Proof of Theorem pclem
Step | Hyp | Ref
| Expression |
1 | | pclem.1 |
. . . . 5
⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} |
2 | | ssrab2 3687 |
. . . . 5
⊢ {𝑛 ∈ ℕ0
∣ (𝑃↑𝑛) ∥ 𝑁} ⊆
ℕ0 |
3 | 1, 2 | eqsstri 3635 |
. . . 4
⊢ 𝐴 ⊆
ℕ0 |
4 | | nn0ssz 11398 |
. . . 4
⊢
ℕ0 ⊆ ℤ |
5 | 3, 4 | sstri 3612 |
. . 3
⊢ 𝐴 ⊆
ℤ |
6 | 5 | a1i 11 |
. 2
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ⊆ ℤ) |
7 | | 0nn0 11307 |
. . . . 5
⊢ 0 ∈
ℕ0 |
8 | 7 | a1i 11 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈
ℕ0) |
9 | | eluzelcn 11699 |
. . . . . . 7
⊢ (𝑃 ∈
(ℤ≥‘2) → 𝑃 ∈ ℂ) |
10 | 9 | adantr 481 |
. . . . . 6
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℂ) |
11 | 10 | exp0d 13002 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) = 1) |
12 | | 1dvds 14996 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → 1 ∥
𝑁) |
13 | 12 | ad2antrl 764 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 ∥ 𝑁) |
14 | 11, 13 | eqbrtrd 4675 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) ∥ 𝑁) |
15 | | oveq2 6658 |
. . . . . 6
⊢ (𝑛 = 0 → (𝑃↑𝑛) = (𝑃↑0)) |
16 | 15 | breq1d 4663 |
. . . . 5
⊢ (𝑛 = 0 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑0) ∥ 𝑁)) |
17 | 16, 1 | elrab2 3366 |
. . . 4
⊢ (0 ∈
𝐴 ↔ (0 ∈
ℕ0 ∧ (𝑃↑0) ∥ 𝑁)) |
18 | 8, 14, 17 | sylanbrc 698 |
. . 3
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ 𝐴) |
19 | | ne0i 3921 |
. . 3
⊢ (0 ∈
𝐴 → 𝐴 ≠ ∅) |
20 | 18, 19 | syl 17 |
. 2
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ≠ ∅) |
21 | | nnssz 11397 |
. . 3
⊢ ℕ
⊆ ℤ |
22 | | zcn 11382 |
. . . . . . 7
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℂ) |
23 | 22 | abscld 14175 |
. . . . . 6
⊢ (𝑁 ∈ ℤ →
(abs‘𝑁) ∈
ℝ) |
24 | 23 | ad2antrl 764 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈
ℝ) |
25 | | eluzelre 11698 |
. . . . . 6
⊢ (𝑃 ∈
(ℤ≥‘2) → 𝑃 ∈ ℝ) |
26 | 25 | adantr 481 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℝ) |
27 | | eluz2b2 11761 |
. . . . . . 7
⊢ (𝑃 ∈
(ℤ≥‘2) ↔ (𝑃 ∈ ℕ ∧ 1 < 𝑃)) |
28 | 27 | simprbi 480 |
. . . . . 6
⊢ (𝑃 ∈
(ℤ≥‘2) → 1 < 𝑃) |
29 | 28 | adantr 481 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 < 𝑃) |
30 | | expnbnd 12993 |
. . . . 5
⊢
(((abs‘𝑁)
∈ ℝ ∧ 𝑃
∈ ℝ ∧ 1 < 𝑃) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥)) |
31 | 24, 26, 29, 30 | syl3anc 1326 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥)) |
32 | | simprr 796 |
. . . . . . . . . . . . 13
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴) |
33 | | oveq2 6658 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑦 → (𝑃↑𝑛) = (𝑃↑𝑦)) |
34 | 33 | breq1d 4663 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑦 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑦) ∥ 𝑁)) |
35 | 34, 1 | elrab2 3366 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℕ0 ∧ (𝑃↑𝑦) ∥ 𝑁)) |
36 | 32, 35 | sylib 208 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ ℕ0 ∧ (𝑃↑𝑦) ∥ 𝑁)) |
37 | 36 | simprd 479 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∥ 𝑁) |
38 | | eluz2nn 11726 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈
(ℤ≥‘2) → 𝑃 ∈ ℕ) |
39 | 38 | ad2antrr 762 |
. . . . . . . . . . . . . 14
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℕ) |
40 | 36 | simpld 475 |
. . . . . . . . . . . . . 14
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℕ0) |
41 | 39, 40 | nnexpcld 13030 |
. . . . . . . . . . . . 13
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℕ) |
42 | 41 | nnzd 11481 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℤ) |
43 | | simplrl 800 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ∈ ℤ) |
44 | | simplrr 801 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ≠ 0) |
45 | | dvdsleabs 15033 |
. . . . . . . . . . . 12
⊢ (((𝑃↑𝑦) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁))) |
46 | 42, 43, 44, 45 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁))) |
47 | 37, 46 | mpd 15 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ≤ (abs‘𝑁)) |
48 | 41 | nnred 11035 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℝ) |
49 | 24 | adantr 481 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (abs‘𝑁) ∈ ℝ) |
50 | 25 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℝ) |
51 | | nnnn0 11299 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℕ0) |
52 | 51 | ad2antrl 764 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ0) |
53 | 50, 52 | reexpcld 13025 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑥) ∈ ℝ) |
54 | | lelttr 10128 |
. . . . . . . . . . 11
⊢ (((𝑃↑𝑦) ∈ ℝ ∧ (abs‘𝑁) ∈ ℝ ∧ (𝑃↑𝑥) ∈ ℝ) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥))) |
55 | 48, 49, 53, 54 | syl3anc 1326 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥))) |
56 | 47, 55 | mpand 711 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → (𝑃↑𝑦) < (𝑃↑𝑥))) |
57 | 40 | nn0zd 11480 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℤ) |
58 | | nnz 11399 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℤ) |
59 | 58 | ad2antrl 764 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℤ) |
60 | 28 | ad2antrr 762 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 1 < 𝑃) |
61 | 50, 57, 59, 60 | ltexp2d 13038 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥))) |
62 | 56, 61 | sylibrd 249 |
. . . . . . . 8
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 < 𝑥)) |
63 | 40 | nn0red 11352 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ) |
64 | | nnre 11027 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℝ) |
65 | 64 | ad2antrl 764 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ) |
66 | | ltle 10126 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥)) |
67 | 63, 65, 66 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥)) |
68 | 62, 67 | syld 47 |
. . . . . . 7
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥)) |
69 | 68 | anassrs 680 |
. . . . . 6
⊢ ((((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ 𝐴) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥)) |
70 | 69 | ralrimdva 2969 |
. . . . 5
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) → ((abs‘𝑁) < (𝑃↑𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
71 | 70 | reximdva 3017 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
72 | 31, 71 | mpd 15 |
. . 3
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
73 | | ssrexv 3667 |
. . 3
⊢ (ℕ
⊆ ℤ → (∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
74 | 21, 72, 73 | mpsyl 68 |
. 2
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
75 | 6, 20, 74 | 3jca 1242 |
1
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |