Step | Hyp | Ref
| Expression |
1 | | 0nn0 9150 |
. . . . . . . . 9
⊢ 0 ∈
ℕ0 |
2 | | nn0uz 9521 |
. . . . . . . . 9
⊢
ℕ0 = (ℤ≥‘0) |
3 | 1, 2 | eleqtri 2245 |
. . . . . . . 8
⊢ 0 ∈
(ℤ≥‘0) |
4 | 3 | a1i 9 |
. . . . . . 7
⊢ (⊤
→ 0 ∈ (ℤ≥‘0)) |
5 | | elnn0uz 9524 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
↔ 𝑘 ∈
(ℤ≥‘0)) |
6 | 5 | biimpri 132 |
. . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘0) → 𝑘 ∈ ℕ0) |
7 | | faccl 10669 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ (!‘𝑘) ∈
ℕ) |
8 | 7 | nnrecred 8925 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ (1 / (!‘𝑘))
∈ ℝ) |
9 | | fveq2 5496 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 𝑘 → (!‘𝑛) = (!‘𝑘)) |
10 | 9 | oveq2d 5869 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (1 / (!‘𝑛)) = (1 / (!‘𝑘))) |
11 | | erelem1.2 |
. . . . . . . . . . . 12
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦ (1 /
(!‘𝑛))) |
12 | 10, 11 | fvmptg 5572 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ0
∧ (1 / (!‘𝑘))
∈ ℝ) → (𝐺‘𝑘) = (1 / (!‘𝑘))) |
13 | 8, 12 | mpdan 419 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ0
→ (𝐺‘𝑘) = (1 / (!‘𝑘))) |
14 | 13, 8 | eqeltrd 2247 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ0
→ (𝐺‘𝑘) ∈
ℝ) |
15 | 6, 14 | syl 14 |
. . . . . . . 8
⊢ (𝑘 ∈
(ℤ≥‘0) → (𝐺‘𝑘) ∈ ℝ) |
16 | 15 | adantl 275 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ (ℤ≥‘0)) → (𝐺‘𝑘) ∈ ℝ) |
17 | | readdcl 7900 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℝ ∧ 𝑦 ∈ ℝ) → (𝑘 + 𝑦) ∈ ℝ) |
18 | 17 | adantl 275 |
. . . . . . 7
⊢
((⊤ ∧ (𝑘
∈ ℝ ∧ 𝑦
∈ ℝ)) → (𝑘
+ 𝑦) ∈
ℝ) |
19 | 4, 16, 18 | seq3p1 10418 |
. . . . . 6
⊢ (⊤
→ (seq0( + , 𝐺)‘(0 + 1)) = ((seq0( + , 𝐺)‘0) + (𝐺‘(0 + 1)))) |
20 | | 0zd 9224 |
. . . . . . . . 9
⊢ (⊤
→ 0 ∈ ℤ) |
21 | 20, 16, 18 | seq3-1 10416 |
. . . . . . . 8
⊢ (⊤
→ (seq0( + , 𝐺)‘0) = (𝐺‘0)) |
22 | | fveq2 5496 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 0 → (!‘𝑛) =
(!‘0)) |
23 | | fac0 10662 |
. . . . . . . . . . . . 13
⊢
(!‘0) = 1 |
24 | 22, 23 | eqtrdi 2219 |
. . . . . . . . . . . 12
⊢ (𝑛 = 0 → (!‘𝑛) = 1) |
25 | 24 | oveq2d 5869 |
. . . . . . . . . . 11
⊢ (𝑛 = 0 → (1 / (!‘𝑛)) = (1 / 1)) |
26 | | ax-1cn 7867 |
. . . . . . . . . . . 12
⊢ 1 ∈
ℂ |
27 | 26 | div1i 8657 |
. . . . . . . . . . 11
⊢ (1 / 1) =
1 |
28 | 25, 27 | eqtrdi 2219 |
. . . . . . . . . 10
⊢ (𝑛 = 0 → (1 / (!‘𝑛)) = 1) |
29 | | 1ex 7915 |
. . . . . . . . . 10
⊢ 1 ∈
V |
30 | 28, 11, 29 | fvmpt 5573 |
