Step | Hyp | Ref
| Expression |
1 | | oveq1 6657 |
. . . . . . . . . . 11
⊢ (𝑥 = 0 → (𝑥 + 1) = (0 + 1)) |
2 | | 0p1e1 11132 |
. . . . . . . . . . 11
⊢ (0 + 1) =
1 |
3 | 1, 2 | syl6eq 2672 |
. . . . . . . . . 10
⊢ (𝑥 = 0 → (𝑥 + 1) = 1) |
4 | 3 | oveq2d 6666 |
. . . . . . . . 9
⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) =
(2↑1)) |
5 | | 2cn 11091 |
. . . . . . . . . . 11
⊢ 2 ∈
ℂ |
6 | | exp1 12866 |
. . . . . . . . . . 11
⊢ (2 ∈
ℂ → (2↑1) = 2) |
7 | 5, 6 | ax-mp 5 |
. . . . . . . . . 10
⊢
(2↑1) = 2 |
8 | | df-2 11079 |
. . . . . . . . . 10
⊢ 2 = (1 +
1) |
9 | 7, 8 | eqtri 2644 |
. . . . . . . . 9
⊢
(2↑1) = (1 + 1) |
10 | 4, 9 | syl6eq 2672 |
. . . . . . . 8
⊢ (𝑥 = 0 → (2↑(𝑥 + 1)) = (1 +
1)) |
11 | 10 | oveq1d 6665 |
. . . . . . 7
⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) = ((1 + 1)
− 1)) |
12 | | ax-1cn 9994 |
. . . . . . . 8
⊢ 1 ∈
ℂ |
13 | 12, 12 | pncan3oi 10297 |
. . . . . . 7
⊢ ((1 + 1)
− 1) = 1 |
14 | 11, 13 | syl6eq 2672 |
. . . . . 6
⊢ (𝑥 = 0 → ((2↑(𝑥 + 1)) − 1) =
1) |
15 | 14 | fveq2d 6195 |
. . . . 5
⊢ (𝑥 = 0 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + ,
𝐹)‘1)) |
16 | | fveq2 6191 |
. . . . 5
⊢ (𝑥 = 0 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘0)) |
17 | 15, 16 | breq12d 4666 |
. . . 4
⊢ (𝑥 = 0 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0))) |
18 | 17 | imbi2d 330 |
. . 3
⊢ (𝑥 = 0 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0)))) |
19 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑥 = 𝑗 → (𝑥 + 1) = (𝑗 + 1)) |
20 | 19 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = 𝑗 → (2↑(𝑥 + 1)) = (2↑(𝑗 + 1))) |
21 | 20 | oveq1d 6665 |
. . . . . 6
⊢ (𝑥 = 𝑗 → ((2↑(𝑥 + 1)) − 1) = ((2↑(𝑗 + 1)) −
1)) |
22 | 21 | fveq2d 6195 |
. . . . 5
⊢ (𝑥 = 𝑗 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) −
1))) |
23 | | fveq2 6191 |
. . . . 5
⊢ (𝑥 = 𝑗 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑗)) |
24 | 22, 23 | breq12d 4666 |
. . . 4
⊢ (𝑥 = 𝑗 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗))) |
25 | 24 | imbi2d 330 |
. . 3
⊢ (𝑥 = 𝑗 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑗)))) |
26 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑥 = (𝑗 + 1) → (𝑥 + 1) = ((𝑗 + 1) + 1)) |
27 | 26 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = (𝑗 + 1) → (2↑(𝑥 + 1)) = (2↑((𝑗 + 1) + 1))) |
28 | 27 | oveq1d 6665 |
. . . . . 6
⊢ (𝑥 = (𝑗 + 1) → ((2↑(𝑥 + 1)) − 1) = ((2↑((𝑗 + 1) + 1)) −
1)) |
29 | 28 | fveq2d 6195 |
. . . . 5
⊢ (𝑥 = (𝑗 + 1) → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) −
1))) |
30 | | fveq2 6191 |
. . . . 5
⊢ (𝑥 = (𝑗 + 1) → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘(𝑗 + 1))) |
31 | 29, 30 | breq12d 4666 |
. . . 4
⊢ (𝑥 = (𝑗 + 1) → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1)))) |
32 | 31 | imbi2d 330 |
. . 3
⊢ (𝑥 = (𝑗 + 1) → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1))))) |
33 | | oveq1 6657 |
. . . . . . . 8
⊢ (𝑥 = 𝑁 → (𝑥 + 1) = (𝑁 + 1)) |
34 | 33 | oveq2d 6666 |
. . . . . . 7
⊢ (𝑥 = 𝑁 → (2↑(𝑥 + 1)) = (2↑(𝑁 + 1))) |
35 | 34 | oveq1d 6665 |
. . . . . 6
⊢ (𝑥 = 𝑁 → ((2↑(𝑥 + 1)) − 1) = ((2↑(𝑁 + 1)) −
1)) |
36 | 35 | fveq2d 6195 |
. . . . 5
⊢ (𝑥 = 𝑁 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) = (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) −
1))) |
37 | | fveq2 6191 |
. . . . 5
⊢ (𝑥 = 𝑁 → (seq0( + , 𝐺)‘𝑥) = (seq0( + , 𝐺)‘𝑁)) |
38 | 36, 37 | breq12d 4666 |
. . . 4
⊢ (𝑥 = 𝑁 → ((seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥) ↔ (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁))) |
39 | 38 | imbi2d 330 |
. . 3
⊢ (𝑥 = 𝑁 → ((𝜑 → (seq1( + , 𝐹)‘((2↑(𝑥 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑥)) ↔ (𝜑 → (seq1( + , 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( + , 𝐺)‘𝑁)))) |
40 | | 1nn 11031 |
. . . . . . 7
⊢ 1 ∈
ℕ |
41 | | climcnds.1 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℝ) |
42 | 41 | ralrimiva 2966 |
. . . . . . 7
⊢ (𝜑 → ∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ) |
43 | | fveq2 6191 |
. . . . . . . . 9
⊢ (𝑘 = 1 → (𝐹‘𝑘) = (𝐹‘1)) |
44 | 43 | eleq1d 2686 |
. . . . . . . 8
⊢ (𝑘 = 1 → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘1) ∈ ℝ)) |
45 | 44 | rspcv 3305 |
. . . . . . 7
⊢ (1 ∈
ℕ → (∀𝑘
∈ ℕ (𝐹‘𝑘) ∈ ℝ → (𝐹‘1) ∈ ℝ)) |
46 | 40, 42, 45 | mpsyl 68 |
. . . . . 6
⊢ (𝜑 → (𝐹‘1) ∈ ℝ) |
47 | 46 | leidd 10594 |
. . . . 5
⊢ (𝜑 → (𝐹‘1) ≤ (𝐹‘1)) |
48 | 46 | recnd 10068 |
. . . . . 6
⊢ (𝜑 → (𝐹‘1) ∈ ℂ) |
49 | 48 | mulid2d 10058 |
. . . . 5
⊢ (𝜑 → (1 · (𝐹‘1)) = (𝐹‘1)) |
50 | 47, 49 | breqtrrd 4681 |
. . . 4
⊢ (𝜑 → (𝐹‘1) ≤ (1 · (𝐹‘1))) |
51 | | 1z 11407 |
. . . . 5
⊢ 1 ∈
ℤ |
52 | | eqidd 2623 |
. . . . 5
⊢ (𝜑 → (𝐹‘1) = (𝐹‘1)) |
53 | 51, 52 | seq1i 12815 |
. . . 4
⊢ (𝜑 → (seq1( + , 𝐹)‘1) = (𝐹‘1)) |
54 | | 0z 11388 |
. . . . 5
⊢ 0 ∈
ℤ |
55 | | 0nn0 11307 |
. . . . . 6
⊢ 0 ∈
ℕ0 |
56 | | climcnds.4 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛)))) |
57 | 56 | ralrimiva 2966 |
. . . . . 6
⊢ (𝜑 → ∀𝑛 ∈ ℕ0 (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛)))) |
58 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑛 = 0 → (𝐺‘𝑛) = (𝐺‘0)) |
59 | | oveq2 6658 |
. . . . . . . . . 10
⊢ (𝑛 = 0 → (2↑𝑛) = (2↑0)) |
60 | | exp0 12864 |
. . . . . . . . . . 11
⊢ (2 ∈
ℂ → (2↑0) = 1) |
61 | 5, 60 | ax-mp 5 |
. . . . . . . . . 10
⊢
(2↑0) = 1 |
62 | 59, 61 | syl6eq 2672 |
. . . . . . . . 9
⊢ (𝑛 = 0 → (2↑𝑛) = 1) |
63 | 62 | fveq2d 6195 |
. . . . . . . . 9
⊢ (𝑛 = 0 → (𝐹‘(2↑𝑛)) = (𝐹‘1)) |
64 | 62, 63 | oveq12d 6668 |
. . . . . . . 8
⊢ (𝑛 = 0 → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = (1 · (𝐹‘1))) |
65 | 58, 64 | eqeq12d 2637 |
. . . . . . 7
⊢ (𝑛 = 0 → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘0) = (1 · (𝐹‘1)))) |
66 | 65 | rspcv 3305 |
. . . . . 6
⊢ (0 ∈
ℕ0 → (∀𝑛 ∈ ℕ0 (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) → (𝐺‘0) = (1 · (𝐹‘1)))) |
67 | 55, 57, 66 | mpsyl 68 |
. . . . 5
⊢ (𝜑 → (𝐺‘0) = (1 · (𝐹‘1))) |
68 | 54, 67 | seq1i 12815 |
. . . 4
⊢ (𝜑 → (seq0( + , 𝐺)‘0) = (1 · (𝐹‘1))) |
69 | 50, 53, 68 | 3brtr4d 4685 |
. . 3
⊢ (𝜑 → (seq1( + , 𝐹)‘1) ≤ (seq0( + , 𝐺)‘0)) |
70 | | fzfid 12772 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1)) ∈ Fin) |
71 | | simpl 473 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝜑) |
72 | 71 | adantr 481 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) →
𝜑) |
73 | | 2nn 11185 |
. . . . . . . . . . . 12
⊢ 2 ∈
ℕ |
74 | | peano2nn0 11333 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
ℕ0) |
75 | 74 | adantl 482 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝑗 + 1) ∈
ℕ0) |
76 | | nnexpcl 12873 |
. . . . . . . . . . . 12
⊢ ((2
∈ ℕ ∧ (𝑗 +
1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℕ) |
77 | 73, 75, 76 | sylancr 695 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ∈
ℕ) |
78 | | elfzuz 12338 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) →
𝑘 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) |
79 | | eluznn 11758 |
. . . . . . . . . . 11
⊢
(((2↑(𝑗 + 1))
∈ ℕ ∧ 𝑘
∈ (ℤ≥‘(2↑(𝑗 + 1)))) → 𝑘 ∈ ℕ) |
80 | 77, 78, 79 | syl2an 494 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) →
𝑘 ∈
ℕ) |
81 | 72, 80, 41 | syl2anc 693 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) →
(𝐹‘𝑘) ∈ ℝ) |
82 | 42 | adantr 481 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
∀𝑘 ∈ ℕ
(𝐹‘𝑘) ∈ ℝ) |
83 | | fveq2 6191 |
. . . . . . . . . . . . 13
⊢ (𝑘 = (2↑(𝑗 + 1)) → (𝐹‘𝑘) = (𝐹‘(2↑(𝑗 + 1)))) |
