| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prmind.5 | . 2
⊢ (𝑥 = 𝐴 → (𝜑 ↔ 𝜂)) | 
| 2 |  | oveq2 7439 | . . . 4
⊢ (𝑛 = 1 → (1...𝑛) = (1...1)) | 
| 3 | 2 | raleqdv 3326 | . . 3
⊢ (𝑛 = 1 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...1)𝜑)) | 
| 4 |  | oveq2 7439 | . . . 4
⊢ (𝑛 = 𝑘 → (1...𝑛) = (1...𝑘)) | 
| 5 | 4 | raleqdv 3326 | . . 3
⊢ (𝑛 = 𝑘 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝑘)𝜑)) | 
| 6 |  | oveq2 7439 | . . . 4
⊢ (𝑛 = (𝑘 + 1) → (1...𝑛) = (1...(𝑘 + 1))) | 
| 7 | 6 | raleqdv 3326 | . . 3
⊢ (𝑛 = (𝑘 + 1) → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...(𝑘 + 1))𝜑)) | 
| 8 |  | oveq2 7439 | . . . 4
⊢ (𝑛 = 𝐴 → (1...𝑛) = (1...𝐴)) | 
| 9 | 8 | raleqdv 3326 | . . 3
⊢ (𝑛 = 𝐴 → (∀𝑥 ∈ (1...𝑛)𝜑 ↔ ∀𝑥 ∈ (1...𝐴)𝜑)) | 
| 10 |  | prmind.6 | . . . . 5
⊢ 𝜓 | 
| 11 |  | elfz1eq 13575 | . . . . . 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 258 | . . . 4
⊢ (𝑥 ∈ (1...1) → 𝜑) | 
| 15 | 14 | rgen 3063 | . . 3
⊢
∀𝑥 ∈
(1...1)𝜑 | 
| 16 |  | peano2nn 12278 | . . . . . . . . . . . . 13
⊢ (𝑘 ∈ ℕ → (𝑘 + 1) ∈
ℕ) | 
| 17 | 16 | ad2antrr 726 | . . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℕ) | 
| 18 | 17 | nncnd 12282 | . . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℂ) | 
| 19 |  | elfzuz 13560 | . . . . . . . . . . . . . 14
⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ∈
(ℤ≥‘2)) | 
| 20 | 19 | ad2antrl 728 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈
(ℤ≥‘2)) | 
| 21 |  | eluz2nn 12924 | . . . . . . . . . . . . 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 12282 | . . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℂ) | 
| 24 | 22 | nnne0d 12316 | . . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≠ 0) | 
| 25 | 18, 23, 24 | divcan2d 12045 | . . . . . . . . . 10
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 · ((𝑘 + 1) / 𝑦)) = (𝑘 + 1)) | 
| 26 |  | simprr 773 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∥ (𝑘 + 1)) | 
| 27 | 22 | nnzd 12640 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℤ) | 
| 28 | 17 | nnzd 12640 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℤ) | 
| 29 |  | dvdsval2 16293 | . . . . . . . . . . . . . 14
⊢ ((𝑦 ∈ ℤ ∧ 𝑦 ≠ 0 ∧ (𝑘 + 1) ∈ ℤ) →
(𝑦 ∥ (𝑘 + 1) ↔ ((𝑘 + 1) / 𝑦) ∈ ℤ)) | 
| 30 | 27, 24, 28, 29 | syl3anc 1373 | . . . . . . . . . . . . 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 232 | . . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℤ) | 
| 32 | 23 | mullidd 11279 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) = 𝑦) | 
| 33 |  | elfzle2 13568 | . . . . . . . . . . . . . . . . 17
⊢ (𝑦 ∈ (2...((𝑘 + 1) − 1)) → 𝑦 ≤ ((𝑘 + 1) − 1)) | 
| 34 | 33 | ad2antrl 728 | . . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ ((𝑘 + 1) − 1)) | 
| 35 |  | nncn 12274 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℂ) | 
| 36 | 35 | ad2antrr 726 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℂ) | 
| 37 |  | ax-1cn 11213 | . . . . . . . . . . . . . . . . 17
⊢ 1 ∈
ℂ | 
| 38 |  | pncan 11514 | . . . . . . . . . . . . . . . . 17
⊢ ((𝑘 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝑘 + 1)
− 1) = 𝑘) | 
| 39 | 36, 37, 38 | sylancl 586 | . . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) − 1) = 𝑘) | 
