
121 | | 1exp 10448 |
. . . . . . . . . . . . 13
⊢ ((𝑄 + 1) ∈ ℤ →
(1↑(𝑄 + 1)) =
1) |
122 | 120, 121 | syl 14 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1↑(𝑄 + 1)) = 1) |
123 | 119, 122 | syl5eq 2202 |
. . . . . . . . . . 11
⊢ (𝜑 → ((abs‘1)↑(𝑄 + 1)) = 1) |
124 | 123 | oveq1d 5839 |
. . . . . . . . . 10
⊢ (𝜑 →
(((abs‘1)↑(𝑄 +
1)) · (((𝑄 + 1) + 1)
/ ((!‘(𝑄 + 1))
· (𝑄 + 1)))) = (1
· (((𝑄 + 1) + 1) /
((!‘(𝑄 + 1)) ·
(𝑄 +
1))))) |
125 | 104 | recnd 7906 |
. . . . . . . . . . 11
⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) ∈ ℂ) |
126 | 125 | mulid2d 7896 |
. . . . . . . . . 10
⊢ (𝜑 → (1 · (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) = (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) |
127 | 124, 126 | eqtrd 2190 |
. . . . . . . . 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 3995 |
. . . . . . . 8
⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) ≤ (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1)))) |
129 | 14 | nnred 8846 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄 + 1) ∈ ℝ) |
130 | 129, 129 | readdcld 7907 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ∈ ℝ) |
131 | 129, 129 | remulcld 7908 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) ∈ ℝ) |
132 | | 1red 7893 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℝ) |
133 | 13 | nnge1d 8876 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 1 ≤ 𝑄) |
134 | | 1nn 8844 |
. . . . . . . . . . . . . 14
⊢ 1 ∈
ℕ |
135 | | nnleltp1 9226 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℕ ∧ 𝑄
∈ ℕ) → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1))) |
136 | 134, 13, 135 | sylancr 411 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (1 ≤ 𝑄 ↔ 1 < (𝑄 + 1))) |
137 | 133, 136 | mpbid 146 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 < (𝑄 + 1)) |
138 | 132, 129,
129, 137 | ltadd2dd 8297 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) + (𝑄 + 1))) |
139 | 14 | nncnd 8847 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (𝑄 + 1) ∈ ℂ) |
140 | 139 | 2timesd 9075 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 · (𝑄 + 1)) = ((𝑄 + 1) + (𝑄 + 1))) |
141 | | df-2 8892 |
. . . . . . . . . . . . . 14
⊢ 2 = (1 +
1) |
142 | 132, 75, 132, 133 | leadd1dd 8434 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (1 + 1) ≤ (𝑄 + 1)) |
143 | 141, 142 | eqbrtrid 3999 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 2 ≤ (𝑄 + 1)) |
144 | | 2re 8903 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ |
145 | 144 | a1i 9 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 2 ∈
ℝ) |
146 | 14 | nngt0d 8877 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 0 < (𝑄 + 1)) |
147 | | lemul1 8468 |
. . . . . . . . . . . . . 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 1224 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (2 ≤ (𝑄 + 1) ↔ (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1)))) |
149 | 143, 148 | mpbid 146 |
. . . . . . . . . . . 12
⊢ (𝜑 → (2 · (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1))) |
150 | 140, 149 | eqbrtrrd 3988 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 + 1) + (𝑄 + 1)) ≤ ((𝑄 + 1) · (𝑄 + 1))) |
151 | 101, 130,
131, 138, 150 | ltletrd 8298 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 + 1) + 1) < ((𝑄 + 1) · (𝑄 + 1))) |
152 | | facp1 10604 |
. . . . . . . . . . . . . . 15
⊢ (𝑄 ∈ ℕ0
→ (!‘(𝑄 + 1)) =
((!‘𝑄) ·
(𝑄 + 1))) |
153 | 60, 152 | syl 14 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (!‘(𝑄 + 1)) = ((!‘𝑄) · (𝑄 + 1))) |
154 | 153 | oveq1d 5839 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄))) |