84 | 83 | eleq1d 2686 |
. . . . . . . . . . . 12
⊢ (𝑘 = (2↑(𝑗 + 1)) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)) |
85 | 84 | rspcv 3305 |
. . . . . . . . . . 11
⊢
((2↑(𝑗 + 1))
∈ ℕ → (∀𝑘 ∈ ℕ (𝐹‘𝑘) ∈ ℝ → (𝐹‘(2↑(𝑗 + 1))) ∈ ℝ)) |
86 | 77, 82, 85 | sylc 65 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘(2↑(𝑗 + 1))) ∈
ℝ) |
87 | 86 | adantr 481 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) →
(𝐹‘(2↑(𝑗 + 1))) ∈
ℝ) |
88 | | simpr 477 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) → 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) |
89 | | simplll 798 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝜑) |
90 | 77 | adantr 481 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) → (2↑(𝑗 + 1)) ∈ ℕ) |
91 | | elfzuz 12338 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...𝑛) → 𝑘 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) |
92 | 90, 91, 79 | syl2an 494 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → 𝑘 ∈ ℕ) |
93 | 89, 92, 41 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...𝑛)) → (𝐹‘𝑘) ∈ ℝ) |
94 | | simplll 798 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝜑) |
95 | | elfzuz 12338 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1)) → 𝑘 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) |
96 | 90, 95, 79 | syl2an 494 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → 𝑘 ∈ ℕ) |
97 | | climcnds.3 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
98 | 94, 96, 97 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...(𝑛 − 1))) → (𝐹‘(𝑘 + 1)) ≤ (𝐹‘𝑘)) |
99 | 88, 93, 98 | monoord2 12832 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1)))) |
100 | 99 | ralrimiva 2966 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
∀𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1)))) |
101 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
102 | 101 | breq1d 4663 |
. . . . . . . . . . 11
⊢ (𝑛 = 𝑘 → ((𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ↔ (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1))))) |
103 | 102 | rspccva 3308 |
. . . . . . . . . 10
⊢
((∀𝑛 ∈
(ℤ≥‘(2↑(𝑗 + 1)))(𝐹‘𝑛) ≤ (𝐹‘(2↑(𝑗 + 1))) ∧ 𝑘 ∈
(ℤ≥‘(2↑(𝑗 + 1)))) → (𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1)))) |
104 | 100, 78, 103 | syl2an 494 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) →
(𝐹‘𝑘) ≤ (𝐹‘(2↑(𝑗 + 1)))) |
105 | 70, 81, 87, 104 | fsumle 14531 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘𝑘) ≤ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1)))) |
106 | | fzfid 12772 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(1...((2↑(𝑗 + 1))
− 1)) ∈ Fin) |
107 | | hashcl 13147 |
. . . . . . . . . . . . 13
⊢
((1...((2↑(𝑗 +
1)) − 1)) ∈ Fin → (#‘(1...((2↑(𝑗 + 1)) − 1))) ∈
ℕ0) |
108 | 106, 107 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘(1...((2↑(𝑗 +
1)) − 1))) ∈ ℕ0) |
109 | 108 | nn0cnd 11353 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘(1...((2↑(𝑗 +
1)) − 1))) ∈ ℂ) |
110 | 77 | nnred 11035 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ∈
ℝ) |
111 | 110 | recnd 10068 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ∈
ℂ) |
112 | | hashcl 13147 |
. . . . . . . . . . . . 13
⊢
(((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1)) ∈ Fin → (#‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) ∈
ℕ0) |
113 | 70, 112 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))) ∈ ℕ0) |
114 | 113 | nn0cnd 11353 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))) ∈ ℂ) |
115 | | 2z 11409 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 2 ∈
ℤ |
116 | | zexpcl 12875 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((2
∈ ℤ ∧ (𝑗 +
1) ∈ ℕ0) → (2↑(𝑗 + 1)) ∈ ℤ) |
