Step | Hyp | Ref
| Expression |
1 | | trilpolemeq1.a |
. . . . 5
⊢ (𝜑 → 𝐴 = 1) |
2 | 1 | ad2antrr 488 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 = 1) |
3 | | trilpolemgt1.f |
. . . . . . . 8
⊢ (𝜑 → 𝐹:ℕ⟶{0, 1}) |
4 | | trilpolemgt1.a |
. . . . . . . 8
⊢ 𝐴 = Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) |
5 | 3, 4 | trilpolemcl 14441 |
. . . . . . 7
⊢ (𝜑 → 𝐴 ∈ ℝ) |
6 | 5 | ad2antrr 488 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 ∈ ℝ) |
7 | | nnuz 9552 |
. . . . . . . . . . . . . 14
⊢ ℕ =
(ℤ≥‘1) |
8 | | eqid 2177 |
. . . . . . . . . . . . . 14
⊢
(ℤ≥‘(𝑥 + 1)) = (ℤ≥‘(𝑥 + 1)) |
9 | | simplr 528 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ ℕ) |
10 | 9 | peano2nnd 8923 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝑥 + 1) ∈ ℕ) |
11 | | eqid 2177 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ ↦ ((1 /
(2↑𝑛)) · (𝐹‘𝑛))) = (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛))) |
12 | | oveq2 5877 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 𝑖 → (2↑𝑛) = (2↑𝑖)) |
13 | 12 | oveq2d 5885 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑖 → (1 / (2↑𝑛)) = (1 / (2↑𝑖))) |
14 | | fveq2 5511 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑖 → (𝐹‘𝑛) = (𝐹‘𝑖)) |
15 | 13, 14 | oveq12d 5887 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑖 → ((1 / (2↑𝑛)) · (𝐹‘𝑛)) = ((1 / (2↑𝑖)) · (𝐹‘𝑖))) |
16 | | simpr 110 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℕ) |
17 | | 2rp 9645 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 ∈
ℝ+ |
18 | 17 | a1i 9 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 2 ∈
ℝ+) |
19 | 16 | nnzd 9363 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 𝑖 ∈ ℤ) |
20 | 18, 19 | rpexpcld 10663 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (2↑𝑖) ∈
ℝ+) |
21 | 20 | rpreccld 9694 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈
ℝ+) |
22 | 21 | rpred 9683 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈
ℝ) |
23 | | 0re 7948 |
. . . . . . . . . . . . . . . . . 18
⊢ 0 ∈
ℝ |
24 | | 1re 7947 |
. . . . . . . . . . . . . . . . . 18
⊢ 1 ∈
ℝ |
25 | | prssi 3749 |
. . . . . . . . . . . . . . . . . 18
⊢ ((0
∈ ℝ ∧ 1 ∈ ℝ) → {0, 1} ⊆
ℝ) |
26 | 23, 24, 25 | mp2an 426 |
. . . . . . . . . . . . . . . . 17
⊢ {0, 1}
⊆ ℝ |
27 | 3 | ad3antrrr 492 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → 𝐹:ℕ⟶{0, 1}) |
28 | 27, 16 | ffvelcdmd 5648 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ {0, 1}) |
29 | 26, 28 | sselid 3153 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (𝐹‘𝑖) ∈ ℝ) |
30 | 22, 29 | remulcld 7978 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ) |
31 | 11, 15, 16, 30 | fvmptd3 5605 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖))) |
32 | 30 | recnd 7976 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℂ) |
33 | 3, 11 | trilpolemclim 14440 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → seq1( + , (𝑛 ∈ ℕ ↦ ((1 /
(2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ) |
34 | 33 | ad2antrr 488 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ) |
35 | 7, 8, 10, 31, 32, 34 | isumsplit 11483 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)))) |
36 | 9 | nncnd 8922 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ ℂ) |
37 | | 1cnd 7964 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 1 ∈
ℂ) |
38 | 36, 37 | pncand 8259 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((𝑥 + 1) − 1) = 𝑥) |
39 | 38 | oveq2d 5885 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1...((𝑥 + 1) − 1)) = (1...𝑥)) |
40 | 9, 7 | eleqtrdi 2270 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈
(ℤ≥‘1)) |
