Step | Hyp | Ref
| Expression |
1 | | prmind.5 |
. 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) |
2 | | oveq2 6823 |
. . . 4
⊢ (𝑛 = 1 → (1...𝑛) = (1...1)) |
3 | 2 | raleqdv 3284 |
. . 3
⊢ (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑)) |
4 | | oveq2 6823 |
. . . 4
⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘)) |
5 | 4 | raleqdv 3284 |
. . 3
⊢ (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑)) |
6 | | oveq2 6823 |
. . . 4
⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1))) |
7 | 6 | raleqdv 3284 |
. . 3
⊢ (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑)) |
8 | | oveq2 6823 |
. . . 4
⊢ (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴)) |
9 | 8 | raleqdv 3284 |
. . 3
⊢ (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑)) |
10 | | prmind.6 |
. . . . 5
⊢ 𝜓 |
11 | | elfz1eq 12566 |
. . . . . 6
⊢ (𝑥 ∈ (1...1) → 𝑥 = 1) |
12 | | prmind.1 |
. . . . . 6
⊢ (𝑥 = 1 → (𝜑 ↔ 𝜓)) |
13 | 11, 12 | syl 17 |
. . . . 5
⊢ (𝑥 ∈ (1...1) → (𝜑 ↔ 𝜓)) |
14 | 10, 13 | mpbiri 248 |
. . . 4
⊢ (𝑥 ∈ (1...1) → 𝜑) |
15 | 14 | rgen 3061 |
. . 3
⊢
∀𝑥 ∈
(1...1)𝜑 |
16 | | peano2nn 11245 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) |
17 | 16 | ad2antrr 764 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ) |
18 | 17 | nncnd 11249 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ) |
19 | | elfzuz 12552 |
. . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈
(ℤ≥‘2)) |
20 | 19 | ad2antrl 766 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈
(ℤ≥‘2)) |
21 | | eluz2nn 11940 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈
(ℤ≥‘2) → 𝑦 ∈ ℕ) |
22 | 20, 21 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℕ) |
23 | 22 | nncnd 11249 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ) |
24 | 22 | nnne0d 11278 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0) |
25 | 18, 23, 24 | divcan2d 11016 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1)) |
26 | | simprr 813 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1)) |
27 | 22 | nnzd 11694 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ) |
28 | 17 | nnzd 11694 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ) |
29 | | dvdsval2 15206 |
. . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) →
(𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ)) |
30 | 27, 24, 28, 29 | syl3anc 1477 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ)) |
31 | 26, 30 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ) |
32 | 23 | mulid2d 10271 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦) |
33 | | elfzle2 12559 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1)) |
34 | 33 | ad2antrl 766 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1)) |
35 | | nncn 11241 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) |
36 | 35 | ad2antrr 764 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ) |
37 | | ax-1cn 10207 |
. . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ |
38 | | pncan 10500 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 + 1)
− 1) = 𝑘) |
39 | 36, 37, 38 | sylancl 697 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘) |
40 | 34, 39 | breqtrd 4831 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ 𝑘) |
41 | | nnz 11612 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℤ) |
42 | 41 | ad2antrr 764 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ) |
43 | | zleltp1 11641 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1))) |
44 | 27, 42, 43 | syl2anc 696 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1))) |
45 | 40, 44 | mpbid 222 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1)) |
46 | 32, 45 | eqbrtrd 4827 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1)) |
47 | | 1red 10268 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈
ℝ) |
48 | 17 | nnred 11248 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ) |
49 | 22 | nnred 11248 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ) |
50 | 22 | nngt0d 11277 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦) |
51 | | ltmuldiv 11109 |
. . . . . . . . . . . . . 14
⊢ ((1
∈ ℝ ∧ (𝑘 +
1) ∈ ℝ ∧ (𝑦
∈ ℝ ∧ 0 < 𝑦)) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦))) |
52 | 47, 48, 49, 50, 51 | syl112anc 1481 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((1 · 𝑦) < (𝑘 + 1) ↔ 1 < ((𝑘 + 1) / 𝑦))) |
53 | 46, 52 | mpbid 222 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦)) |
54 | | eluz2b1 11973 |
. . . . . . . . . . . 12
⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ≥‘2)
↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 <
((𝑘 + 1) / 𝑦))) |
55 | 31, 53, 54 | sylanbrc 701 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈
(ℤ≥‘2)) |
56 | | prmind.2 |
. . . . . . . . . . . . 13
⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒)) |
57 | | simplr 809 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑) |
58 | | fznn 12622 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℤ → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘))) |
59 | 42, 58 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ∈ (1...𝑘) ↔ (𝑦 ∈ ℕ ∧ 𝑦 ≤ 𝑘))) |
60 | 22, 40, 59 | mpbir2and 995 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘)) |
61 | 56, 57, 60 | rspcdva 3456 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒) |
62 | | vex 3344 |
. . . . . . . . . . . . . . 15
⊢ 𝑧 ∈ V |
63 | | prmind.3 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) |
64 | 62, 63 | sbcie 3612 |
. . . . . . . . . . . . . 14
⊢
([𝑧 / 𝑥]𝜑 ↔ 𝜃) |
65 | | dfsbcq 3579 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
66 | 64, 65 | syl5bbr 274 |
. . . . . . . . . . . . 13
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
67 | 63 | cbvralv 3311 |
. . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
(1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃) |
68 | 57, 67 | sylib 208 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃) |
69 | 17 | nnrpd 12084 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈
ℝ+) |
70 | 22 | nnrpd 12084 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ+) |
71 | 69, 70 | rpdivcld 12103 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈
ℝ+) |
72 | 71 | rpgt0d 12089 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦)) |
73 | | elnnz 11600 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦))) |
74 | 31, 72, 73 | sylanbrc 701 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ) |
75 | 17 | nnne0d 11278 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ≠ 0) |
76 | 18, 75 | dividd 11012 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1) |
77 | | eluz2b2 11975 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 ∈
(ℤ≥‘2) ↔ (𝑦 ∈ ℕ ∧ 1 < 𝑦)) |
78 | 77 | simprbi 483 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈
(ℤ≥‘2) → 1 < 𝑦) |
79 | 20, 78 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦) |
80 | 76, 79 | eqbrtrd 4827 |
. . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦) |
81 | 17 | nngt0d 11277 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1)) |
82 | | ltdiv23 11127 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑘 + 1) ∈ ℝ ∧
((𝑘 + 1) ∈ ℝ
∧ 0 < (𝑘 + 1)) ∧
(𝑦 ∈ ℝ ∧ 0
< 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
83 | 48, 48, 81, 49, 50, 82 | syl122anc 1486 |
. . . . . . . . . . . . . . . 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))) |
84 | 80, 83 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1)) |
85 | | zleltp1 11641 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
86 | 31, 42, 85 | syl2anc 696 |
. . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) |
87 | 84, 86 | mpbird 247 |
. . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘) |
88 | | fznn 12622 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘))) |
89 | 42, 88 | 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))) |
90 | 74, 87, 89 | mpbir2and 995 |
. . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘)) |
91 | 66, 68, 90 | rspcdva 3456 |
. . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) |
92 | 61, 91 | jca 555 |
. . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) |
93 | 66 | anbi2d 742 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))) |
94 | | ovex 6843 |
. . . . . . . . . . . . . . . 16
⊢ (𝑦 · 𝑧) ∈ V |
95 | | prmind.4 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) |
96 | 94, 95 | sbcie 3612 |
. . . . . . . . . . . . . . 15
⊢
([(𝑦 ·
𝑧) / 𝑥]𝜑 ↔ 𝜏) |
97 | | oveq2 6823 |
. . . . . . . . . . . . . . . 16
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦))) |
98 | 97 | sbceq1d 3582 |
. . . . . . . . . . . . . . 15
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)) |
99 | 96, 98 | syl5bbr 274 |
. . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜏 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)) |
100 | 93, 99 | imbi12d 333 |
. . . . . . . . . . . . 13
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒 ∧ 𝜃) → 𝜏) ↔ ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))) |
101 | 100 | imbi2d 329 |
. . . . . . . . . . . 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)))) |
102 | | prmind2.8 |
. . . . . . . . . . . . 13
⊢ ((𝑦 ∈
(ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((𝜒 ∧ 𝜃) → 𝜏)) |
103 | 102 | expcom 450 |
. . . . . . . . . . . 12
⊢ (𝑧 ∈
(ℤ≥‘2) → (𝑦 ∈ (ℤ≥‘2)
→ ((𝜒 ∧ 𝜃) → 𝜏))) |
104 | 101, 103 | vtoclga 3413 |
. . . . . . . . . . 11
⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ≥‘2)
→ (𝑦 ∈
(ℤ≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))) |
105 | 55, 20, 92, 104 | syl3c 66 |
. . . . . . . . . 10
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑) |
106 | 25, 105 | sbceq1dd 3583 |
. . . . . . . . 9
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑) |
107 | 106 | rexlimdvaa 3171 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) |