. . . . . . . . 9
⊢ (0 ∈
ℕ0 → (𝐺‘0) = 1) |
31 | 1, 30 | mp1i 10 |
. . . . . . . 8
⊢ (⊤
→ (𝐺‘0) =
1) |
32 | 21, 31 | eqtrd 2203 |
. . . . . . 7
⊢ (⊤
→ (seq0( + , 𝐺)‘0) = 1) |
33 | | 1e0p1 9384 |
. . . . . . . . 9
⊢ 1 = (0 +
1) |
34 | 33 | fveq2i 5499 |
. . . . . . . 8
⊢ (𝐺‘1) = (𝐺‘(0 + 1)) |
35 | | 1nn0 9151 |
. . . . . . . . 9
⊢ 1 ∈
ℕ0 |
36 | | fveq2 5496 |
. . . . . . . . . . . . 13
⊢ (𝑛 = 1 → (!‘𝑛) =
(!‘1)) |
37 | | fac1 10663 |
. . . . . . . . . . . . 13
⊢
(!‘1) = 1 |
38 | 36, 37 | eqtrdi 2219 |
. . . . . . . . . . . 12
⊢ (𝑛 = 1 → (!‘𝑛) = 1) |
39 | 38 | oveq2d 5869 |
. . . . . . . . . . 11
⊢ (𝑛 = 1 → (1 / (!‘𝑛)) = (1 / 1)) |
40 | 39, 27 | eqtrdi 2219 |
. . . . . . . . . 10
⊢ (𝑛 = 1 → (1 / (!‘𝑛)) = 1) |
41 | 40, 11, 29 | fvmpt 5573 |
. . . . . . . . 9
⊢ (1 ∈
ℕ0 → (𝐺‘1) = 1) |
42 | 35, 41 | mp1i 10 |
. . . . . . . 8
⊢ (⊤
→ (𝐺‘1) =
1) |
43 | 34, 42 | eqtr3id 2217 |
. . . . . . 7
⊢ (⊤
→ (𝐺‘(0 + 1)) =
1) |
44 | 32, 43 | oveq12d 5871 |
. . . . . 6
⊢ (⊤
→ ((seq0( + , 𝐺)‘0) + (𝐺‘(0 + 1))) = (1 + 1)) |
45 | 19, 44 | eqtrd 2203 |
. . . . 5
⊢ (⊤
→ (seq0( + , 𝐺)‘(0 + 1)) = (1 + 1)) |
46 | 33 | fveq2i 5499 |
. . . . 5
⊢ (seq0( +
, 𝐺)‘1) = (seq0( + ,
𝐺)‘(0 +
1)) |
47 | | df-2 8937 |
. . . . 5
⊢ 2 = (1 +
1) |
48 | 45, 46, 47 | 3eqtr4g 2228 |
. . . 4
⊢ (⊤
→ (seq0( + , 𝐺)‘1) = 2) |
49 | 35 | a1i 9 |
. . . . 5
⊢ (⊤
→ 1 ∈ ℕ0) |
50 | | nn0z 9232 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℕ0
→ 𝑛 ∈
ℤ) |
51 | | 1exp 10505 |
. . . . . . . . . . . 12
⊢ (𝑛 ∈ ℤ →
(1↑𝑛) =
1) |
52 | 50, 51 | syl 14 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ ℕ0
→ (1↑𝑛) =
1) |
53 | 52 | oveq1d 5868 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ0
→ ((1↑𝑛) /
(!‘𝑛)) = (1 /
(!‘𝑛))) |
54 | 53 | mpteq2ia 4075 |
. . . . . . . . 9
⊢ (𝑛 ∈ ℕ0
↦ ((1↑𝑛) /
(!‘𝑛))) = (𝑛 ∈ ℕ0
↦ (1 / (!‘𝑛))) |
55 | 11, 54 | eqtr4i 2194 |
. . . . . . . 8
⊢ 𝐺 = (𝑛 ∈ ℕ0 ↦
((1↑𝑛) /
(!‘𝑛))) |
56 | 55 | efcvg 11629 |
. . . . . . 7
⊢ (1 ∈
ℂ → seq0( + , 𝐺)
⇝ (exp‘1)) |
57 | 26, 56 | mp1i 10 |
. . . . . 6
⊢ (⊤
→ seq0( + , 𝐺) ⇝
(exp‘1)) |
58 | | df-e 11612 |
. . . . . 6
⊢ e =
(exp‘1) |
59 | 57, 58 | breqtrrdi 4031 |
. . . . 5
⊢ (⊤
→ seq0( + , 𝐺) ⇝
e) |
60 | 13 | adantl 275 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝐺‘𝑘) = (1 / (!‘𝑘))) |
61 | 7 | adantl 275 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
62 | 61 | nnrecred 8925 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (1 / (!‘𝑘)) ∈ ℝ) |
