Step | Hyp | Ref
| Expression |
1 | | 2re 11887 |
. . . . 5
⊢ 2 ∈
ℝ |
2 | 1 | a1i 11 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 2 ∈
ℝ) |
3 | | bitsss 15966 |
. . . . 5
⊢
(bits‘(𝐴‘𝑡)) ⊆
ℕ0 |
4 | | simprr 773 |
. . . . 5
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ (bits‘(𝐴‘𝑡))) |
5 | 3, 4 | sseldi 3889 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑛 ∈ ℕ0) |
6 | 2, 5 | reexpcld 13716 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℝ) |
7 | | simprl 771 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℕ) |
8 | 7 | nnred 11828 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ) |
9 | 6, 8 | remulcld 10846 |
. 2
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ∈ ℝ) |
10 | | eulerpartlems.r |
. . . . . . . 8
⊢ 𝑅 = {𝑓 ∣ (◡𝑓 “ ℕ) ∈
Fin} |
11 | | eulerpartlems.s |
. . . . . . . 8
⊢ 𝑆 = (𝑓 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ↦ Σ𝑘 ∈ ℕ ((𝑓‘𝑘) · 𝑘)) |
12 | 10, 11 | eulerpartlemelr 32008 |
. . . . . . 7
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → (𝐴:ℕ⟶ℕ0 ∧
(◡𝐴 “ ℕ) ∈
Fin)) |
13 | 12 | simpld 498 |
. . . . . 6
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → 𝐴:ℕ⟶ℕ0) |
14 | 13 | ffvelrnda 6893 |
. . . . 5
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈
ℕ0) |
15 | 14 | adantrr 717 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈
ℕ0) |
16 | 15 | nn0red 12134 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ) |
17 | 16, 8 | remulcld 10846 |
. 2
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ∈ ℝ) |
18 | 10, 11 | eulerpartlemsf 32010 |
. . . . 5
⊢ 𝑆:((ℕ0
↑m ℕ) ∩ 𝑅)⟶ℕ0 |
19 | 18 | ffvelrni 6892 |
. . . 4
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) ∈
ℕ0) |
20 | 19 | adantr 484 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈
ℕ0) |
21 | 20 | nn0red 12134 |
. 2
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑆‘𝐴) ∈ ℝ) |
22 | 14 | nn0red 12134 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝐴‘𝑡) ∈ ℝ) |
23 | 22 | adantrr 717 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝐴‘𝑡) ∈ ℝ) |
24 | 7 | nnrpd 12609 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → 𝑡 ∈ ℝ+) |
25 | 24 | rprege0d 12618 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) |
26 | | bitsfi 15977 |
. . . . . 6
⊢ ((𝐴‘𝑡) ∈ ℕ0 →
(bits‘(𝐴‘𝑡)) ∈ Fin) |
27 | 15, 26 | syl 17 |
. . . . 5
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (bits‘(𝐴‘𝑡)) ∈ Fin) |
28 | 1 | a1i 11 |
. . . . . 6
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 2 ∈ ℝ) |
29 | 3 | a1i 11 |
. . . . . . 7
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (bits‘(𝐴‘𝑡)) ⊆
ℕ0) |
30 | 29 | sselda 3891 |
. . . . . 6
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 𝑖 ∈ ℕ0) |
31 | 28, 30 | reexpcld 13716 |
. . . . 5
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → (2↑𝑖) ∈ ℝ) |
32 | | 0le2 11915 |
. . . . . . 7
⊢ 0 ≤
2 |
33 | 32 | a1i 11 |
. . . . . 6
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ 2) |
34 | 28, 30, 33 | expge0d 13717 |
. . . . 5
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) ∧ 𝑖 ∈ (bits‘(𝐴‘𝑡))) → 0 ≤ (2↑𝑖)) |
35 | 4 | snssd 4712 |