117 | 115, 75, 116 | sylancr 695 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ∈
ℤ) |
118 | | nn0p1nn 11332 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑗 ∈ ℕ0
→ (𝑗 + 1) ∈
ℕ) |
119 | 118 | adantl 482 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝑗 + 1) ∈
ℕ) |
120 | | nnuz 11723 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ℕ =
(ℤ≥‘1) |
121 | 119, 120 | syl6eleq 2711 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝑗 + 1) ∈
(ℤ≥‘1)) |
122 | | 2re 11090 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 2 ∈
ℝ |
123 | | 1le2 11241 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ 1 ≤
2 |
124 | | leexp2a 12916 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((2
∈ ℝ ∧ 1 ≤ 2 ∧ (𝑗 + 1) ∈
(ℤ≥‘1)) → (2↑1) ≤ (2↑(𝑗 + 1))) |
125 | 122, 123,
124 | mp3an12 1414 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑗 + 1) ∈
(ℤ≥‘1) → (2↑1) ≤ (2↑(𝑗 + 1))) |
126 | 121, 125 | syl 17 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑1) ≤ (2↑(𝑗
+ 1))) |
127 | 7, 126 | syl5eqbrr 4689 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 2 ≤
(2↑(𝑗 +
1))) |
128 | 115 | eluz1i 11695 |
. . . . . . . . . . . . . . . . . . 19
⊢
((2↑(𝑗 + 1))
∈ (ℤ≥‘2) ↔ ((2↑(𝑗 + 1)) ∈ ℤ ∧ 2 ≤
(2↑(𝑗 +
1)))) |
129 | 117, 127,
128 | sylanbrc 698 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ∈
(ℤ≥‘2)) |
130 | | uz2m1nn 11763 |
. . . . . . . . . . . . . . . . . 18
⊢
((2↑(𝑗 + 1))
∈ (ℤ≥‘2) → ((2↑(𝑗 + 1)) − 1) ∈
ℕ) |
131 | 129, 130 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) ∈ ℕ) |
132 | 131, 120 | syl6eleq 2711 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) ∈ (ℤ≥‘1)) |
133 | | peano2zm 11420 |
. . . . . . . . . . . . . . . . . 18
⊢
((2↑(𝑗 + 1))
∈ ℤ → ((2↑(𝑗 + 1)) − 1) ∈
ℤ) |
134 | 117, 133 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) ∈ ℤ) |
135 | | peano2nn0 11333 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑗 + 1) ∈ ℕ0
→ ((𝑗 + 1) + 1) ∈
ℕ0) |
136 | 75, 135 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((𝑗 + 1) + 1) ∈
ℕ0) |
137 | | zexpcl 12875 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((2
∈ ℤ ∧ ((𝑗 +
1) + 1) ∈ ℕ0) → (2↑((𝑗 + 1) + 1)) ∈ ℤ) |
138 | 115, 136,
137 | sylancr 695 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑((𝑗 + 1) + 1))
∈ ℤ) |
139 | | peano2zm 11420 |
. . . . . . . . . . . . . . . . . 18
⊢
((2↑((𝑗 + 1) +
1)) ∈ ℤ → ((2↑((𝑗 + 1) + 1)) − 1) ∈
ℤ) |
140 | 138, 139 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) ∈ ℤ) |
141 | 117 | zred 11482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ∈
ℝ) |
142 | 138 | zred 11482 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑((𝑗 + 1) + 1))
∈ ℝ) |
143 | | 1red 10055 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 1 ∈
ℝ) |
144 | 75 | nn0zd 11480 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝑗 + 1) ∈
ℤ) |
145 | | uzid 11702 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑗 + 1) ∈ ℤ →
(𝑗 + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
146 | | peano2uz 11741 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑗 + 1) ∈
(ℤ≥‘(𝑗 + 1)) → ((𝑗 + 1) + 1) ∈
(ℤ≥‘(𝑗 + 1))) |
147 | | leexp2a 12916 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((2
∈ ℝ ∧ 1 ≤ 2 ∧ ((𝑗 + 1) + 1) ∈
(ℤ≥‘(𝑗 + 1))) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1))) |
148 | 122, 123,
147 | mp3an12 1414 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑗 + 1) + 1) ∈
(ℤ≥‘(𝑗 + 1)) → (2↑(𝑗 + 1)) ≤ (2↑((𝑗 + 1) + 1))) |
149 | 144, 145,
146, 148 | 4syl 19 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) ≤
(2↑((𝑗 + 1) +
1))) |
150 | 141, 142,
143, 149 | lesub1dd 10643 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) ≤ ((2↑((𝑗 + 1) +
1)) − 1)) |
