Step | Hyp | Ref
| Expression |
1 | | breq2 5034 |
. . . 4
⊢ (𝐴 = 𝐵 → ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
2 | 1 | a1d 25 |
. . 3
⊢ (𝐴 = 𝐵 → ((𝑝 ∈ ℙ ∧ 𝑛 ∈ ℕ) → ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
3 | 2 | ralrimivv 3102 |
. 2
⊢ (𝐴 = 𝐵 → ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
4 | | elnn0 11978 |
. . 3
⊢ (𝐴 ∈ ℕ0
↔ (𝐴 ∈ ℕ
∨ 𝐴 =
0)) |
5 | | elnn0 11978 |
. . . . . . 7
⊢ (𝐵 ∈ ℕ0
↔ (𝐵 ∈ ℕ
∨ 𝐵 =
0)) |
6 | | nnre 11723 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℕ → 𝐴 ∈
ℝ) |
7 | | nnre 11723 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈ ℕ → 𝐵 ∈
ℝ) |
8 | | lttri2 10801 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
9 | 6, 7, 8 | syl2an 599 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ ∧ 𝐵 ∈ ℕ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
10 | 9 | ancoms 462 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐴 ≠ 𝐵 ↔ (𝐴 < 𝐵 ∨ 𝐵 < 𝐴))) |
11 | | nn0prpwlem 34149 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈ ℕ →
∀𝑘 ∈ ℕ
(𝑘 < 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
12 | | breq1 5033 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝐴 → (𝑘 < 𝐵 ↔ 𝐴 < 𝐵)) |
13 | | breq2 5034 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝐴 → ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴)) |
14 | 13 | bibi1d 347 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝐴 → (((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
15 | 14 | notbid 321 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝐴 → (¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
16 | 15 | 2rexbidv 3210 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝐴 → (∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
17 | 12, 16 | imbi12d 348 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝐴 → ((𝑘 < 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵)) ↔ (𝐴 < 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)))) |
18 | 17 | rspcv 3521 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℕ →
(∀𝑘 ∈ ℕ
(𝑘 < 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵)) → (𝐴 < 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)))) |
19 | 11, 18 | mpan9 510 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐴 < 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
20 | | breq1 5033 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝐵 → (𝑘 < 𝐴 ↔ 𝐵 < 𝐴)) |
21 | | breq2 5034 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑘 = 𝐵 → ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
22 | 21 | bibi1d 347 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑘 = 𝐵 → (((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴) ↔ ((𝑝↑𝑛) ∥ 𝐵 ↔ (𝑝↑𝑛) ∥ 𝐴))) |
23 | | bicom 225 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑝↑𝑛) ∥ 𝐵 ↔ (𝑝↑𝑛) ∥ 𝐴) ↔ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
24 | 22, 23 | bitrdi 290 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 = 𝐵 → (((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴) ↔ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
25 | 24 | notbid 321 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 = 𝐵 → (¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴) ↔ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
26 | 25 | 2rexbidv 3210 |
. . . . . . . . . . . . . . . 16