63 | 60, 62 | eqeltrd 2247 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝐺‘𝑘) ∈ ℝ) |
64 | 61 | nnred 8891 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (!‘𝑘) ∈ ℝ) |
65 | 61 | nngt0d 8922 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 < (!‘𝑘)) |
66 | | 1re 7919 |
. . . . . . . 8
⊢ 1 ∈
ℝ |
67 | | 0le1 8400 |
. . . . . . . 8
⊢ 0 ≤
1 |
68 | | divge0 8789 |
. . . . . . . 8
⊢ (((1
∈ ℝ ∧ 0 ≤ 1) ∧ ((!‘𝑘) ∈ ℝ ∧ 0 < (!‘𝑘))) → 0 ≤ (1 /
(!‘𝑘))) |
69 | 66, 67, 68 | mpanl12 434 |
. . . . . . 7
⊢
(((!‘𝑘) ∈
ℝ ∧ 0 < (!‘𝑘)) → 0 ≤ (1 / (!‘𝑘))) |
70 | 64, 65, 69 | syl2anc 409 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 ≤ (1 / (!‘𝑘))) |
71 | 70, 60 | breqtrrd 4017 |
. . . . 5
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 0 ≤ (𝐺‘𝑘)) |
72 | 2, 49, 59, 63, 71 | climserle 11308 |
. . . 4
⊢ (⊤
→ (seq0( + , 𝐺)‘1) ≤ e) |
73 | 48, 72 | eqbrtrrd 4013 |
. . 3
⊢ (⊤
→ 2 ≤ e) |
74 | 73 | mptru 1357 |
. 2
⊢ 2 ≤
e |
75 | | nnuz 9522 |
. . . . . 6
⊢ ℕ =
(ℤ≥‘1) |
76 | | 1zzd 9239 |
. . . . . 6
⊢ (⊤
→ 1 ∈ ℤ) |
77 | 1 | a1i 9 |
. . . . . . . 8
⊢ (⊤
→ 0 ∈ ℕ0) |
78 | 63 | recnd 7948 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (𝐺‘𝑘) ∈ ℂ) |
79 | 2, 77, 78, 59 | clim2ser 11300 |
. . . . . . 7
⊢ (⊤
→ seq(0 + 1)( + , 𝐺)
⇝ (e − (seq0( + , 𝐺)‘0))) |
80 | | 0p1e1 8992 |
. . . . . . . 8
⊢ (0 + 1) =
1 |
81 | | seqeq1 10404 |
. . . . . . . 8
⊢ ((0 + 1)
= 1 → seq(0 + 1)( + , 𝐺) = seq1( + , 𝐺)) |
82 | 80, 81 | ax-mp 5 |
. . . . . . 7
⊢ seq(0 +
1)( + , 𝐺) = seq1( + ,
𝐺) |
83 | 32 | mptru 1357 |
. . . . . . . 8
⊢ (seq0( +
, 𝐺)‘0) =
1 |
84 | 83 | oveq2i 5864 |
. . . . . . 7
⊢ (e
− (seq0( + , 𝐺)‘0)) = (e − 1) |
85 | 79, 82, 84 | 3brtr3g 4022 |
. . . . . 6
⊢ (⊤
→ seq1( + , 𝐺) ⇝
(e − 1)) |
86 | | 2cnd 8951 |
. . . . . . . 8
⊢ (⊤
→ 2 ∈ ℂ) |
87 | | halfre 9091 |
. . . . . . . . . . . . . . 15
⊢ (1 / 2)
∈ ℝ |
88 | 87 | a1i 9 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ (1 / 2) ∈ ℝ) |
89 | | id 19 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℕ0) |
90 | 88, 89 | reexpcld 10626 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ ((1 / 2)↑𝑘)
∈ ℝ) |
91 | | oveq2 5861 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑘 → ((1 / 2)↑𝑛) = ((1 / 2)↑𝑘)) |
92 | | eqid 2170 |
. . . . . . . . . . . . . 14
⊢ (𝑛 ∈ ℕ0
↦ ((1 / 2)↑𝑛)) =
(𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛)) |
93 | 91, 92 | fvmptg 5572 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ0
∧ ((1 / 2)↑𝑘)
∈ ℝ) → ((𝑛
∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑘) = ((1 / 2)↑𝑘)) |
94 | 90, 93 | mpdan 419 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑘) = ((1 / 2)↑𝑘)) |
95 | 94 | adantl 275 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))‘𝑘) = ((1 / 2)↑𝑘)) |
96 | | simpr 109 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 𝑘 ∈ ℕ0) |
97 | | reexpcl 10493 |
. . . . . . . . . . . . 13
⊢ (((1 / 2)
∈ ℝ ∧ 𝑘
∈ ℕ0) → ((1 / 2)↑𝑘) ∈ ℝ) |
98 | 87, 96, 97 | sylancr 412 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((1 / 2)↑𝑘) ∈ ℝ) |
99 | 98 | recnd 7948 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((1 / 2)↑𝑘) ∈ ℂ) |
100 | 95, 99 | eqeltrd 2247 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))‘𝑘) ∈
ℂ) |
101 | | 1lt2 9047 |
. . . . . . . . . . . . . 14
⊢ 1 <
2 |
102 | | 2re 8948 |
. . . . . . . . . . . . . . 15
⊢ 2 ∈
ℝ |
103 | | 0le2 8968 |
. . . . . . . . . . . . . . 15
⊢ 0 ≤
2 |
104 | | absid 11035 |
. . . . . . . . . . . . . . 15
⊢ ((2
∈ ℝ ∧ 0 ≤ 2) → (abs‘2) = 2) |
105 | 102, 103,
104 | mp2an 424 |
. . . . . . . . . . . . . 14
⊢
(abs‘2) = 2 |
106 | 101, 105 | breqtrri 4016 |
. . . . . . . . . . . . 13
⊢ 1 <
(abs‘2) |
107 | 106 | a1i 9 |
. . . . . . . . . . . 12
⊢ (⊤
→ 1 < (abs‘2)) |
108 | 86, 107, 95 | georeclim 11476 |
. . . . . . . . . . 11
⊢ (⊤
→ seq0( + , (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))) ⇝ (2 / (2 −
1))) |
109 | | 2m1e1 8996 |
. . . . . . . . . . . . 13
⊢ (2
− 1) = 1 |
110 | 109 | oveq2i 5864 |
. . . . . . . . . . . 12
⊢ (2 / (2
− 1)) = (2 / 1) |
111 | | 2cn 8949 |
. . . . . . . . . . . . 13
⊢ 2 ∈
ℂ |
112 | 111 | div1i 8657 |
. . . . . . . . . . . 12
⊢ (2 / 1) =
2 |
113 | 110, 112 | eqtri 2191 |
. . . . . . . . . . 11
⊢ (2 / (2
− 1)) = 2 |
114 | 108, 113 | breqtrdi 4030 |
. . . . . . . . . 10
⊢ (⊤
→ seq0( + , (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))) ⇝ 2) |
115 | 2, 77, 100, 114 | clim2ser 11300 |
. . . . . . . . 9
⊢ (⊤
→ seq(0 + 1)( + , (𝑛
∈ ℕ0 ↦ ((1 / 2)↑𝑛))) ⇝ (2 − (seq0( + , (𝑛 ∈ ℕ0
↦ ((1 / 2)↑𝑛)))‘0))) |
116 | | seqeq1 10404 |
. . . . . . . . . 10
⊢ ((0 + 1)
= 1 → seq(0 + 1)( + , (𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))) = seq1( + ,
(𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛)))) |
117 | 80, 116 | ax-mp 5 |
. . . . . . . . 9
⊢ seq(0 +
1)( + , (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))) = seq1( + , (𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))) |
118 | 6 | adantl 275 |
. . . . . . . . . . . . . . 15
⊢
((⊤ ∧ 𝑘
∈ (ℤ≥‘0)) → 𝑘 ∈ ℕ0) |
119 | 94, 90 | eqeltrd 2247 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℕ0
→ ((𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑘) ∈ ℝ) |
120 | 118, 119 | syl 14 |
. . . . . . . . . . . . . 14
⊢
((⊤ ∧ 𝑘
∈ (ℤ≥‘0)) → ((𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))‘𝑘) ∈
ℝ) |
121 | 20, 120, 18 | seq3-1 10416 |
. . . . . . . . . . . . 13
⊢ (⊤
→ (seq0( + , (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛)))‘0) = ((𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))‘0)) |
122 | | halfcn 9092 |
. . . . . . . . . . . . . . . . 17
⊢ (1 / 2)
∈ ℂ |
123 | | exp0 10480 |
. . . . . . . . . . . . . . . . 17
⊢ ((1 / 2)
∈ ℂ → ((1 / 2)↑0) = 1) |
124 | 122, 123 | ax-mp 5 |
. . . . . . . . . . . . . . . 16
⊢ ((1 /
2)↑0) = 1 |
125 | 124, 35 | eqeltri 2243 |
. . . . . . . . . . . . . . 15
⊢ ((1 /
2)↑0) ∈ ℕ0 |
126 | | oveq2 5861 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 0 → ((1 / 2)↑𝑛) = ((1 /
2)↑0)) |
127 | 126, 92 | fvmptg 5572 |
. . . . . . . . . . . . . . 15
⊢ ((0
∈ ℕ0 ∧ ((1 / 2)↑0) ∈ ℕ0)
→ ((𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))‘0) = ((1 /
2)↑0)) |
128 | 1, 125, 127 | mp2an 424 |
. . . . . . . . . . . . . 14
⊢ ((𝑛 ∈ ℕ0
↦ ((1 / 2)↑𝑛))‘0) = ((1 /
2)↑0) |
129 | 128, 124 | eqtri 2191 |
. . . . . . . . . . . . 13
⊢ ((𝑛 ∈ ℕ0
↦ ((1 / 2)↑𝑛))‘0) = 1 |
130 | 121, 129 | eqtrdi 2219 |
. . . . . . . . . . . 12
⊢ (⊤
→ (seq0( + , (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛)))‘0) = 1) |
131 | 130 | mptru 1357 |
. . . . . . . . . . 11
⊢ (seq0( +
, (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛)))‘0) = 1 |
132 | 131 | oveq2i 5864 |
. . . . . . . . . 10
⊢ (2
− (seq0( + , (𝑛
∈ ℕ0 ↦ ((1 / 2)↑𝑛)))‘0)) = (2 −
1) |
133 | 132, 109 | eqtri 2191 |
. . . . . . . . 9
⊢ (2
− (seq0( + , (𝑛
∈ ℕ0 ↦ ((1 / 2)↑𝑛)))‘0)) = 1 |
134 | 115, 117,
133 | 3brtr3g 4022 |
. . . . . . . 8
⊢ (⊤
→ seq1( + , (𝑛 ∈
ℕ0 ↦ ((1 / 2)↑𝑛))) ⇝ 1) |
135 | | nnnn0 9142 |
. . . . . . . . 9
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℕ0) |
136 | 135, 100 | sylan2 284 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑘) ∈ ℂ) |
137 | 102 | a1i 9 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → 2 ∈
ℝ) |
138 | 135, 90 | syl 14 |
. . . . . . . . . . . 12
⊢ (𝑘 ∈ ℕ → ((1 /
2)↑𝑘) ∈
ℝ) |
139 | 137, 138 | remulcld 7950 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ → (2