41 | | fzisfzounsn 10222 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 ∈
(ℤ≥‘1) → (1...𝑥) = ((1..^𝑥) ∪ {𝑥})) |
42 | 40, 41 | syl 14 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1...𝑥) = ((1..^𝑥) ∪ {𝑥})) |
43 | 39, 42 | eqtrd 2210 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1...((𝑥 + 1) − 1)) = ((1..^𝑥) ∪ {𝑥})) |
44 | 43 | sumeq1d 11358 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) = Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐹‘𝑖))) |
45 | | nfv 1528 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑖((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) |
46 | | nfcv 2319 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑖((1 /
(2↑𝑥)) · (𝐹‘𝑥)) |
47 | | 1zzd 9269 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 1 ∈
ℤ) |
48 | 9 | nnzd 9363 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝑥 ∈ ℤ) |
49 | | fzofig 10418 |
. . . . . . . . . . . . . . . . 17
⊢ ((1
∈ ℤ ∧ 𝑥
∈ ℤ) → (1..^𝑥) ∈ Fin) |
50 | 47, 48, 49 | syl2anc 411 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1..^𝑥) ∈ Fin) |
51 | | fzonel 10146 |
. . . . . . . . . . . . . . . . 17
⊢ ¬
𝑥 ∈ (1..^𝑥) |
52 | 51 | a1i 9 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ¬ 𝑥 ∈ (1..^𝑥)) |
53 | 17 | a1i 9 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 2 ∈
ℝ+) |
54 | | elfzoelz 10133 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ ℤ) |
55 | 54 | adantl 277 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 𝑖 ∈ ℤ) |
56 | 53, 55 | rpexpcld 10663 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (2↑𝑖) ∈
ℝ+) |
57 | 56 | rpreccld 9694 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (1 / (2↑𝑖)) ∈
ℝ+) |
58 | 57 | rpred 9683 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (1 / (2↑𝑖)) ∈ ℝ) |
59 | | elfzouz 10137 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈
(ℤ≥‘1)) |
60 | 59, 7 | eleqtrrdi 2271 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑖 ∈ (1..^𝑥) → 𝑖 ∈ ℕ) |
61 | 60, 29 | sylan2 286 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (𝐹‘𝑖) ∈ ℝ) |
62 | 58, 61 | remulcld 7978 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ) |
63 | 62 | recnd 7976 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℂ) |
64 | | oveq2 5877 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑖 = 𝑥 → (2↑𝑖) = (2↑𝑥)) |
65 | 64 | oveq2d 5885 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑥 → (1 / (2↑𝑖)) = (1 / (2↑𝑥))) |
66 | | fveq2 5511 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑖 = 𝑥 → (𝐹‘𝑖) = (𝐹‘𝑥)) |
67 | 65, 66 | oveq12d 5887 |
. . . . . . . . . . . . . . . 16
⊢ (𝑖 = 𝑥 → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑥)) · (𝐹‘𝑥))) |
68 | 17 | a1i 9 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 2 ∈
ℝ+) |
69 | 68, 48 | rpexpcld 10663 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (2↑𝑥) ∈
ℝ+) |
70 | 69 | rpreccld 9694 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1 / (2↑𝑥)) ∈
ℝ+) |
71 | 70 | rpred 9683 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1 / (2↑𝑥)) ∈ ℝ) |
72 | 3 | ad2antrr 488 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐹:ℕ⟶{0, 1}) |
73 | 72, 9 | ffvelcdmd 5648 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝐹‘𝑥) ∈ {0, 1}) |
74 | 26, 73 | sselid 3153 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝐹‘𝑥) ∈ ℝ) |
75 | 71, 74 | remulcld 7978 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) ∈ ℝ) |
76 | 75 | recnd 7976 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) ∈ ℂ) |
77 | 45, 46, 50, 9, 52, 63, 67, 76 | fsumsplitsn 11402 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + ((1 / (2↑𝑥)) · (𝐹‘𝑥)))) |