| 40 | 34, 39 | breqtrd 5169 | . . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ≤ 𝑘) | 
| 41 |  | nnz 12634 | . . . . . . . . . . . . . . . . 17
⊢ (𝑘 ∈ ℕ → 𝑘 ∈
ℤ) | 
| 42 | 41 | ad2antrr 726 | . . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑘 ∈ ℤ) | 
| 43 |  | zleltp1 12668 | . . . . . . . . . . . . . . . 16
⊢ ((𝑦 ∈ ℤ ∧ 𝑘 ∈ ℤ) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1))) | 
| 44 | 27, 42, 43 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑦 ≤ 𝑘 ↔ 𝑦 < (𝑘 + 1))) | 
| 45 | 40, 44 | mpbid 232 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 < (𝑘 + 1)) | 
| 46 | 32, 45 | eqbrtrd 5165 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (1 · 𝑦) < (𝑘 + 1)) | 
| 47 |  | 1red 11262 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 ∈
ℝ) | 
| 48 | 17 | nnred 12281 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈ ℝ) | 
| 49 | 22 | nnred 12281 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ) | 
| 50 | 22 | nngt0d 12315 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < 𝑦) | 
| 51 |  | ltmuldiv 12141 | . . . . . . . . . . . . . 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 1376 | . . . . . . . . . . . . 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 232 | . . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < ((𝑘 + 1) / 𝑦)) | 
| 54 |  | eluz2b1 12961 | . . . . . . . . . . . 12
⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ≥‘2)
↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 1 <
((𝑘 + 1) / 𝑦))) | 
| 55 | 31, 53, 54 | sylanbrc 583 | . . . . . . . . . . 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 769 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑥 ∈ (1...𝑘)𝜑) | 
| 58 |  | fznn 13632 | . . . . . . . . . . . . . . 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 713 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ (1...𝑘)) | 
| 61 | 56, 57, 60 | rspcdva 3623 | . . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝜒) | 
| 62 |  | vex 3484 | . . . . . . . . . . . . . . 15
⊢ 𝑧 ∈ V | 
| 63 |  | prmind.3 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = 𝑧 → (𝜑 ↔ 𝜃)) | 
| 64 | 62, 63 | sbcie 3830 | . . . . . . . . . . . . . 14
⊢
([𝑧 / 𝑥]𝜑 ↔ 𝜃) | 
| 65 |  | dfsbcq 3790 | . . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([𝑧 / 𝑥]𝜑 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) | 
| 66 | 64, 65 | bitr3id 285 | . . . . . . . . . . . . 13
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜃 ↔ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) | 
| 67 | 63 | cbvralvw 3237 | . . . . . . . . . . . . . 14
⊢
(∀𝑥 ∈
(1...𝑘)𝜑 ↔ ∀𝑧 ∈ (1...𝑘)𝜃) | 
| 68 | 57, 67 | sylib 218 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ∀𝑧 ∈ (1...𝑘)𝜃) | 
| 69 | 17 | nnrpd 13075 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ∈
ℝ+) | 
| 70 | 22 | nnrpd 13075 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 𝑦 ∈ ℝ+) | 
| 71 | 69, 70 | rpdivcld 13094 | . . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈
ℝ+) | 
| 72 | 71 | rpgt0d 13080 | . . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < ((𝑘 + 1) / 𝑦)) | 
| 73 |  | elnnz 12623 | . . . . . . . . . . . . . . 15
