Proof of Theorem pclem
Step | Hyp | Ref
| Expression |
1 | | pclem.1 |
. . . . 5
⊢ 𝐴 = {𝑛 ∈ ℕ0 ∣ (𝑃↑𝑛) ∥ 𝑁} |
2 | 1 | ssrab3 3995 |
. . . 4
⊢ 𝐴 ⊆
ℕ0 |
3 | | nn0ssz 12198 |
. . . 4
⊢
ℕ0 ⊆ ℤ |
4 | 2, 3 | sstri 3910 |
. . 3
⊢ 𝐴 ⊆
ℤ |
5 | 4 | a1i 11 |
. 2
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ⊆ ℤ) |
6 | | 0nn0 12105 |
. . . . 5
⊢ 0 ∈
ℕ0 |
7 | 6 | a1i 11 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈
ℕ0) |
8 | | eluzelcn 12450 |
. . . . . . 7
⊢ (𝑃 ∈
(ℤ≥‘2) → 𝑃 ∈ ℂ) |
9 | 8 | adantr 484 |
. . . . . 6
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℂ) |
10 | 9 | exp0d 13710 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) = 1) |
11 | | 1dvds 15832 |
. . . . . 6
⊢ (𝑁 ∈ ℤ → 1 ∥
𝑁) |
12 | 11 | ad2antrl 728 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 ∥ 𝑁) |
13 | 10, 12 | eqbrtrd 5075 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝑃↑0) ∥ 𝑁) |
14 | | oveq2 7221 |
. . . . . 6
⊢ (𝑛 = 0 → (𝑃↑𝑛) = (𝑃↑0)) |
15 | 14 | breq1d 5063 |
. . . . 5
⊢ (𝑛 = 0 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑0) ∥ 𝑁)) |
16 | 15, 1 | elrab2 3605 |
. . . 4
⊢ (0 ∈
𝐴 ↔ (0 ∈
ℕ0 ∧ (𝑃↑0) ∥ 𝑁)) |
17 | 7, 13, 16 | sylanbrc 586 |
. . 3
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 0 ∈ 𝐴) |
18 | 17 | ne0d 4250 |
. 2
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝐴 ≠ ∅) |
19 | | nnssz 12197 |
. . 3
⊢ ℕ
⊆ ℤ |
20 | | zcn 12181 |
. . . . . . 7
⊢ (𝑁 ∈ ℤ → 𝑁 ∈
ℂ) |
21 | 20 | abscld 15000 |
. . . . . 6
⊢ (𝑁 ∈ ℤ →
(abs‘𝑁) ∈
ℝ) |
22 | 21 | ad2antrl 728 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (abs‘𝑁) ∈
ℝ) |
23 | | eluzelre 12449 |
. . . . . 6
⊢ (𝑃 ∈
(ℤ≥‘2) → 𝑃 ∈ ℝ) |
24 | 23 | adantr 484 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 𝑃 ∈ ℝ) |
25 | | eluz2gt1 12516 |
. . . . . 6
⊢ (𝑃 ∈
(ℤ≥‘2) → 1 < 𝑃) |
26 | 25 | adantr 484 |
. . . . 5
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → 1 < 𝑃) |
27 | | expnbnd 13799 |
. . . . 5
⊢
(((abs‘𝑁)
∈ ℝ ∧ 𝑃
∈ ℝ ∧ 1 < 𝑃) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥)) |
28 | 22, 24, 26, 27 | syl3anc 1373 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥)) |
29 | | simprr 773 |
. . . . . . . . . . . . 13
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ 𝐴) |
30 | | oveq2 7221 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑦 → (𝑃↑𝑛) = (𝑃↑𝑦)) |
31 | 30 | breq1d 5063 |
. . . . . . . . . . . . . 14
⊢ (𝑛 = 𝑦 → ((𝑃↑𝑛) ∥ 𝑁 ↔ (𝑃↑𝑦) ∥ 𝑁)) |
32 | 31, 1 | elrab2 3605 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝐴 ↔ (𝑦 ∈ ℕ0 ∧ (𝑃↑𝑦) ∥ 𝑁)) |
33 | 29, 32 | sylib 221 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 ∈ ℕ0 ∧ (𝑃↑𝑦) ∥ 𝑁)) |
34 | 33 | simprd 499 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∥ 𝑁) |
35 | | eluz2nn 12480 |
. . . . . . . . . . . . . . 15
⊢ (𝑃 ∈
(ℤ≥‘2) → 𝑃 ∈ ℕ) |
36 | 35 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℕ) |
37 | 33 | simpld 498 |
. . . . . . . . . . . . . 14
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℕ0) |
38 | 36, 37 | nnexpcld 13812 |
. . . . . . . . . . . . 13
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℕ) |
39 | 38 | nnzd 12281 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℤ) |
40 | | simplrl 777 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ∈ ℤ) |
41 | | simplrr 778 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑁 ≠ 0) |
42 | | dvdsleabs 15872 |
. . . . . . . . . . . 12