78 | | simpr 110 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝐹‘𝑥) = 0) |
79 | 78 | oveq2d 5885 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) = ((1 / (2↑𝑥)) · 0)) |
80 | 70 | rpcnd 9685 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (1 / (2↑𝑥)) ∈ ℂ) |
81 | 80 | mul01d 8340 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · 0) =
0) |
82 | 79, 81 | eqtrd 2210 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((1 / (2↑𝑥)) · (𝐹‘𝑥)) = 0) |
83 | 82 | oveq2d 5885 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + ((1 / (2↑𝑥)) · (𝐹‘𝑥))) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0)) |
84 | 44, 77, 83 | 3eqtrd 2214 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0)) |
85 | 84 | oveq1d 5884 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))) = ((Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)))) |
86 | 35, 85 | eqtrd 2210 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)))) |
87 | 50, 62 | fsumrecl 11393 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ) |
88 | | 0red 7949 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 0 ∈
ℝ) |
89 | 87, 88 | readdcld 7977 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) ∈ ℝ) |
90 | 10 | nnzd 9363 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (𝑥 + 1) ∈ ℤ) |
91 | | eluznn 9589 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑥 + 1) ∈ ℕ ∧ 𝑖 ∈
(ℤ≥‘(𝑥 + 1))) → 𝑖 ∈ ℕ) |
92 | 10, 91 | sylan 283 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ≥‘(𝑥 + 1))) → 𝑖 ∈
ℕ) |
93 | 92, 30 | syldan 282 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ≥‘(𝑥 + 1))) → ((1 /
(2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ) |
94 | 11, 15, 92, 93 | fvmptd3 5605 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ≥‘(𝑥 + 1))) → ((𝑛 ∈ ℕ ↦ ((1 /
(2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) = ((1 / (2↑𝑖)) · (𝐹‘𝑖))) |
95 | 31, 32 | eqeltrd 2254 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))‘𝑖) ∈ ℂ) |
96 | 7, 10, 95 | iserex 11331 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (seq1( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ↔ seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ )) |
97 | 34, 96 | mpbid 147 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ ((1 / (2↑𝑛)) · (𝐹‘𝑛)))) ∈ dom ⇝ ) |
98 | 8, 90, 94, 93, 97 | isumrecl 11421 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) ∈ ℝ) |
99 | 50, 58 | fsumrecl 11393 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) ∈ ℝ) |
100 | 99, 71 | readdcld 7977 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) ∈ ℝ) |
101 | | eqid 2177 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ ℕ ↦ (1 /
(2↑𝑛))) = (𝑛 ∈ ℕ ↦ (1 /
(2↑𝑛))) |
102 | 92, 21 | syldan 282 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ≥‘(𝑥 + 1))) → (1 /
(2↑𝑖)) ∈
ℝ+) |
103 | 101, 13, 92, 102 | fvmptd3 5605 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ≥‘(𝑥 + 1))) → ((𝑛 ∈ ℕ ↦ (1 /
(2↑𝑛)))‘𝑖) = (1 / (2↑𝑖))) |
104 | 92, 22 | syldan 282 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (ℤ≥‘(𝑥 + 1))) → (1 /
(2↑𝑖)) ∈
ℝ) |
105 | | seqex 10433 |
. . . . . . . . . . . . . . . . 17
⊢ seq1( + ,
(𝑛 ∈ ℕ ↦
(1 / (2↑𝑛)))) ∈
V |
106 | | ax-1cn 7895 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ |
107 | 101 | geo2lim 11508 |
. . . . . . . . . . . . . . . . . 18
⊢ (1 ∈
ℂ → seq1( + , (𝑛
∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ 1) |
108 | 106, 107 | ax-mp 5 |
. . . . . . . . . . . . . . . . 17
⊢ seq1( + ,
(𝑛 ∈ ℕ ↦
(1 / (2↑𝑛)))) ⇝
1 |
109 | | breldmg 4829 |
. . . . . . . . . . . . . . . . 17
⊢ ((seq1( +
, (𝑛 ∈ ℕ ↦
(1 / (2↑𝑛)))) ∈ V
∧ 1 ∈ ℂ ∧ seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ⇝ 1) → seq1( + ,