⊢ (𝑘 = 𝐵 → (∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴) ↔ ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
27 | 20, 26 | imbi12d 348 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 = 𝐵 → ((𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴)) ↔ (𝐵 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)))) |
28 | 27 | rspcv 3521 |
. . . . . . . . . . . . . 14
⊢ (𝐵 ∈ ℕ →
(∀𝑘 ∈ ℕ
(𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴)) → (𝐵 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)))) |
29 | | nn0prpwlem 34149 |
. . . . . . . . . . . . . 14
⊢ (𝐴 ∈ ℕ →
∀𝑘 ∈ ℕ
(𝑘 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝑘 ↔ (𝑝↑𝑛) ∥ 𝐴))) |
30 | 28, 29 | impel 509 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐵 < 𝐴 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
31 | 19, 30 | jaod 858 |
. . . . . . . . . . . 12
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → ((𝐴 < 𝐵 ∨ 𝐵 < 𝐴) → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
32 | 10, 31 | sylbid 243 |
. . . . . . . . . . 11
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (𝐴 ≠ 𝐵 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
33 | | df-ne 2935 |
. . . . . . . . . . 11
⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) |
34 | | rexnal2 3170 |
. . . . . . . . . . 11
⊢
(∃𝑝 ∈
ℙ ∃𝑛 ∈
ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ¬ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
35 | 32, 33, 34 | 3imtr3g 298 |
. . . . . . . . . 10
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) → (¬
𝐴 = 𝐵 → ¬ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
36 | 35 | con4d 115 |
. . . . . . . . 9
⊢ ((𝐵 ∈ ℕ ∧ 𝐴 ∈ ℕ) →
(∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵)) |
37 | 36 | ex 416 |
. . . . . . . 8
⊢ (𝐵 ∈ ℕ → (𝐴 ∈ ℕ →
(∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
38 | | prmunb 16350 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℕ →
∃𝑝 ∈ ℙ
𝐴 < 𝑝) |
39 | | 1nn 11727 |
. . . . . . . . . . . . . . 15
⊢ 1 ∈
ℕ |
40 | | prmz 16116 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℤ) |
41 | | 1nn0 11992 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 1 ∈
ℕ0 |
42 | | zexpcl 13536 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑝 ∈ ℤ ∧ 1 ∈
ℕ0) → (𝑝↑1) ∈ ℤ) |
43 | 40, 41, 42 | sylancl 589 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 ∈ ℙ → (𝑝↑1) ∈
ℤ) |
44 | | dvds0 15717 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑝↑1) ∈ ℤ →
(𝑝↑1) ∥
0) |
45 | 43, 44 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑝 ∈ ℙ → (𝑝↑1) ∥
0) |
46 | 45 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐴 < 𝑝) → (𝑝↑1) ∥ 0) |
47 | | dvdsle 15755 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑝↑1) ∈ ℤ ∧
𝐴 ∈ ℕ) →
((𝑝↑1) ∥ 𝐴 → (𝑝↑1) ≤ 𝐴)) |
48 | 43, 47 | sylan 583 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝↑1) ∥ 𝐴 → (𝑝↑1) ≤ 𝐴)) |
49 | | prmnn 16115 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℕ) |
50 | | nnre 11723 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 ∈ ℕ → 𝑝 ∈
ℝ) |
51 | 49, 50 | syl 17 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℝ) |
52 | | reexpcl 13538 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑝 ∈ ℝ ∧ 1 ∈
ℕ0) → (𝑝↑1) ∈ ℝ) |
53 | 51, 41, 52 | sylancl 589 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 ∈ ℙ → (𝑝↑1) ∈
ℝ) |