. . . . 5
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → {𝑛} ⊆ (bits‘(𝐴‘𝑡))) |
36 | 27, 31, 34, 35 | fsumless 15341 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) ≤ Σ𝑖 ∈ (bits‘(𝐴‘𝑡))(2↑𝑖)) |
37 | 6 | recnd 10844 |
. . . . 5
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ∈ ℂ) |
38 | | oveq2 7210 |
. . . . . 6
⊢ (𝑖 = 𝑛 → (2↑𝑖) = (2↑𝑛)) |
39 | 38 | sumsn 15291 |
. . . . 5
⊢ ((𝑛 ∈ (bits‘(𝐴‘𝑡)) ∧ (2↑𝑛) ∈ ℂ) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛)) |
40 | 4, 37, 39 | syl2anc 587 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ {𝑛} (2↑𝑖) = (2↑𝑛)) |
41 | | bitsinv1 15982 |
. . . . 5
⊢ ((𝐴‘𝑡) ∈ ℕ0 →
Σ𝑖 ∈
(bits‘(𝐴‘𝑡))(2↑𝑖) = (𝐴‘𝑡)) |
42 | 15, 41 | syl 17 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → Σ𝑖 ∈ (bits‘(𝐴‘𝑡))(2↑𝑖) = (𝐴‘𝑡)) |
43 | 36, 40, 42 | 3brtr3d 5074 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → (2↑𝑛) ≤ (𝐴‘𝑡)) |
44 | | lemul1a 11669 |
. . 3
⊢
((((2↑𝑛) ∈
ℝ ∧ (𝐴‘𝑡) ∈ ℝ ∧ (𝑡 ∈ ℝ ∧ 0 ≤ 𝑡)) ∧ (2↑𝑛) ≤ (𝐴‘𝑡)) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡)) |
45 | 6, 23, 25, 43, 44 | syl31anc 1375 |
. 2
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ ((𝐴‘𝑡) · 𝑡)) |
46 | | fzfid 13529 |
. . . . . . 7
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → (1...(𝑆‘𝐴)) ∈ Fin) |
47 | | elfznn 13124 |
. . . . . . . . . . 11
⊢ (𝑘 ∈ (1...(𝑆‘𝐴)) → 𝑘 ∈ ℕ) |
48 | | ffvelrn 6891 |
. . . . . . . . . . 11
⊢ ((𝐴:ℕ⟶ℕ0 ∧
𝑘 ∈ ℕ) →
(𝐴‘𝑘) ∈
ℕ0) |
49 | 13, 47, 48 | syl2an 599 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈
ℕ0) |
50 | 49 | nn0red 12134 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → (𝐴‘𝑘) ∈ ℝ) |
51 | 47 | adantl 485 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℕ) |
52 | 51 | nnred 11828 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 𝑘 ∈ ℝ) |
53 | 50, 52 | remulcld 10846 |
. . . . . . . 8
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ) |
54 | 53 | adantlr 715 |
. . . . . . 7
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑘) · 𝑘) ∈ ℝ) |
55 | 49 | nn0ge0d 12136 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ (𝐴‘𝑘)) |
56 | | 0red 10819 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ∈
ℝ) |
57 | 51 | nngt0d 11862 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 < 𝑘) |
58 | 56, 52, 57 | ltled 10963 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ 𝑘) |
59 | 50, 52, 55, 58 | mulge0d 11392 |
. . . . . . . 8
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘)) |
60 | 59 | adantlr 715 |
. . . . . . 7
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) ∧ 𝑘 ∈ (1...(𝑆‘𝐴))) → 0 ≤ ((𝐴‘𝑘) · 𝑘)) |
61 | | fveq2 6706 |
. . . . . . . 8
⊢ (𝑘 = 𝑡 → (𝐴‘𝑘) = (𝐴‘𝑡)) |
62 | | id 22 |
. . . . . . . 8
⊢ (𝑘 = 𝑡 → 𝑘 = 𝑡) |
63 | 61, 62 | oveq12d 7220 |
. . . . . . 7
⊢ (𝑘 = 𝑡 → ((𝐴‘𝑘) · 𝑘) = ((𝐴‘𝑡) · 𝑡)) |
64 | | simpr 488 |
. . . . . . 7
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → 𝑡 ∈ (1...(𝑆‘𝐴))) |
65 | 46, 54, 60, 63, 64 | fsumge1 15342 |
. . . . . 6
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
66 | 65 | adantlr 715 |
. . . . 5
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