(𝑛 ∈ ℕ ↦
(1 / (2↑𝑛)))) ∈
dom ⇝ ) |
110 | 105, 106,
108, 109 | mp3an 1337 |
. . . . . . . . . . . . . . . 16
⊢ seq1( + ,
(𝑛 ∈ ℕ ↦
(1 / (2↑𝑛)))) ∈
dom ⇝ |
111 | 110 | a1i 9 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝
) |
112 | 101, 13, 16, 21 | fvmptd3 5605 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) = (1 / (2↑𝑖))) |
113 | 21 | rpcnd 9685 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → (1 / (2↑𝑖)) ∈
ℂ) |
114 | 112, 113 | eqeltrd 2254 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ ℕ) → ((𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))‘𝑖) ∈ ℂ) |
115 | 7, 10, 114 | iserex 11331 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (seq1( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝ ↔
seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ (1 /
(2↑𝑛)))) ∈ dom
⇝ )) |
116 | 111, 115 | mpbid 147 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → seq(𝑥 + 1)( + , (𝑛 ∈ ℕ ↦ (1 / (2↑𝑛)))) ∈ dom ⇝
) |
117 | 8, 90, 103, 104, 116 | isumrecl 11421 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)) ∈
ℝ) |
118 | | simpr 110 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → (𝐹‘𝑖) = 0) |
119 | 118 | oveq2d 5885 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 0)) |
120 | 57 | adantr 276 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈
ℝ+) |
121 | 120 | rpcnd 9685 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → (1 / (2↑𝑖)) ∈ ℂ) |
122 | 121 | mul01d 8340 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · 0) =
0) |
123 | 119, 122 | eqtrd 2210 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = 0) |
124 | 120 | rpge0d 9687 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → 0 ≤ (1 / (2↑𝑖))) |
125 | 123, 124 | eqbrtrd 4022 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 0) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖))) |
126 | | simpr 110 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (𝐹‘𝑖) = 1) |
127 | 126 | oveq2d 5885 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = ((1 / (2↑𝑖)) · 1)) |
128 | 58 | adantr 276 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℝ) |
129 | 128 | recnd 7976 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ∈ ℂ) |
130 | 129 | mulid1d 7965 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · 1) = (1 /
(2↑𝑖))) |
131 | 127, 130 | eqtrd 2210 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) = (1 / (2↑𝑖))) |
132 | 128 | leidd 8461 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → (1 / (2↑𝑖)) ≤ (1 / (2↑𝑖))) |
133 | 131, 132 | eqbrtrd 4022 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) ∧ (𝐹‘𝑖) = 1) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖))) |
134 | 72 | adantr 276 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 𝐹:ℕ⟶{0, 1}) |
135 | 60 | adantl 277 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → 𝑖 ∈ ℕ) |
136 | 134, 135 | ffvelcdmd 5648 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (𝐹‘𝑖) ∈ {0, 1}) |
137 | | elpri 3614 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐹‘𝑖) ∈ {0, 1} → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1)) |
138 | 136, 137 | syl 14 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((𝐹‘𝑖) = 0 ∨ (𝐹‘𝑖) = 1)) |
139 | 125, 133,
138 | mpjaodan 798 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → ((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ (1 / (2↑𝑖))) |
140 | 50, 62, 58, 139 | fsumle 11455 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖))) |
141 | 70 | rpgt0d 9686 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 0 < (1 / (2↑𝑥))) |
142 | 87, 88, 99, 71, 140, 141 | leltaddd 8513 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) < (Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥)))) |