151 | | eluz2 11693 |
. . . . . . . . . . . . . . . . 17
⊢
(((2↑((𝑗 + 1) +
1)) − 1) ∈ (ℤ≥‘((2↑(𝑗 + 1)) − 1)) ↔
(((2↑(𝑗 + 1)) −
1) ∈ ℤ ∧ ((2↑((𝑗 + 1) + 1)) − 1) ∈ ℤ ∧
((2↑(𝑗 + 1)) −
1) ≤ ((2↑((𝑗 + 1) +
1)) − 1))) |
152 | 134, 140,
150, 151 | syl3anbrc 1246 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) ∈ (ℤ≥‘((2↑(𝑗 + 1)) − 1))) |
153 | | elfzuzb 12336 |
. . . . . . . . . . . . . . . 16
⊢
(((2↑(𝑗 + 1))
− 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) ↔
(((2↑(𝑗 + 1)) −
1) ∈ (ℤ≥‘1) ∧ ((2↑((𝑗 + 1) + 1)) − 1) ∈
(ℤ≥‘((2↑(𝑗 + 1)) − 1)))) |
154 | 132, 152,
153 | sylanbrc 698 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) ∈ (1...((2↑((𝑗
+ 1) + 1)) − 1))) |
155 | | fzsplit 12367 |
. . . . . . . . . . . . . . 15
⊢
(((2↑(𝑗 + 1))
− 1) ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) →
(1...((2↑((𝑗 + 1) +
1)) − 1)) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((((2↑(𝑗 + 1)) − 1) +
1)...((2↑((𝑗 + 1) +
1)) − 1)))) |
156 | 154, 155 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(1...((2↑((𝑗 + 1) +
1)) − 1)) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((((2↑(𝑗 + 1)) − 1) +
1)...((2↑((𝑗 + 1) +
1)) − 1)))) |
157 | | npcan 10290 |
. . . . . . . . . . . . . . . . 17
⊢
(((2↑(𝑗 + 1))
∈ ℂ ∧ 1 ∈ ℂ) → (((2↑(𝑗 + 1)) − 1) + 1) = (2↑(𝑗 + 1))) |
158 | 111, 12, 157 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(((2↑(𝑗 + 1)) −
1) + 1) = (2↑(𝑗 +
1))) |
159 | 158 | oveq1d 6665 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((((2↑(𝑗 + 1)) −
1) + 1)...((2↑((𝑗 + 1)
+ 1)) − 1)) = ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) |
160 | 159 | uneq2d 3767 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((1...((2↑(𝑗 + 1))
− 1)) ∪ ((((2↑(𝑗 + 1)) − 1) + 1)...((2↑((𝑗 + 1) + 1)) − 1))) =
((1...((2↑(𝑗 + 1))
− 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))) |
161 | 156, 160 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(1...((2↑((𝑗 + 1) +
1)) − 1)) = ((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) −
1)))) |
162 | 161 | fveq2d 6195 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘(1...((2↑((𝑗
+ 1) + 1)) − 1))) = (#‘((1...((2↑(𝑗 + 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) −
1))))) |
163 | | expp1 12867 |
. . . . . . . . . . . . . . . . 17
⊢ ((2
∈ ℂ ∧ (𝑗 +
1) ∈ ℕ0) → (2↑((𝑗 + 1) + 1)) = ((2↑(𝑗 + 1)) · 2)) |
164 | 5, 75, 163 | sylancr 695 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑((𝑗 + 1) + 1)) =
((2↑(𝑗 + 1)) ·
2)) |
165 | 111 | times2d 11276 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) ·
2) = ((2↑(𝑗 + 1)) +
(2↑(𝑗 +
1)))) |
166 | 164, 165 | eqtrd 2656 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑((𝑗 + 1) + 1)) =
((2↑(𝑗 + 1)) +
(2↑(𝑗 +
1)))) |
167 | 166 | oveq1d 6665 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) = (((2↑(𝑗 +
1)) + (2↑(𝑗 + 1)))
− 1)) |
168 | | 1cnd 10056 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 1 ∈
ℂ) |
169 | 111, 111,
168 | addsubd 10413 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(((2↑(𝑗 + 1)) +
(2↑(𝑗 + 1))) −
1) = (((2↑(𝑗 + 1))
− 1) + (2↑(𝑗 +
1)))) |
170 | 167, 169 | eqtrd 2656 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) = (((2↑(𝑗 +
1)) − 1) + (2↑(𝑗
+ 1)))) |
171 | | uztrn 11704 |
. . . . . . . . . . . . . . . . 17
⊢
((((2↑((𝑗 + 1)
+ 1)) − 1) ∈ (ℤ≥‘((2↑(𝑗 + 1)) − 1)) ∧
((2↑(𝑗 + 1)) −
1) ∈ (ℤ≥‘1)) → ((2↑((𝑗 + 1) + 1)) − 1) ∈
(ℤ≥‘1)) |
172 | 152, 132,
171 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) ∈ (ℤ≥‘1)) |
173 | 172, 120 | syl6eleqr 2712 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) ∈ ℕ) |
174 | 173 | nnnn0d 11351 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑((𝑗 + 1) + 1))
− 1) ∈ ℕ0) |
175 | | hashfz1 13134 |
