Step | Hyp | Ref
| Expression |
1 | | elfz1end 12371 |
. . 3
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) |
2 | 1 | biimpi 206 |
. 2
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴)) |
3 | | oveq2 6658 |
. . . 4
⊢ (𝑛 = 1 → (1...𝑛) = (1...1)) |
4 | 3 | raleqdv 3144 |
. . 3
⊢ (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑)) |
5 | | oveq2 6658 |
. . . 4
⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘)) |
6 | 5 | raleqdv 3144 |
. . 3
⊢ (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑)) |
7 | | oveq2 6658 |
. . . 4
⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1))) |
8 | 7 | raleqdv 3144 |
. . 3
⊢ (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑)) |
9 | | oveq2 6658 |
. . . 4
⊢ (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴)) |
10 | 9 | raleqdv 3144 |
. . 3
⊢ (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑)) |
11 | | prmind.6 |
. . . . 5
⊢ 𝜓 |
12 | | elfz1eq 12352 |
. . . . . 6
⊢ (𝑥 ∈ (1...1) → 𝑥 = 1) |
13 | | prmind.1 |
. . . . . 6
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
14 | 12, 13 | syl 17 |
. . . . 5
⊢ (𝑥 ∈ (1...1) → (𝜑 ↔ 𝜓)) |
15 | 11, 14 | mpbiri 248 |
. . . 4
⊢ (𝑥 ∈ (1...1) → 𝜑) |
16 | 15 | rgen 2922 |
. . 3
⊢
∀𝑥 ∈
(1...1)𝜑 |
17 | | peano2nn 11032 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
18 | 17 | ad2antrr 762 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ) |
19 | 18 | nncnd 11036 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ) |
20 | | elfzuz 12338 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈
(ℤ≥‘2)) |
21 | 20 | ad2antrl 764 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈
(ℤ≥‘2)) |
22 | | eluz2nn 11726 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈
(ℤ≥‘2) → 𝑦 ∈ ℕ) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℕ) |
24 | 23 | nncnd 11036 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ) |
25 | 23 | nnne0d 11065 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0) |
26 | 19, 24, 25 | divcan2d 10803 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1)) |
27 | | simprr 796 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1)) |
28 | 23 | nnzd 11481 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ) |
29 | 18 | nnzd 11481 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ) |
30 | | dvdsval2 14986 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) →
(𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ)) |
31 | 28, 25, 29, 30 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ)) |
32 | 27, 31 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ) |
33 | 24 | mulid2d 10058 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦) |
34 | | elfzle2 12345 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1)) |
35 | 34 | ad2antrl 764 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1)) |
36 | | nncn 11028 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
37 | 36 | ad2antrr 762 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ) |
38 | | ax-1cn 9994 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ |
39 | | pncan 10287 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 + 1)
− 1) = 𝑘) |
40 | 37, 38, 39 | sylancl 694 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘) |
41 | 35, 40 | breqtrd 4679 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ 𝑘) |
42 | | nnz 11399 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℤ) |
43 | 42 | ad2antrr 762 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ) |
44 | | zleltp1 11428 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1))) |
45 | 28, 43, 44 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1))) |
46 | 41, 45 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1)) |
47 | 33, 46 | eqbrtrd 4675 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1)) |
48 | | 1red 10055 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈
ℝ) |
49 | 18 | nnred 11035 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ) |
50 | 23 | nnred 11035 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ) |
51 | 23 | nngt0d 11064 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦) |
52 | | ltmuldiv 10896 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℝ ∧ (𝑘 +
1) ∈ ℝ ∧ (𝑦
∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦))) |
53 | 48, 49, 50, 51, 52 | syl112anc 1330 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦))) |
54 | 47, 53 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦)) |
55 | | eluz2b1 11759 |
. . . . . . . . . . . 12
⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ≥‘2)
↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 <
((𝑘 + 1) / 𝑦))) |
56 | 32, 54, 55 | sylanbrc 698 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈
(ℤ≥‘2)) |
57 | | fznn 12408 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℤ → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘))) |
58 | 43, 57 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘))) |