⊢ (((𝑘 + 1) / 𝑦) ∈ ℕ ↔ (((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 0 < ((𝑘 + 1) / 𝑦))) | 
| 74 | 31, 72, 73 | sylanbrc 583 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ ℕ) | 
| 75 | 17 | nnne0d 12316 | . . . . . . . . . . . . . . . . . 18
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝑘 + 1) ≠ 0) | 
| 76 | 18, 75 | dividd 12041 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) = 1) | 
| 77 |  | eluz2gt1 12962 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑦 ∈
(ℤ≥‘2) → 1 < 𝑦) | 
| 78 | 20, 77 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 1 < 𝑦) | 
| 79 | 76, 78 | eqbrtrd 5165 | . . . . . . . . . . . . . . . 16
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / (𝑘 + 1)) < 𝑦) | 
| 80 | 17 | nngt0d 12315 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → 0 < (𝑘 + 1)) | 
| 81 |  | ltdiv23 12159 | . . . . . . . . . . . . . . . . 17
⊢ (((𝑘 + 1) ∈ ℝ ∧
((𝑘 + 1) ∈ ℝ
∧ 0 < (𝑘 + 1)) ∧
(𝑦 ∈ ℝ ∧ 0
< 𝑦)) → (((𝑘 + 1) / (𝑘 + 1)) < 𝑦 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) | 
| 82 | 48, 48, 80, 49, 50, 81 | syl122anc 1381 | . . . . . . . . . . . . . . . 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))) | 
| 83 | 79, 82 | mpbid 232 | . . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) < (𝑘 + 1)) | 
| 84 |  | zleltp1 12668 | . . . . . . . . . . . . . . . 16
⊢ ((((𝑘 + 1) / 𝑦) ∈ ℤ ∧ 𝑘 ∈ ℤ) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) | 
| 85 | 31, 42, 84 | syl2anc 584 | . . . . . . . . . . . . . . 15
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (((𝑘 + 1) / 𝑦) ≤ 𝑘 ↔ ((𝑘 + 1) / 𝑦) < (𝑘 + 1))) | 
| 86 | 83, 85 | mpbird 257 | . . . . . . . . . . . . . 14
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ≤ 𝑘) | 
| 87 |  | fznn 13632 | . . . . . . . . . . . . . . 15
⊢ (𝑘 ∈ ℤ → (((𝑘 + 1) / 𝑦) ∈ (1...𝑘) ↔ (((𝑘 + 1) / 𝑦) ∈ ℕ ∧ ((𝑘 + 1) / 𝑦) ≤ 𝑘))) | 
| 88 | 42, 87 | 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))) | 
| 89 | 74, 86, 88 | mpbir2and 713 | . . . . . . . . . . . . 13
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → ((𝑘 + 1) / 𝑦) ∈ (1...𝑘)) | 
| 90 | 66, 68, 89 | rspcdva 3623 | . . . . . . . . . . . 12
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) | 
| 91 | 61, 90 | jca 511 | . . . . . . . . . . 11
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑)) | 
| 92 | 66 | anbi2d 630 | . . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ((𝜒 ∧ 𝜃) ↔ (𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑))) | 
| 93 |  | ovex 7464 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 · 𝑧) ∈ V | 
| 94 |  | prmind.4 | . . . . . . . . . . . . . . . 16
⊢ (𝑥 = (𝑦 · 𝑧) → (𝜑 ↔ 𝜏)) | 
| 95 | 93, 94 | sbcie 3830 | . . . . . . . . . . . . . . 15
⊢
([(𝑦 ·
𝑧) / 𝑥]𝜑 ↔ 𝜏) | 
| 96 |  | oveq2 7439 | . . . . . . . . . . . . . . . 16
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝑦 · 𝑧) = (𝑦 · ((𝑘 + 1) / 𝑦))) | 
| 97 | 96 | sbceq1d 3793 | . . . . . . . . . . . . . . 15
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → ([(𝑦 · 𝑧) / 𝑥]𝜑 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)) | 
| 98 | 95, 97 | bitr3id 285 | . . . . . . . . . . . . . 14
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (𝜏 ↔ [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑)) | 