54 | | lenlt 10797 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑝↑1) ∈ ℝ ∧
𝐴 ∈ ℝ) →
((𝑝↑1) ≤ 𝐴 ↔ ¬ 𝐴 < (𝑝↑1))) |
55 | 53, 6, 54 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝↑1) ≤ 𝐴 ↔ ¬ 𝐴 < (𝑝↑1))) |
56 | 49 | nncnd 11732 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑝 ∈ ℙ → 𝑝 ∈
ℂ) |
57 | 56 | exp1d 13597 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 ∈ ℙ → (𝑝↑1) = 𝑝) |
58 | 57 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝑝↑1) = 𝑝) |
59 | 58 | breq2d 5042 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (𝐴 < (𝑝↑1) ↔ 𝐴 < 𝑝)) |
60 | 59 | notbid 321 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → (¬
𝐴 < (𝑝↑1) ↔ ¬ 𝐴 < 𝑝)) |
61 | 55, 60 | bitrd 282 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝↑1) ≤ 𝐴 ↔ ¬ 𝐴 < 𝑝)) |
62 | 48, 61 | sylibd 242 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑝 ∈ ℙ ∧ 𝐴 ∈ ℕ) → ((𝑝↑1) ∥ 𝐴 → ¬ 𝐴 < 𝑝)) |
63 | 62 | ancoms 462 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → ((𝑝↑1) ∥ 𝐴 → ¬ 𝐴 < 𝑝)) |
64 | 63 | con2d 136 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝐴 < 𝑝 → ¬ (𝑝↑1) ∥ 𝐴)) |
65 | 64 | 3impia 1118 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐴 < 𝑝) → ¬ (𝑝↑1) ∥ 𝐴) |
66 | 46, 65 | jcnd 166 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐴 < 𝑝) → ¬ ((𝑝↑1) ∥ 0 → (𝑝↑1) ∥ 𝐴)) |
67 | | biimpr 223 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑝↑1) ∥ 𝐴 ↔ (𝑝↑1) ∥ 0) → ((𝑝↑1) ∥ 0 → (𝑝↑1) ∥ 𝐴)) |
68 | 66, 67 | nsyl 142 |
. . . . . . . . . . . . . . 15
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐴 < 𝑝) → ¬ ((𝑝↑1) ∥ 𝐴 ↔ (𝑝↑1) ∥ 0)) |
69 | | oveq2 7178 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑛 = 1 → (𝑝↑𝑛) = (𝑝↑1)) |
70 | 69 | breq1d 5040 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑1) ∥ 𝐴)) |
71 | 69 | breq1d 5040 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑1) ∥ 0)) |
72 | 70, 71 | bibi12d 349 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 1 → (((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0) ↔ ((𝑝↑1) ∥ 𝐴 ↔ (𝑝↑1) ∥ 0))) |
73 | 72 | notbid 321 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 1 → (¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0) ↔ ¬ ((𝑝↑1) ∥ 𝐴 ↔ (𝑝↑1) ∥ 0))) |
74 | 73 | rspcev 3526 |
. . . . . . . . . . . . . . 15
⊢ ((1
∈ ℕ ∧ ¬ ((𝑝↑1) ∥ 𝐴 ↔ (𝑝↑1) ∥ 0)) → ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0)) |
75 | 39, 68, 74 | sylancr 590 |
. . . . . . . . . . . . . 14
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐴 < 𝑝) → ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0)) |
76 | 75 | 3expia 1122 |
. . . . . . . . . . . . 13
⊢ ((𝐴 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝐴 < 𝑝 → ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0))) |
77 | 76 | reximdva 3184 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℕ →
(∃𝑝 ∈ ℙ
𝐴 < 𝑝 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0))) |
78 | 38, 77 | mpd 15 |
. . . . . . . . . . 11
⊢ (𝐴 ∈ ℕ →
∃𝑝 ∈ ℙ
∃𝑛 ∈ ℕ
¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0)) |
79 | | rexnal2 3170 |
. . . . . . . . . . 11
⊢
(∃𝑝 ∈
ℙ ∃𝑛 ∈
ℕ ¬ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0) ↔ ¬ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0)) |
80 | 78, 79 | sylib 221 |
. . . . . . . . . 10
⊢ (𝐴 ∈ ℕ → ¬
∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0)) |