143 | 72, 4, 8, 10 | trilpolemisumle 14442 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖)) ≤ Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖))) |
144 | 89, 98, 100, 117, 142, 143 | ltleaddd 8512 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ((Σ𝑖 ∈ (1..^𝑥)((1 / (2↑𝑖)) · (𝐹‘𝑖)) + 0) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))((1 / (2↑𝑖)) · (𝐹‘𝑖))) < ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
145 | 86, 144 | eqbrtrd 4022 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ ((1 / (2↑𝑖)) · (𝐹‘𝑖)) < ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
146 | 4, 145 | eqbrtrid 4035 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
147 | | nfcv 2319 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑖(1 /
(2↑𝑥)) |
148 | 57 | rpcnd 9685 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) ∧ 𝑖 ∈ (1..^𝑥)) → (1 / (2↑𝑖)) ∈ ℂ) |
149 | 45, 147, 50, 9, 52, 148, 65, 80 | fsumsplitsn 11402 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) = (Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥)))) |
150 | 149 | oveq1d 5884 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖))) = ((Σ𝑖 ∈ (1..^𝑥)(1 / (2↑𝑖)) + (1 / (2↑𝑥))) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
151 | 146, 150 | breqtrrd 4028 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < (Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
152 | 42 | sumeq1d 11358 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) = Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖))) |
153 | 152 | oveq1d 5884 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖))) = (Σ𝑖 ∈ ((1..^𝑥) ∪ {𝑥})(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
154 | 151, 153 | breqtrrd 4028 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
155 | 7, 8, 10, 112, 113, 111 | isumsplit 11483 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) = (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
156 | 39 | sumeq1d 11358 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ (1...((𝑥 + 1) − 1))(1 / (2↑𝑖)) = Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖))) |
157 | 156 | oveq1d 5884 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → (Σ𝑖 ∈ (1...((𝑥 + 1) − 1))(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖))) = (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
158 | 155, 157 | eqtrd 2210 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → Σ𝑖 ∈ ℕ (1 / (2↑𝑖)) = (Σ𝑖 ∈ (1...𝑥)(1 / (2↑𝑖)) + Σ𝑖 ∈ (ℤ≥‘(𝑥 + 1))(1 / (2↑𝑖)))) |
159 | 154, 158 | breqtrrd 4028 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < Σ𝑖 ∈ ℕ (1 / (2↑𝑖))) |
160 | | geoihalfsum 11514 |
. . . . . . 7
⊢
Σ𝑖 ∈
ℕ (1 / (2↑𝑖)) =
1 |
161 | 159, 160 | breqtrdi 4041 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 < 1) |
162 | 6, 161 | ltned 8061 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → 𝐴 ≠ 1) |
163 | 162 | neneqd 2368 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ ℕ) ∧ (𝐹‘𝑥) = 0) → ¬ 𝐴 = 1) |
164 | 2, 163 | pm2.65da 661 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ¬ (𝐹‘𝑥) = 0) |
165 | 3 | ffvelcdmda 5647 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) ∈ {0, 1}) |
166 | | elpri 3614 |
. . . . 5
⊢ ((𝐹‘𝑥) ∈ {0, 1} → ((𝐹‘𝑥) = 0 ∨ (𝐹‘𝑥) = 1)) |
167 | 165, 166 | syl 14 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝐹‘𝑥) = 0 ∨ (𝐹‘𝑥) = 1)) |
168 | 167 | orcomd 729 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → ((𝐹‘𝑥) = 1 ∨ (𝐹‘𝑥) = 0)) |
169 | 164, 168 | ecased 1349 |
. 2
⊢ ((𝜑 ∧ 𝑥 ∈ ℕ) → (𝐹‘𝑥) = 1) |
170 | 169 | ralrimiva 2550 |
1
⊢ (𝜑 → ∀𝑥 ∈ ℕ (𝐹‘𝑥) = 1) |