| 99 | 92, 98 | imbi12d 344 | . . . . . . . . . . . . 13
⊢ (𝑧 = ((𝑘 + 1) / 𝑦) → (((𝜒 ∧ 𝜃) → 𝜏) ↔ ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))) | 
| 100 | 99 | imbi2d 340 | . . . . . . . . . . . 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)))) | 
| 101 |  | prmind2.8 | . . . . . . . . . . . . 13
⊢ ((𝑦 ∈
(ℤ≥‘2) ∧ 𝑧 ∈ (ℤ≥‘2))
→ ((𝜒 ∧ 𝜃) → 𝜏)) | 
| 102 | 101 | expcom 413 | . . . . . . . . . . . 12
⊢ (𝑧 ∈
(ℤ≥‘2) → (𝑦 ∈ (ℤ≥‘2)
→ ((𝜒 ∧ 𝜃) → 𝜏))) | 
| 103 | 100, 102 | vtoclga 3577 | . . . . . . . . . . 11
⊢ (((𝑘 + 1) / 𝑦) ∈ (ℤ≥‘2)
→ (𝑦 ∈
(ℤ≥‘2) → ((𝜒 ∧ [((𝑘 + 1) / 𝑦) / 𝑥]𝜑) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑))) | 
| 104 | 55, 20, 91, 103 | syl3c 66 | . . . . . . . . . 10
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑦 · ((𝑘 + 1) / 𝑦)) / 𝑥]𝜑) | 
| 105 | 25, 104 | sbceq1dd 3794 | . . . . . . . . 9
⊢ (((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) ∧ (𝑦 ∈ (2...((𝑘 + 1) − 1)) ∧ 𝑦 ∥ (𝑘 + 1))) → [(𝑘 + 1) / 𝑥]𝜑) | 
| 106 | 105 | rexlimdvaa 3156 | . . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) | 
| 107 |  | ralnex 3072 | . . . . . . . . 9
⊢
(∀𝑦 ∈
(2...((𝑘 + 1) − 1))
¬ 𝑦 ∥ (𝑘 + 1) ↔ ¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1)) | 
| 108 |  | simpl 482 | . . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℕ) | 
| 109 |  | elnnuz 12922 | . . . . . . . . . . . . . 14
⊢ (𝑘 ∈ ℕ ↔ 𝑘 ∈
(ℤ≥‘1)) | 
| 110 | 108, 109 | sylib 218 | . . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈
(ℤ≥‘1)) | 
| 111 |  | eluzp1p1 12906 | . . . . . . . . . . . . 13
⊢ (𝑘 ∈
(ℤ≥‘1) → (𝑘 + 1) ∈ (ℤ≥‘(1
+ 1))) | 
| 112 | 110, 111 | syl 17 | . . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈ (ℤ≥‘(1
+ 1))) | 
| 113 |  | df-2 12329 | . . . . . . . . . . . . 13
⊢ 2 = (1 +
1) | 
| 114 | 113 | fveq2i 6909 | . . . . . . . . . . . 12
⊢
(ℤ≥‘2) = (ℤ≥‘(1 +
1)) | 
| 115 | 112, 114 | eleqtrrdi 2852 | . . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (𝑘 + 1) ∈
(ℤ≥‘2)) | 
| 116 |  | isprm3 16720 | . . . . . . . . . . . 12
⊢ ((𝑘 + 1) ∈ ℙ ↔
((𝑘 + 1) ∈
(ℤ≥‘2) ∧ ∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1))) | 
| 117 | 116 | baibr 536 | . . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈
(ℤ≥‘2) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ)) | 
| 118 | 115, 117 | syl 17 | . . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) ↔ (𝑘 + 1) ∈ ℙ)) | 
| 119 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑥 ∈ (1...𝑘)𝜑) | 
| 120 | 56 | cbvralvw 3237 | . . . . . . . . . . . . 13
⊢
(∀𝑥 ∈
(1...𝑘)𝜑 ↔ ∀𝑦 ∈ (1...𝑘)𝜒) | 
| 121 | 119, 120 | sylib 218 | . . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...𝑘)𝜒) | 
| 122 | 108 | nncnd 12282 | . . . . . . . . . . . . . 14
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → 𝑘 ∈ ℂ) | 
| 123 | 122, 37, 38 | sylancl 586 | . . . . . . . . . . . . 13
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) − 1) = 𝑘) | 
| 124 | 123 | oveq2d 7447 | . . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (1...((𝑘 + 1) − 1)) = (1...𝑘)) | 
| 125 | 121, 124 | raleqtrrdv 3330 | . . . . . . . . . . 11
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒) | 
| 126 |  | nfcv 2905 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥(𝑘 + 1) | 