67 | | eldif 3867 |
. . . . . . 7
⊢ (𝑡 ∈ (ℕ ∖
(1...(𝑆‘𝐴))) ↔ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))) |
68 | | nndiffz1 30799 |
. . . . . . . . . . . . . 14
⊢ ((𝑆‘𝐴) ∈ ℕ0 → (ℕ
∖ (1...(𝑆‘𝐴))) =
(ℤ≥‘((𝑆‘𝐴) + 1))) |
69 | 68 | eleq2d 2819 |
. . . . . . . . . . . . 13
⊢ ((𝑆‘𝐴) ∈ ℕ0 → (𝑡 ∈ (ℕ ∖
(1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1)))) |
70 | 19, 69 | syl 17 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → (𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴))) ↔ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1)))) |
71 | 70 | pm5.32i 578 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) ↔ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1)))) |
72 | 10, 11 | eulerpartlems 32011 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℤ≥‘((𝑆‘𝐴) + 1))) → (𝐴‘𝑡) = 0) |
73 | 71, 72 | sylbi 220 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (𝐴‘𝑡) = 0) |
74 | 73 | oveq1d 7217 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = (0 · 𝑡)) |
75 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) |
76 | 75 | eldifad 3869 |
. . . . . . . . . . 11
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℕ) |
77 | 76 | nncnd 11829 |
. . . . . . . . . 10
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 𝑡 ∈ ℂ) |
78 | 77 | mul02d 11013 |
. . . . . . . . 9
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → (0 · 𝑡) = 0) |
79 | 74, 78 | eqtrd 2774 |
. . . . . . . 8
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) = 0) |
80 | | fzfid 13529 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → (1...(𝑆‘𝐴)) ∈ Fin) |
81 | 80, 53, 59 | fsumge0 15340 |
. . . . . . . . 9
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → 0 ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
82 | 81 | adantr 484 |
. . . . . . . 8
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → 0 ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
83 | 79, 82 | eqbrtrd 5065 |
. . . . . . 7
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ (ℕ ∖ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
84 | 67, 83 | sylan2br 598 |
. . . . . 6
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴)))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
85 | 84 | anassrs 471 |
. . . . 5
⊢ (((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) ∧ ¬ 𝑡 ∈ (1...(𝑆‘𝐴))) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
86 | 66, 85 | pm2.61dan 813 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
87 | 10, 11 | eulerpartlemsv3 32012 |
. . . . 5
⊢ (𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
88 | 87 | adantr 484 |
. . . 4
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → (𝑆‘𝐴) = Σ𝑘 ∈ (1...(𝑆‘𝐴))((𝐴‘𝑘) · 𝑘)) |
89 | 86, 88 | breqtrrd 5071 |
. . 3
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ 𝑡 ∈ ℕ) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴)) |
90 | 89 | adantrr 717 |
. 2
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((𝐴‘𝑡) · 𝑡) ≤ (𝑆‘𝐴)) |
91 | 9, 17, 21, 45, 90 | letrd 10972 |
1
⊢ ((𝐴 ∈ ((ℕ0
↑m ℕ) ∩ 𝑅) ∧ (𝑡 ∈ ℕ ∧ 𝑛 ∈ (bits‘(𝐴‘𝑡)))) → ((2↑𝑛) · 𝑡) ≤ (𝑆‘𝐴)) |