155 | 102 | nncnd 8847 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (!‘(𝑄 + 1)) ∈
ℂ) |
156 | 61 | nnap0d 8879 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → (!‘𝑄) # 0) |
157 | 155, 62, 156 | divrecapd 8666 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((!‘(𝑄 + 1)) / (!‘𝑄)) = ((!‘(𝑄 + 1)) · (1 /
(!‘𝑄)))) |
158 | 139, 62, 156 | divcanap3d 8668 |
. . . . . . . . . . . . 13
⊢ (𝜑 → (((!‘𝑄) · (𝑄 + 1)) / (!‘𝑄)) = (𝑄 + 1)) |
159 | 154, 157,
158 | 3eqtr3rd 2199 |
. . . . . . . . . . . 12
⊢ (𝜑 → (𝑄 + 1) = ((!‘(𝑄 + 1)) · (1 / (!‘𝑄)))) |
160 | 159 | oveq1d 5839 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (1 / (!‘𝑄))) · (𝑄 + 1))) |
161 | 105 | recnd 7906 |
. . . . . . . . . . . 12
⊢ (𝜑 → (1 / (!‘𝑄)) ∈
ℂ) |
162 | 155, 161,
139 | mul32d 8028 |
. . . . . . . . . . 11
⊢ (𝜑 → (((!‘(𝑄 + 1)) · (1 /
(!‘𝑄))) ·
(𝑄 + 1)) =
(((!‘(𝑄 + 1))
· (𝑄 + 1)) ·
(1 / (!‘𝑄)))) |
163 | 160, 162 | eqtrd 2190 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝑄 + 1) · (𝑄 + 1)) = (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 / (!‘𝑄)))) |
164 | 151, 163 | breqtrd 3990 |
. . . . . . . . 9
⊢ (𝜑 → ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 /
(!‘𝑄)))) |
165 | 103 | nnred 8846 |
. . . . . . . . . 10
⊢ (𝜑 → ((!‘(𝑄 + 1)) · (𝑄 + 1)) ∈
ℝ) |
166 | 103 | nngt0d 8877 |
. . . . . . . . . 10
⊢ (𝜑 → 0 < ((!‘(𝑄 + 1)) · (𝑄 + 1))) |
167 | | ltdivmul 8747 |
. . . . . . . . . 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 1224 |
. . . . . . . . 9
⊢ (𝜑 → ((((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄)) ↔ ((𝑄 + 1) + 1) < (((!‘(𝑄 + 1)) · (𝑄 + 1)) · (1 /
(!‘𝑄))))) |
169 | 164, 168 | mpbird 166 |
. . . . . . . 8
⊢ (𝜑 → (((𝑄 + 1) + 1) / ((!‘(𝑄 + 1)) · (𝑄 + 1))) < (1 / (!‘𝑄))) |
170 | 55, 104, 105, 128, 169 | lelttrd 8000 |
. . . . . . 7
⊢ (𝜑 → Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄))) |
171 | 55, 132, 68 | ltmuldiv2d 9652 |
. . . . . . 7
⊢ (𝜑 → (((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < 1 ↔ Σ𝑘 ∈ (ℤ≥‘(𝑄 + 1))(𝐹‘𝑘) < (1 / (!‘𝑄)))) |
172 | 170, 171 | mpbird 166 |
. . . . . 6
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < 1) |
173 | | 0p1e1 8947 |
. . . . . 6
⊢ (0 + 1) =
1 |
174 | 172, 173 | breqtrrdi 4006 |
. . . . 5
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) < (0 + 1)) |
175 | 43, 70, 98, 99, 174 | btwnapz 9294 |
. . . 4
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈
(ℤ≥‘(𝑄 + 1))(𝐹‘𝑘)) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))) |
176 | 67, 175 | eqbrtrrd 3988 |
. . 3
⊢ (𝜑 → (((!‘𝑄) · e) −
((!‘𝑄) ·
Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘))) # (((!‘𝑄) · (𝑃 / 𝑄)) − ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)))) |
177 | 62, 65 | mulcld 7898 |
. . . 4
⊢ (𝜑 → ((!‘𝑄) · e) ∈
ℂ) |
178 | 81 | zcnd 9287 |
. . . 4
⊢ (𝜑 → ((!‘𝑄) · (𝑃 / 𝑄)) ∈ ℂ) |
179 | 62, 48 | mulcld 7898 |
. . . 4
⊢ (𝜑 → ((!‘𝑄) · Σ𝑘 ∈ (0...𝑄)(𝐹‘𝑘)) ∈ ℂ) |
180 | | apsub1 8517 |
. . . 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 1220 |
. . 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 166 |
. 2
⊢ (𝜑 → ((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄))) |
183 | | znq 9533 |
. . . . 5
⊢ ((𝑃 ∈ ℤ ∧ 𝑄 ∈ ℕ) → (𝑃 / 𝑄) ∈ ℚ) |
184 | 71, 13, 183 | syl2anc 409 |
. . . 4
⊢ (𝜑 → (𝑃 / 𝑄) ∈ ℚ) |
185 | | qcn 9543 |
. . . 4
⊢ ((𝑃 / 𝑄) ∈ ℚ → (𝑃 / 𝑄) ∈ ℂ) |
186 | 184, 185 | syl 14 |
. . 3
⊢ (𝜑 → (𝑃 / 𝑄) ∈ ℂ) |
187 | | apmul2 8662 |
. . 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 1224 |
. 2
⊢ (𝜑 → (e # (𝑃 / 𝑄) ↔ ((!‘𝑄) · e) # ((!‘𝑄) · (𝑃 / 𝑄)))) |
189 | 182, 188 | mpbird 166 |
1
⊢ (𝜑 → e # (𝑃 / 𝑄)) |