59 | 23, 41, 58 | mpbir2and 957 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘)) |
60 | | simplr 792 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑) |
61 | | prmind.2 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
62 | 61 | rspcv 3305 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (1...𝑘) → (∀𝑥 ∈ (1...𝑘)𝜑 → 𝜒)) |
63 | 59, 60, 62 | sylc 65 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒) |
64 | 18 | nnrpd 11870 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈
ℝ+) |
65 | 23 | nnrpd 11870 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ+) |
66 | 64, 65 | rpdivcld 11889 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈
ℝ+) |
67 | 66 | rpgt0d 11875 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦)) |
68 | | elnnz 11387 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦))) |
69 | 32, 67, 68 | sylanbrc 698 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ) |
70 | 18 | nnne0d 11065 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ≠ 0) |
71 | 19, 70 | dividd 10799 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1) |
72 | | eluz2b2 11761 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈
(ℤ≥‘2) ↔ (𝑦 ∈ ℕ ∧ 1 < 𝑦)) |
73 | 72 | simprbi 480 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈
(ℤ≥‘2) → 1 < 𝑦) |
74 | 21, 73 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦) |
75 | 71, 74 | eqbrtrd 4675 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦) |
76 | 18 | nngt0d 11064 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1)) |
77 | | ltdiv23 10914 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 + 1) ∈ ℝ ∧
((𝑘 + 1) ∈ ℝ
∧ 0 < (𝑘 + 1)) ∧
(𝑦 ∈ ℝ ∧ 0
< 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
78 | 49, 49, 76, 50, 51, 77 | syl122anc 1335 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
79 | 75, 78 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1)) |
80 | | zleltp1 11428 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
81 | 32, 43, 80 | syl2anc 693 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
82 | 79, 81 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘) |
83 | | fznn 12408 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘))) |
84 | 43, 83 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘))) |
85 | 69, 82, 84 | mpbir2and 957 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘)) |
86 | | prmind.3 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) |
87 | 86 | cbvralv 3171 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
(1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃) |
88 | 60, 87 | sylib 208 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃) |
89 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑧 ∈ V |
90 | 89, 86 | sbcie 3470 |
. . . . . . . . . . . . . . 15
⊢
([𝑧 / 𝑥]𝜑 ↔ 𝜃) |
91 | | dfsbcq 3437 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
92 | 90, 91 | syl5bbr 274 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
93 | 92 | rspcv 3305 |
. . . . . . . . . . . . 13
⊢ (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) → (∀𝑧 ∈ (1...𝑘)𝜃 → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
94 | 85, 88, 93 | sylc 65 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) |
95 | 63, 94 | jca 554 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
96 | 92 | anbi2d 740 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))) |
97 | | ovex 6678 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 · 𝑧) ∈ V |
98 | | prmind.4 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) |
99 | 97, 98 | sbcie 3470 |
. . . . . . . . . . . . . . 15
⊢
([(𝑦 ·
𝑧) / 𝑥]𝜑 ↔ 𝜏) |
100 | | oveq2 6658 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦))) |
101 | 100 | sbceq1d 3440 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)) |
102 | 99, 101 | syl5bbr 274 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜏 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)) |
103 | 96, 102 | imbi12d 334 |
. . . . . . . . . . . . 13
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒 ∧ 𝜃) → 𝜏) ↔ ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))) |
104 | 103 | imbi2d 330 |
. . . . . . . . . . . 12
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝑦 ∈ (ℤ≥‘2)
→ ((𝜒 ∧ 𝜃) → 𝜏)) ↔ (𝑦 ∈ (ℤ≥‘2)
→ ((𝜒 ∧
[((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)))) |
105 | | prmind2.8 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈
(ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((𝜒 ∧ 𝜃) → 𝜏)) |
106 | 105 | expcom 451 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈
(ℤ≥‘2) → (𝑦 ∈ (ℤ≥‘2)
→ ((𝜒 ∧ 𝜃) → 𝜏))) |
107 | 104, 106 | vtoclga 3272 |
. . . . . . . . . . 11
⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ≥‘2)
→ (𝑦 ∈
(ℤ≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))) |
108 | 56, 21, 95, 107 | syl3c 66 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑) |
109 | 26, 108 | sbceq1dd 3441 |