. . . . . . . . . . . . . 14
⊢
(((2↑((𝑗 + 1) +
1)) − 1) ∈ ℕ0 → (#‘(1...((2↑((𝑗 + 1) + 1)) − 1))) =
((2↑((𝑗 + 1) + 1))
− 1)) |
176 | 174, 175 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘(1...((2↑((𝑗
+ 1) + 1)) − 1))) = ((2↑((𝑗 + 1) + 1)) − 1)) |
177 | 131 | nnnn0d 11351 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) ∈ ℕ0) |
178 | | hashfz1 13134 |
. . . . . . . . . . . . . . 15
⊢
(((2↑(𝑗 + 1))
− 1) ∈ ℕ0 → (#‘(1...((2↑(𝑗 + 1)) − 1))) =
((2↑(𝑗 + 1)) −
1)) |
179 | 177, 178 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘(1...((2↑(𝑗 +
1)) − 1))) = ((2↑(𝑗 + 1)) − 1)) |
180 | 179 | oveq1d 6665 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((#‘(1...((2↑(𝑗
+ 1)) − 1))) + (2↑(𝑗 + 1))) = (((2↑(𝑗 + 1)) − 1) + (2↑(𝑗 + 1)))) |
181 | 170, 176,
180 | 3eqtr4d 2666 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘(1...((2↑((𝑗
+ 1) + 1)) − 1))) = ((#‘(1...((2↑(𝑗 + 1)) − 1))) + (2↑(𝑗 + 1)))) |
182 | 110 | ltm1d 10956 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) −
1) < (2↑(𝑗 +
1))) |
183 | | fzdisj 12368 |
. . . . . . . . . . . . . 14
⊢
(((2↑(𝑗 + 1))
− 1) < (2↑(𝑗
+ 1)) → ((1...((2↑(𝑗 + 1)) − 1)) ∩ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) =
∅) |
184 | 182, 183 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((1...((2↑(𝑗 + 1))
− 1)) ∩ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) =
∅) |
185 | | hashun 13171 |
. . . . . . . . . . . . 13
⊢
(((1...((2↑(𝑗 +
1)) − 1)) ∈ Fin ∧ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)) ∈ Fin ∧
((1...((2↑(𝑗 + 1))
− 1)) ∩ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))) = ∅) →
(#‘((1...((2↑(𝑗
+ 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))) =
((#‘(1...((2↑(𝑗
+ 1)) − 1))) + (#‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))) |
186 | 106, 70, 184, 185 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(#‘((1...((2↑(𝑗
+ 1)) − 1)) ∪ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))) =
((#‘(1...((2↑(𝑗
+ 1)) − 1))) + (#‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))))) |
187 | 162, 181,
186 | 3eqtr3d 2664 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((#‘(1...((2↑(𝑗
+ 1)) − 1))) + (2↑(𝑗 + 1))) = ((#‘(1...((2↑(𝑗 + 1)) − 1))) +
(#‘((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))))) |
188 | 109, 111,
114, 187 | addcanad 10241 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(2↑(𝑗 + 1)) =
(#‘((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1)))) |
189 | 188 | oveq1d 6665 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) ·
(𝐹‘(2↑(𝑗 + 1)))) =
((#‘((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))) · (𝐹‘(2↑(𝑗 + 1))))) |
190 | 57 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
∀𝑛 ∈
ℕ0 (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛)))) |
191 | | fveq2 6191 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑗 + 1) → (𝐺‘𝑛) = (𝐺‘(𝑗 + 1))) |
192 | | oveq2 6658 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑗 + 1) → (2↑𝑛) = (2↑(𝑗 + 1))) |
193 | 192 | fveq2d 6195 |
. . . . . . . . . . . . 13
⊢ (𝑛 = (𝑗 + 1) → (𝐹‘(2↑𝑛)) = (𝐹‘(2↑(𝑗 + 1)))) |
194 | 192, 193 | oveq12d 6668 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑗 + 1) → ((2↑𝑛) · (𝐹‘(2↑𝑛))) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))) |
195 | 191, 194 | eqeq12d 2637 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑗 + 1) → ((𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) ↔ (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))) |
196 | 195 | rspcv 3305 |
. . . . . . . . . 10
⊢ ((𝑗 + 1) ∈ ℕ0
→ (∀𝑛 ∈
ℕ0 (𝐺‘𝑛) = ((2↑𝑛) · (𝐹‘(2↑𝑛))) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1)))))) |