⊢ (((𝑃↑𝑦) ∈ ℤ ∧ 𝑁 ∈ ℤ ∧ 𝑁 ≠ 0) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁))) |
43 | 39, 40, 41, 42 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((𝑃↑𝑦) ∥ 𝑁 → (𝑃↑𝑦) ≤ (abs‘𝑁))) |
44 | 34, 43 | mpd 15 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ≤ (abs‘𝑁)) |
45 | 38 | nnred 11845 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑦) ∈ ℝ) |
46 | 22 | adantr 484 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (abs‘𝑁) ∈ ℝ) |
47 | 23 | ad2antrr 726 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑃 ∈ ℝ) |
48 | | nnnn0 12097 |
. . . . . . . . . . . . 13
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℕ0) |
49 | 48 | ad2antrl 728 |
. . . . . . . . . . . 12
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℕ0) |
50 | 47, 49 | reexpcld 13733 |
. . . . . . . . . . 11
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑃↑𝑥) ∈ ℝ) |
51 | | lelttr 10923 |
. . . . . . . . . . 11
⊢ (((𝑃↑𝑦) ∈ ℝ ∧ (abs‘𝑁) ∈ ℝ ∧ (𝑃↑𝑥) ∈ ℝ) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥))) |
52 | 45, 46, 50, 51 | syl3anc 1373 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (((𝑃↑𝑦) ≤ (abs‘𝑁) ∧ (abs‘𝑁) < (𝑃↑𝑥)) → (𝑃↑𝑦) < (𝑃↑𝑥))) |
53 | 44, 52 | mpand 695 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → (𝑃↑𝑦) < (𝑃↑𝑥))) |
54 | 37 | nn0zd 12280 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℤ) |
55 | | nnz 12199 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℤ) |
56 | 55 | ad2antrl 728 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℤ) |
57 | 25 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 1 < 𝑃) |
58 | 47, 54, 56, 57 | ltexp2d 13820 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 ↔ (𝑃↑𝑦) < (𝑃↑𝑥))) |
59 | 53, 58 | sylibrd 262 |
. . . . . . . 8
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 < 𝑥)) |
60 | 37 | nn0red 12151 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑦 ∈ ℝ) |
61 | | nnre 11837 |
. . . . . . . . . 10
⊢ (𝑥 ∈ ℕ → 𝑥 ∈
ℝ) |
62 | 61 | ad2antrl 728 |
. . . . . . . . 9
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → 𝑥 ∈ ℝ) |
63 | | ltle 10921 |
. . . . . . . . 9
⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥)) |
64 | 60, 62, 63 | syl2anc 587 |
. . . . . . . 8
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → (𝑦 < 𝑥 → 𝑦 ≤ 𝑥)) |
65 | 59, 64 | syld 47 |
. . . . . . 7
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ (𝑥 ∈ ℕ ∧ 𝑦 ∈ 𝐴)) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥)) |
66 | 65 | anassrs 471 |
. . . . . 6
⊢ ((((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) ∧ 𝑦 ∈ 𝐴) → ((abs‘𝑁) < (𝑃↑𝑥) → 𝑦 ≤ 𝑥)) |
67 | 66 | ralrimdva 3110 |
. . . . 5
⊢ (((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) ∧ 𝑥 ∈ ℕ) → ((abs‘𝑁) < (𝑃↑𝑥) → ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
68 | 67 | reximdva 3193 |
. . . 4
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (∃𝑥 ∈ ℕ (abs‘𝑁) < (𝑃↑𝑥) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
69 | 28, 68 | mpd 15 |
. . 3
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
70 | | ssrexv 3968 |
. . 3
⊢ (ℕ
⊆ ℤ → (∃𝑥 ∈ ℕ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |
71 | 19, 69, 70 | mpsyl 68 |
. 2
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥) |
72 | 5, 18, 71 | 3jca 1130 |
1
⊢ ((𝑃 ∈
(ℤ≥‘2) ∧ (𝑁 ∈ ℤ ∧ 𝑁 ≠ 0)) → (𝐴 ⊆ ℤ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℤ ∀𝑦 ∈ 𝐴 𝑦 ≤ 𝑥)) |