· ((1 / 2)↑𝑘))
∈ ℝ) |
140 | 91 | oveq2d 5869 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (2 · ((1 / 2)↑𝑛)) = (2 · ((1 /
2)↑𝑘))) |
141 | | erelem1.1 |
. . . . . . . . . . . 12
⊢ 𝐹 = (𝑛 ∈ ℕ ↦ (2 · ((1 /
2)↑𝑛))) |
142 | 140, 141 | fvmptg 5572 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧ (2
· ((1 / 2)↑𝑘))
∈ ℝ) → (𝐹‘𝑘) = (2 · ((1 / 2)↑𝑘))) |
143 | 139, 142 | mpdan 419 |
. . . . . . . . . 10
⊢ (𝑘 ∈ ℕ → (𝐹‘𝑘) = (2 · ((1 / 2)↑𝑘))) |
144 | 143 | adantl 275 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) = (2 · ((1 / 2)↑𝑘))) |
145 | 135, 95 | sylan2 284 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ) → ((𝑛
∈ ℕ0 ↦ ((1 / 2)↑𝑛))‘𝑘) = ((1 / 2)↑𝑘)) |
146 | 145 | oveq2d 5869 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · ((𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))‘𝑘)) = (2 · ((1 /
2)↑𝑘))) |
147 | 144, 146 | eqtr4d 2206 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) = (2 · ((𝑛 ∈ ℕ0 ↦ ((1 /
2)↑𝑛))‘𝑘))) |
148 | 75, 76, 86, 134, 136, 147 | isermulc2 11303 |
. . . . . . 7
⊢ (⊤
→ seq1( + , 𝐹) ⇝
(2 · 1)) |
149 | | 2t1e2 9031 |
. . . . . . 7
⊢ (2
· 1) = 2 |
150 | 148, 149 | breqtrdi 4030 |
. . . . . 6
⊢ (⊤
→ seq1( + , 𝐹) ⇝
2) |
151 | 135, 63 | sylan2 284 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ∈ ℝ) |
152 | | remulcl 7902 |
. . . . . . . . 9
⊢ ((2
∈ ℝ ∧ ((1 / 2)↑𝑘) ∈ ℝ) → (2 · ((1 /
2)↑𝑘)) ∈
ℝ) |
153 | 102, 98, 152 | sylancr 412 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · ((1 / 2)↑𝑘)) ∈ ℝ) |
154 | 135, 153 | sylan2 284 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (2 · ((1 / 2)↑𝑘)) ∈ ℝ) |
155 | 144, 154 | eqeltrd 2247 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) |
156 | | faclbnd2 10676 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ℕ0
→ ((2↑𝑘) / 2)
≤ (!‘𝑘)) |
157 | 156 | adantl 275 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2↑𝑘) / 2) ≤ (!‘𝑘)) |
158 | | 2nn 9039 |
. . . . . . . . . . . . . 14
⊢ 2 ∈
ℕ |
159 | | nnexpcl 10489 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ ∧ 𝑘
∈ ℕ0) → (2↑𝑘) ∈ ℕ) |
160 | 158, 96, 159 | sylancr 412 |
. . . . . . . . . . . . 13
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2↑𝑘) ∈ ℕ) |
161 | 160 | nnrpd 9651 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2↑𝑘) ∈
ℝ+) |
162 | 161 | rphalfcld 9666 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((2↑𝑘) / 2) ∈
ℝ+) |
163 | 61 | nnrpd 9651 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (!‘𝑘) ∈
ℝ+) |
164 | 162, 163 | lerecd 9673 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (((2↑𝑘) / 2) ≤ (!‘𝑘) ↔ (1 / (!‘𝑘)) ≤ (1 / ((2↑𝑘) / 2)))) |
165 | 157, 164 | mpbid 146 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (1 / (!‘𝑘)) ≤ (1 / ((2↑𝑘) / 2))) |
166 | | 2cnd 8951 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 2 ∈ ℂ) |
167 | 160 | nncnd 8892 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2↑𝑘) ∈ ℂ) |
168 | 160 | nnap0d 8924 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2↑𝑘) # 0) |
169 | 166, 167,
168 | divrecapd 8710 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 / (2↑𝑘)) = (2 · (1 / (2↑𝑘)))) |
170 | | 2ap0 8971 |
. . . . . . . . . . . 12
⊢ 2 #
0 |
171 | 170 | a1i 9 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 2 # 0) |
172 | 167, 166,
168, 171 | recdivapd 8724 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (1 / ((2↑𝑘) / 2)) = (2 / (2↑𝑘))) |
173 | | nn0z 9232 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ0
→ 𝑘 ∈
ℤ) |
174 | 173 | adantl 275 |
. . . . . . . . . . . 12
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → 𝑘 ∈ ℤ) |
175 | 166, 171,
174 | exprecapd 10617 |
. . . . . . . . . . 11
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → ((1 / 2)↑𝑘) = (1 / (2↑𝑘))) |
176 | 175 | oveq2d 5869 |
. . . . . . . . . 10
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · ((1 / 2)↑𝑘)) = (2 · (1 / (2↑𝑘)))) |
177 | 169, 172,
176 | 3eqtr4rd 2214 |
. . . . . . . . 9
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (2 · ((1 / 2)↑𝑘)) = (1 / ((2↑𝑘) / 2))) |
178 | 165, 177 | breqtrrd 4017 |
. . . . . . . 8
⊢
((⊤ ∧ 𝑘
∈ ℕ0) → (1 / (!‘𝑘)) ≤ (2 · ((1 / 2)↑𝑘))) |
179 | 135, 178 | sylan2 284 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (1 / (!‘𝑘)) ≤ (2 · ((1 / 2)↑𝑘))) |
180 | 135, 60 | sylan2 284 |
. . . . . . 7
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) = (1 / (!‘𝑘))) |
181 | 179, 180,
144 | 3brtr4d 4021 |
. . . . . 6
⊢
((⊤ ∧ 𝑘
∈ ℕ) → (𝐺‘𝑘) ≤ (𝐹‘𝑘)) |
182 | 75, 76, 85, 150, 151, 155, 181 | iserle 11305 |
. . . . 5
⊢ (⊤
→ (e − 1) ≤ 2) |
183 | 182 | mptru 1357 |
. . . 4
⊢ (e
− 1) ≤ 2 |
184 | | ere 11633 |
. . . . 5
⊢ e ∈
ℝ |
185 | 184, 66, 102 | lesubaddi 8425 |
. . . 4
⊢ ((e
− 1) ≤ 2 ↔ e ≤ (2 + 1)) |
186 | 183, 185 | mpbi 144 |
. . 3
⊢ e ≤ (2
+ 1) |
187 | | df-3 8938 |
. . 3
⊢ 3 = (2 +
1) |
188 | 186, 187 | breqtrri 4016 |
. 2
⊢ e ≤
3 |
189 | 74, 188 | pm3.2i 270 |
1
⊢ (2 ≤ e
∧ e ≤ 3) |