81 | 80 | pm2.21d 121 |
. . . . . . . . 9
⊢ (𝐴 ∈ ℕ →
(∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0) → 𝐴 = 0)) |
82 | | breq2 5034 |
. . . . . . . . . . . 12
⊢ (𝐵 = 0 → ((𝑝↑𝑛) ∥ 𝐵 ↔ (𝑝↑𝑛) ∥ 0)) |
83 | 82 | bibi2d 346 |
. . . . . . . . . . 11
⊢ (𝐵 = 0 → (((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0))) |
84 | 83 | 2ralbidv 3111 |
. . . . . . . . . 10
⊢ (𝐵 = 0 → (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0))) |
85 | | eqeq2 2750 |
. . . . . . . . . 10
⊢ (𝐵 = 0 → (𝐴 = 𝐵 ↔ 𝐴 = 0)) |
86 | 84, 85 | imbi12d 348 |
. . . . . . . . 9
⊢ (𝐵 = 0 → ((∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵) ↔ (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0) → 𝐴 = 0))) |
87 | 81, 86 | syl5ibr 249 |
. . . . . . . 8
⊢ (𝐵 = 0 → (𝐴 ∈ ℕ → (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
88 | 37, 87 | jaoi 856 |
. . . . . . 7
⊢ ((𝐵 ∈ ℕ ∨ 𝐵 = 0) → (𝐴 ∈ ℕ → (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
89 | 5, 88 | sylbi 220 |
. . . . . 6
⊢ (𝐵 ∈ ℕ0
→ (𝐴 ∈ ℕ
→ (∀𝑝 ∈
ℙ ∀𝑛 ∈
ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
90 | 89 | com12 32 |
. . . . 5
⊢ (𝐴 ∈ ℕ → (𝐵 ∈ ℕ0
→ (∀𝑝 ∈
ℙ ∀𝑛 ∈
ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
91 | | orcom 869 |
. . . . . . . . . 10
⊢ ((𝐵 ∈ ℕ ∨ 𝐵 = 0) ↔ (𝐵 = 0 ∨ 𝐵 ∈ ℕ)) |
92 | | df-or 847 |
. . . . . . . . . 10
⊢ ((𝐵 = 0 ∨ 𝐵 ∈ ℕ) ↔ (¬ 𝐵 = 0 → 𝐵 ∈ ℕ)) |
93 | 5, 91, 92 | 3bitri 300 |
. . . . . . . . 9
⊢ (𝐵 ∈ ℕ0
↔ (¬ 𝐵 = 0 →
𝐵 ∈
ℕ)) |
94 | | prmunb 16350 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ ℕ →
∃𝑝 ∈ ℙ
𝐵 < 𝑝) |
95 | 45 | 3ad2ant2 1135 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐵 < 𝑝) → (𝑝↑1) ∥ 0) |
96 | | dvdsle 15755 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝑝↑1) ∈ ℤ ∧
𝐵 ∈ ℕ) →
((𝑝↑1) ∥ 𝐵 → (𝑝↑1) ≤ 𝐵)) |
97 | 43, 96 | sylan 583 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → ((𝑝↑1) ∥ 𝐵 → (𝑝↑1) ≤ 𝐵)) |
98 | | lenlt 10797 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝑝↑1) ∈ ℝ ∧
𝐵 ∈ ℝ) →
((𝑝↑1) ≤ 𝐵 ↔ ¬ 𝐵 < (𝑝↑1))) |
99 | 53, 7, 98 | syl2an 599 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → ((𝑝↑1) ≤ 𝐵 ↔ ¬ 𝐵 < (𝑝↑1))) |
100 | 57 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝑝↑1) = 𝑝) |
101 | 100 | breq2d 5042 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (𝐵 < (𝑝↑1) ↔ 𝐵 < 𝑝)) |
102 | 101 | notbid 321 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → (¬
𝐵 < (𝑝↑1) ↔ ¬ 𝐵 < 𝑝)) |
103 | 99, 102 | bitrd 282 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → ((𝑝↑1) ≤ 𝐵 ↔ ¬ 𝐵 < 𝑝)) |
104 | 97, 103 | sylibd 242 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝑝 ∈ ℙ ∧ 𝐵 ∈ ℕ) → ((𝑝↑1) ∥ 𝐵 → ¬ 𝐵 < 𝑝)) |
105 | 104 | ancoms 462 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ) → ((𝑝↑1) ∥ 𝐵 → ¬ 𝐵 < 𝑝)) |
106 | 105 | con2d 136 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝐵 < 𝑝 → ¬ (𝑝↑1) ∥ 𝐵)) |
107 | 106 | 3impia 1118 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐵 < 𝑝) → ¬ (𝑝↑1) ∥ 𝐵) |
108 | 95, 107 | jcnd 166 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐵 < 𝑝) → ¬ ((𝑝↑1) ∥ 0 → (𝑝↑1) ∥ 𝐵)) |
109 | | biimp 218 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑝↑1) ∥ 0 ↔ (𝑝↑1) ∥ 𝐵) → ((𝑝↑1) ∥ 0 → (𝑝↑1) ∥ 𝐵)) |
110 | 108, 109 | nsyl 142 |
. . . . . . . . . . . . . . 15
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐵 < 𝑝) → ¬ ((𝑝↑1) ∥ 0 ↔ (𝑝↑1) ∥ 𝐵)) |