108 | | ralnex 3131 |
. . . . . . . . 9
⊢
(∀𝑦 ∈
(2...((𝑘 + 1) − 1))
¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)) |
109 | | simpl 474 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ) |
110 | | elnnuz 11938 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈
(ℤ≥‘1)) |
111 | 109, 110 | sylib 208 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈
(ℤ≥‘1)) |
112 | | eluzp1p1 11926 |
. . . . . . . . . . . . 13
⊢ (𝑘 ∈
(ℤ≥‘1) → (𝑘 + 1) ∈ (ℤ≥‘(1
+ 1))) |
113 | 111, 112 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ≥‘(1
+ 1))) |
114 | | df-2 11292 |
. . . . . . . . . . . . 13
⊢ 2 = (1 +
1) |
115 | 114 | fveq2i 6357 |
. . . . . . . . . . . 12
⊢
(ℤ≥‘2) = (ℤ≥‘(1 +
1)) |
116 | 113, 115 | syl6eleqr 2851 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈
(ℤ≥‘2)) |
117 | | isprm3 15619 |
. . . . . . . . . . . 12
⊢ ((𝑘 + 1) ∈ ℙ ↔
((𝑘 + 1) ∈
(ℤ≥‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1))) |
118 | 117 | baibr 983 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈
(ℤ≥‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ)) |
119 | 116, 118 | syl 17 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ)) |
120 | | simpr 479 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑) |
121 | 56 | cbvralv 3311 |
. . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
(1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒) |
122 | 120, 121 | sylib 208 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒) |
123 | 109 | nncnd 11249 |
. . . . . . . . . . . . . . 15
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ) |
124 | 123, 37, 38 | sylancl 697 |
. . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘) |
125 | 124 | oveq2d 6831 |
. . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘)) |
126 | 125 | raleqdv 3284 |
. . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 ↔ ∀𝑦 ∈ (1...𝑘)𝜒)) |
127 | 122, 126 | mpbird 247 |
. . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒) |
128 | | nfcv 2903 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑘 + 1) |
129 | | nfv 1993 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 |
130 | | nfsbc1v 3597 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑 |
131 | 129, 130 | nfim 1975 |
. . . . . . . . . . . 12
⊢
Ⅎ𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑) |
132 | | oveq1 6822 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1)) |
133 | 132 | oveq2d 6831 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1))) |
134 | 133 | raleqdv 3284 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)) |
135 | | sbceq1a 3588 |
. . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
136 | 134, 135 | imbi12d 333 |
. . . . . . . . . . . 12
⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑))) |
137 | | prmind2.7 |
. . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℙ ∧
∀𝑦 ∈
(1...(𝑥 − 1))𝜒) → 𝜑) |
138 | 137 | ex 449 |
. . . . . . . . . . . 12
⊢ (𝑥 ∈ ℙ →
(∀𝑦 ∈
(1...(𝑥 − 1))𝜒 → 𝜑)) |
139 | 128, 131,
136, 138 | vtoclgaf 3412 |
. . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈ ℙ →
(∀𝑦 ∈
(1...((𝑘 + 1) −
1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)) |
140 | 127, 139 | syl5com 31 |
. . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑)) |
141 | 119, 140 | sylbid 230 |
. . . . . . . . 9
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) |
142 | 108, 141 | syl5bir 233 |
. . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) |
143 | 107, 142 | pm2.61d 170 |
. . . . . . 7
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑) |
144 | 143 | ex 449 |
. . . . . 6
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → [(𝑘 + 1) / 𝑥]𝜑)) |
145 | | ralsnsg 4361 |
. . . . . . 7
⊢ ((𝑘 + 1) ∈ ℕ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
146 | 16, 145 | syl 17 |
. . . . . 6
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) |
147 | 144, 146 | sylibrd 249 |
. . . . 5
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)) |
148 | 147 | ancld 577 |
. . . 4
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))) |
149 | | fzsuc 12602 |
. . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) |
150 | 110, 149 | sylbi 207 |
. . . . . 6
⊢ (𝑘 ∈ ℕ →
(1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) |
151 | 150 | raleqdv 3284 |
. . . . 5
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑)) |
152 | | ralunb 3938 |
. . . . 5
⊢
(∀𝑥 ∈
((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)) |
153 | 151, 152 | syl6bb 276 |
. . . 4
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))) |
154 | 148, 153 | sylibrd 249 |
. . 3
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑)) |
155 | 3, 5, 7, 9, 15, 154 | nnind 11251 |
. 2
⊢ (𝐴 ∈ ℕ →
∀𝑥 ∈ (1...𝐴)𝜑) |
156 | | elfz1end 12585 |
. . 3
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) |
157 | 156 | biimpi 206 |
. 2
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴)) |
158 | 1, 155, 157 | rspcdva 3456 |
1
⊢ (𝐴 ∈ ℕ → 𝜂) |