197 | 75, 190, 196 | sylc 65 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐺‘(𝑗 + 1)) = ((2↑(𝑗 + 1)) · (𝐹‘(2↑(𝑗 + 1))))) |
198 | 86 | recnd 10068 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐹‘(2↑(𝑗 + 1))) ∈
ℂ) |
199 | | fsumconst 14522 |
. . . . . . . . . 10
⊢
((((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1)) ∈ Fin ∧ (𝐹‘(2↑(𝑗 + 1))) ∈ ℂ) → Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1))) =
((#‘((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))) · (𝐹‘(2↑(𝑗 + 1))))) |
200 | 70, 198, 199 | syl2anc 693 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘(2↑(𝑗 + 1))) = ((#‘((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1)))
· (𝐹‘(2↑(𝑗 + 1))))) |
201 | 189, 197,
200 | 3eqtr4d 2666 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐺‘(𝑗 + 1)) = Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘(2↑(𝑗 + 1)))) |
202 | 105, 201 | breqtrrd 4681 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1))) |
203 | | elfznn 12370 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1)) → 𝑘 ∈
ℕ) |
204 | 71, 203, 41 | syl2an 494 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) →
(𝐹‘𝑘) ∈ ℝ) |
205 | 106, 204 | fsumrecl 14465 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘) ∈
ℝ) |
206 | 70, 81 | fsumrecl 14465 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘𝑘) ∈ ℝ) |
207 | | nn0uz 11722 |
. . . . . . . . . 10
⊢
ℕ0 = (ℤ≥‘0) |
208 | | 0zd 11389 |
. . . . . . . . . 10
⊢ (𝜑 → 0 ∈
ℤ) |
209 | | simpr 477 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → 𝑛 ∈
ℕ0) |
210 | | nnexpcl 12873 |
. . . . . . . . . . . . . 14
⊢ ((2
∈ ℕ ∧ 𝑛
∈ ℕ0) → (2↑𝑛) ∈ ℕ) |
211 | 73, 209, 210 | sylancr 695 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
(2↑𝑛) ∈
ℕ) |
212 | 211 | nnred 11035 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
(2↑𝑛) ∈
ℝ) |
213 | 42 | adantr 481 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
∀𝑘 ∈ ℕ
(𝐹‘𝑘) ∈ ℝ) |
214 | | fveq2 6191 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = (2↑𝑛) → (𝐹‘𝑘) = (𝐹‘(2↑𝑛))) |
215 | 214 | eleq1d 2686 |
. . . . . . . . . . . . . 14
⊢ (𝑘 = (2↑𝑛) → ((𝐹‘𝑘) ∈ ℝ ↔ (𝐹‘(2↑𝑛)) ∈ ℝ)) |
216 | 215 | rspcv 3305 |
. . . . . . . . . . . . 13
⊢
((2↑𝑛) ∈
ℕ → (∀𝑘
∈ ℕ (𝐹‘𝑘) ∈ ℝ → (𝐹‘(2↑𝑛)) ∈ ℝ)) |
217 | 211, 213,
216 | sylc 65 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐹‘(2↑𝑛)) ∈
ℝ) |
218 | 212, 217 | remulcld 10070 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) →
((2↑𝑛) · (𝐹‘(2↑𝑛))) ∈
ℝ) |
219 | 56, 218 | eqeltrd 2701 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ ℕ0) → (𝐺‘𝑛) ∈ ℝ) |
220 | 207, 208,
219 | serfre 12830 |
. . . . . . . . 9
⊢ (𝜑 → seq0( + , 𝐺):ℕ0⟶ℝ) |
221 | 220 | ffvelrnda 6359 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (seq0( +
, 𝐺)‘𝑗) ∈
ℝ) |
222 | 141, 86 | remulcld 10070 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((2↑(𝑗 + 1)) ·
(𝐹‘(2↑(𝑗 + 1)))) ∈
ℝ) |
223 | 197, 222 | eqeltrd 2701 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (𝐺‘(𝑗 + 1)) ∈ ℝ) |
224 | | le2add 10510 |
. . . . . . . 8
⊢
(((Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘) ∈ ℝ ∧
Σ𝑘 ∈
((2↑(𝑗 +
1))...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘𝑘) ∈ ℝ) ∧ ((seq0( + , 𝐺)‘𝑗) ∈ ℝ ∧ (𝐺‘(𝑗 + 1)) ∈ ℝ)) → ((Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗) ∧ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1))) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))) |
225 | 205, 206,
221, 223, 224 | syl22anc 1327 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
((Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗) ∧ Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘) ≤ (𝐺‘(𝑗 + 1))) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))) |