111 | 69 | breq1d 5040 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 = 1 → ((𝑝↑𝑛) ∥ 𝐵 ↔ (𝑝↑1) ∥ 𝐵)) |
112 | 71, 111 | bibi12d 349 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 = 1 → (((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ((𝑝↑1) ∥ 0 ↔ (𝑝↑1) ∥ 𝐵))) |
113 | 112 | notbid 321 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 1 → (¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ¬ ((𝑝↑1) ∥ 0 ↔ (𝑝↑1) ∥ 𝐵))) |
114 | 113 | rspcev 3526 |
. . . . . . . . . . . . . . 15
⊢ ((1
∈ ℕ ∧ ¬ ((𝑝↑1) ∥ 0 ↔ (𝑝↑1) ∥ 𝐵)) → ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
115 | 39, 110, 114 | sylancr 590 |
. . . . . . . . . . . . . 14
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ ∧ 𝐵 < 𝑝) → ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
116 | 115 | 3expia 1122 |
. . . . . . . . . . . . 13
⊢ ((𝐵 ∈ ℕ ∧ 𝑝 ∈ ℙ) → (𝐵 < 𝑝 → ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
117 | 116 | reximdva 3184 |
. . . . . . . . . . . 12
⊢ (𝐵 ∈ ℕ →
(∃𝑝 ∈ ℙ
𝐵 < 𝑝 → ∃𝑝 ∈ ℙ ∃𝑛 ∈ ℕ ¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
118 | 94, 117 | mpd 15 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ ℕ →
∃𝑝 ∈ ℙ
∃𝑛 ∈ ℕ
¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
119 | | rexnal2 3170 |
. . . . . . . . . . 11
⊢
(∃𝑝 ∈
ℙ ∃𝑛 ∈
ℕ ¬ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ¬ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
120 | 118, 119 | sylib 221 |
. . . . . . . . . 10
⊢ (𝐵 ∈ ℕ → ¬
∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵)) |
121 | 120 | imim2i 16 |
. . . . . . . . 9
⊢ ((¬
𝐵 = 0 → 𝐵 ∈ ℕ) → (¬
𝐵 = 0 → ¬
∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
122 | 93, 121 | sylbi 220 |
. . . . . . . 8
⊢ (𝐵 ∈ ℕ0
→ (¬ 𝐵 = 0 →
¬ ∀𝑝 ∈
ℙ ∀𝑛 ∈
ℕ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
123 | 122 | con4d 115 |
. . . . . . 7
⊢ (𝐵 ∈ ℕ0
→ (∀𝑝 ∈
ℙ ∀𝑛 ∈
ℕ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐵 = 0)) |
124 | | eqcom 2745 |
. . . . . . 7
⊢ (𝐵 = 0 ↔ 0 = 𝐵) |
125 | 123, 124 | syl6ib 254 |
. . . . . 6
⊢ (𝐵 ∈ ℕ0
→ (∀𝑝 ∈
ℙ ∀𝑛 ∈
ℕ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵) → 0 = 𝐵)) |
126 | | breq2 5034 |
. . . . . . . . 9
⊢ (𝐴 = 0 → ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 0)) |
127 | 126 | bibi1d 347 |
. . . . . . . 8
⊢ (𝐴 = 0 → (((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
128 | 127 | 2ralbidv 3111 |
. . . . . . 7
⊢ (𝐴 = 0 → (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵))) |
129 | | eqeq1 2742 |
. . . . . . 7
⊢ (𝐴 = 0 → (𝐴 = 𝐵 ↔ 0 = 𝐵)) |
130 | 128, 129 | imbi12d 348 |
. . . . . 6
⊢ (𝐴 = 0 → ((∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵) ↔ (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 0 ↔ (𝑝↑𝑛) ∥ 𝐵) → 0 = 𝐵))) |
131 | 125, 130 | syl5ibr 249 |
. . . . 5
⊢ (𝐴 = 0 → (𝐵 ∈ ℕ0 →
(∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
132 | 90, 131 | jaoi 856 |
. . . 4
⊢ ((𝐴 ∈ ℕ ∨ 𝐴 = 0) → (𝐵 ∈ ℕ0 →
(∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵))) |
133 | 132 | imp 410 |
. . 3
⊢ (((𝐴 ∈ ℕ ∨ 𝐴 = 0) ∧ 𝐵 ∈ ℕ0) →
(∀𝑝 ∈ ℙ
∀𝑛 ∈ ℕ
((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵)) |
134 | 4, 133 | sylanb 584 |
. 2
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵) → 𝐴 = 𝐵)) |
135 | 3, 134 | impbid2 229 |
1
⊢ ((𝐴 ∈ ℕ0
∧ 𝐵 ∈
ℕ0) → (𝐴 = 𝐵 ↔ ∀𝑝 ∈ ℙ ∀𝑛 ∈ ℕ ((𝑝↑𝑛) ∥ 𝐴 ↔ (𝑝↑𝑛) ∥ 𝐵))) |