. . . . . . . . 9
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑) |
110 | 109 | rexlimdvaa 3032 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) |
111 | | ralnex 2992 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
(2...((𝑘 + 1) − 1))
¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)) |
112 | | simpl 473 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ) |
113 | | elnnuz 11724 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈
(ℤ≥‘1)) |
114 | 112, 113 | sylib 208 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈
(ℤ≥‘1)) |
115 | | eluzp1p1 11713 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈
(ℤ≥‘1) → (𝑘 + 1) ∈ (ℤ≥‘(1
+ 1))) |
116 | 114, 115 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ≥‘(1
+ 1))) |
117 | | df-2 11079 |
. . . . . . . . . . . . 13
⊢ 2 = (1 +
1) |
118 | 117 | fveq2i 6194 |
. . . . . . . . . . . 12
⊢
(ℤ≥‘2) = (ℤ≥‘(1 +
1)) |
119 | 116, 118 | syl6eleqr 2712 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈
(ℤ≥‘2)) |
120 | | isprm3 15396 |
. . . . . . . . . . . 12
⊢ ((𝑘 + 1) ∈ ℙ ↔
((𝑘 + 1) ∈
(ℤ≥‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1))) |
121 | 120 | baibr 945 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈
(ℤ≥‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ)) |
122 | 119, 121 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ)) |
123 | | simpr 477 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑) |
124 | 61 | cbvralv 3171 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
(1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒) |
125 | 123, 124 | sylib 208 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒) |
126 | 112 | nncnd 11036 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ) |
127 | 126, 38, 39 | sylancl 694 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘) |
128 | 127 | oveq2d 6666 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘)) |
129 | 128 | raleqdv 3144 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)) |
130 | 125, 129 | mpbird 247 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒) |
131 | | nfcv 2764 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑘 + 1) |
132 | | nfv 1843 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 |
133 | | nfsbc1v 3455 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑 |
134 | 132, 133 | nfim 1825 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑) |
135 | | oveq1 6657 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1)) |
136 | 135 | oveq2d 6666 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1))) |
137 | 136 | raleqdv 3144 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)) |
138 | | sbceq1a 3446 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
139 | 137, 138 | imbi12d 334 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑))) |
140 | | prmind2.7 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℙ ∧
∀𝑦 ∈
(1...(𝑥 − 1))𝜒) → 𝜑) |
141 | 140 | ex 450 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℙ →
(∀𝑦 ∈
(1...(𝑥 − 1))𝜒 → 𝜑)) |
142 | 131, 134,
139, 141 | vtoclgaf 3271 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈ ℙ →
(∀𝑦 ∈
(1...((𝑘 + 1) −
1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)) |
143 | 130, 142 | syl5com 31 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑)) |
144 | 122, 143 | sylbid 230 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) |
145 | 111, 144 | syl5bir 233 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) |
146 | 110, 145 | pm2.61d 170 |
. . . . . . 7
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑) |
147 | 146 | ex 450 |
. . . . . 6
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → [(𝑘 + 1) / 𝑥]𝜑)) |
148 | | ralsnsg 4216 |
. . . . . . 7
⊢ ((𝑘 + 1) ∈ ℕ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
149 | 17, 148 | syl 17 |
. . . . . 6
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
150 | 147, 149 | sylibrd 249 |
. . . . 5
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)) |
151 | 150 | ancld 576 |
. . . 4
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))) |
152 | | fzsuc 12388 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) |
153 | 113, 152 | sylbi 207 |
. . . . . 6
⊢ (𝑘 ∈ ℕ →
(1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) |
154 | 153 | raleqdv 3144 |
. . . . 5
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑)) |
155 | | ralunb 3794 |
. . . . 5
⊢
(∀𝑥 ∈
((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)) |
156 | 154, 155 | syl6bb 276 |
. . . 4
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))) |
157 | 151, 156 | sylibrd 249 |
. . 3
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑)) |
158 | 4, 6, 8, 10, 16, 157 | nnind 11038 |
. 2
⊢ (𝐴 ∈ ℕ →
∀𝑥 ∈ (1...𝐴)𝜑) |
159 | | prmind.5 |
. . 3
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) |
160 | 159 | rspcv 3305 |
. 2
⊢ (𝐴 ∈ (1...𝐴) → (∀𝑥 ∈ (1...𝐴)𝜑 → 𝜂)) |
161 | 2, 158, 160 | sylc 65 |
1
⊢ (𝐴 ∈ ℕ → 𝜂) |