226 | 202, 225 | mpan2d 710 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗) → (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))) |
227 | | eqidd 2623 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) →
(𝐹‘𝑘) = (𝐹‘𝑘)) |
228 | 41 | recnd 10068 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → (𝐹‘𝑘) ∈ ℂ) |
229 | 71, 203, 228 | syl2an 494 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))) →
(𝐹‘𝑘) ∈ ℂ) |
230 | 227, 132,
229 | fsumser 14461 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑(𝑗 + 1)) − 1))) |
231 | 230 | eqcomd 2628 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (seq1( +
, 𝐹)‘((2↑(𝑗 + 1)) − 1)) =
Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘)) |
232 | 231 | breq1d 4663 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((seq1( +
, 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑗) ↔ Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) ≤ (seq0( + , 𝐺)‘𝑗))) |
233 | | eqidd 2623 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) →
(𝐹‘𝑘) = (𝐹‘𝑘)) |
234 | | elfznn 12370 |
. . . . . . . . . 10
⊢ (𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1)) →
𝑘 ∈
ℕ) |
235 | 71, 234, 228 | syl2an 494 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑗 ∈ ℕ0) ∧ 𝑘 ∈ (1...((2↑((𝑗 + 1) + 1)) − 1))) →
(𝐹‘𝑘) ∈ ℂ) |
236 | 233, 172,
235 | fsumser 14461 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
(1...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘𝑘) = (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1))) |
237 | | fzfid 12772 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
(1...((2↑((𝑗 + 1) +
1)) − 1)) ∈ Fin) |
238 | 184, 161,
237, 235 | fsumsplit 14471 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) →
Σ𝑘 ∈
(1...((2↑((𝑗 + 1) +
1)) − 1))(𝐹‘𝑘) = (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘))) |
239 | 236, 238 | eqtr3d 2658 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (seq1( +
, 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) =
(Σ𝑘 ∈
(1...((2↑(𝑗 + 1))
− 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘))) |
240 | | simpr 477 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈
ℕ0) |
241 | 240, 207 | syl6eleq 2711 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → 𝑗 ∈
(ℤ≥‘0)) |
242 | | seqp1 12816 |
. . . . . . . 8
⊢ (𝑗 ∈
(ℤ≥‘0) → (seq0( + , 𝐺)‘(𝑗 + 1)) = ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1)))) |
243 | 241, 242 | syl 17 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → (seq0( +
, 𝐺)‘(𝑗 + 1)) = ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1)))) |
244 | 239, 243 | breq12d 4666 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((seq1( +
, 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤
(seq0( + , 𝐺)‘(𝑗 + 1)) ↔ (Σ𝑘 ∈ (1...((2↑(𝑗 + 1)) − 1))(𝐹‘𝑘) + Σ𝑘 ∈ ((2↑(𝑗 + 1))...((2↑((𝑗 + 1) + 1)) − 1))(𝐹‘𝑘)) ≤ ((seq0( + , 𝐺)‘𝑗) + (𝐺‘(𝑗 + 1))))) |
245 | 226, 232,
244 | 3imtr4d 283 |
. . . . 5
⊢ ((𝜑 ∧ 𝑗 ∈ ℕ0) → ((seq1( +
, 𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑗) → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤
(seq0( + , 𝐺)‘(𝑗 + 1)))) |
246 | 245 | expcom 451 |
. . . 4
⊢ (𝑗 ∈ ℕ0
→ (𝜑 → ((seq1( + ,
𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑗) → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤
(seq0( + , 𝐺)‘(𝑗 + 1))))) |
247 | 246 | a2d 29 |
. . 3
⊢ (𝑗 ∈ ℕ0
→ ((𝜑 → (seq1( + ,
𝐹)‘((2↑(𝑗 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑗)) → (𝜑 → (seq1( + , 𝐹)‘((2↑((𝑗 + 1) + 1)) − 1)) ≤ (seq0( + , 𝐺)‘(𝑗 + 1))))) |
248 | 18, 25, 32, 39, 69, 247 | nn0ind 11472 |
. 2
⊢ (𝑁 ∈ ℕ0
→ (𝜑 → (seq1( + ,
𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑁))) |
249 | 248 | impcom 446 |
1
⊢ ((𝜑 ∧ 𝑁 ∈ ℕ0) → (seq1( +
, 𝐹)‘((2↑(𝑁 + 1)) − 1)) ≤ (seq0( +
, 𝐺)‘𝑁)) |