| 127 |  | nfv 1914 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑥∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 | 
| 128 |  | nfsbc1v 3808 | . . . . . . . . . . . . 13
⊢
Ⅎ𝑥[(𝑘 + 1) / 𝑥]𝜑 | 
| 129 | 127, 128 | nfim 1896 | . . . . . . . . . . . 12
⊢
Ⅎ𝑥(∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑) | 
| 130 |  | oveq1 7438 | . . . . . . . . . . . . . . 15
⊢ (𝑥 = (𝑘 + 1) → (𝑥 − 1) = ((𝑘 + 1) − 1)) | 
| 131 | 130 | oveq2d 7447 | . . . . . . . . . . . . . 14
⊢ (𝑥 = (𝑘 + 1) → (1...(𝑥 − 1)) = (1...((𝑘 + 1) − 1))) | 
| 132 | 131 | raleqdv 3326 | . . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (∀𝑦 ∈ (1...(𝑥 − 1))𝜒 ↔ ∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒)) | 
| 133 |  | sbceq1a 3799 | . . . . . . . . . . . . 13
⊢ (𝑥 = (𝑘 + 1) → (𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) | 
| 134 | 132, 133 | imbi12d 344 | . . . . . . . . . . . 12
⊢ (𝑥 = (𝑘 + 1) → ((∀𝑦 ∈ (1...(𝑥 − 1))𝜒 → 𝜑) ↔ (∀𝑦 ∈ (1...((𝑘 + 1) − 1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑))) | 
| 135 |  | prmind2.7 | . . . . . . . . . . . . 13
⊢ ((𝑥 ∈ ℙ ∧
∀𝑦 ∈
(1...(𝑥 − 1))𝜒) → 𝜑) | 
| 136 | 135 | ex 412 | . . . . . . . . . . . 12
⊢ (𝑥 ∈ ℙ →
(∀𝑦 ∈
(1...(𝑥 − 1))𝜒 → 𝜑)) | 
| 137 | 126, 129,
134, 136 | vtoclgaf 3576 | . . . . . . . . . . 11
⊢ ((𝑘 + 1) ∈ ℙ →
(∀𝑦 ∈
(1...((𝑘 + 1) −
1))𝜒 → [(𝑘 + 1) / 𝑥]𝜑)) | 
| 138 | 125, 137 | syl5com 31 | . . . . . . . . . 10
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → ((𝑘 + 1) ∈ ℙ → [(𝑘 + 1) / 𝑥]𝜑)) | 
| 139 | 118, 138 | sylbid 240 | . . . . . . . . 9
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (∀𝑦 ∈ (2...((𝑘 + 1) − 1)) ¬ 𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) | 
| 140 | 107, 139 | biimtrrid 243 | . . . . . . . 8
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → (¬ ∃𝑦 ∈ (2...((𝑘 + 1) − 1))𝑦 ∥ (𝑘 + 1) → [(𝑘 + 1) / 𝑥]𝜑)) | 
| 141 | 106, 140 | pm2.61d 179 | . . . . . . 7
⊢ ((𝑘 ∈ ℕ ∧
∀𝑥 ∈ (1...𝑘)𝜑) → [(𝑘 + 1) / 𝑥]𝜑) | 
| 142 | 141 | ex 412 | . . . . . 6
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → [(𝑘 + 1) / 𝑥]𝜑)) | 
| 143 |  | ralsnsg 4670 | . . . . . . 7
⊢ ((𝑘 + 1) ∈ ℕ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) | 
| 144 | 16, 143 | syl 17 | . . . . . 6
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈ {(𝑘 + 1)}𝜑 ↔ [(𝑘 + 1) / 𝑥]𝜑)) | 
| 145 | 142, 144 | sylibrd 259 | . . . . 5
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → ∀𝑥 ∈ {(𝑘 + 1)}𝜑)) | 
| 146 | 145 | ancld 550 | . . . 4
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))) | 
| 147 |  | fzsuc 13611 | . . . . . . 7
⊢ (𝑘 ∈
(ℤ≥‘1) → (1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) | 
| 148 | 109, 147 | sylbi 217 | . . . . . 6
⊢ (𝑘 ∈ ℕ →
(1...(𝑘 + 1)) = ((1...𝑘) ∪ {(𝑘 + 1)})) | 
| 149 | 148 | raleqdv 3326 | . . . . 5
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...(𝑘 + 1))𝜑 ↔ ∀𝑥 ∈ ((1...𝑘) ∪ {(𝑘 + 1)})𝜑)) | 
| 150 |  | ralunb 4197 | . . . . 5
⊢
(∀𝑥 ∈
((1...𝑘) ∪ {(𝑘 + 1)})𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑)) | 
| 151 | 149, 150 | bitrdi 287 | . . . 4
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...(𝑘 + 1))𝜑 ↔ (∀𝑥 ∈ (1...𝑘)𝜑 ∧ ∀𝑥 ∈ {(𝑘 + 1)}𝜑))) | 
| 152 | 146, 151 | sylibrd 259 | . . 3
⊢ (𝑘 ∈ ℕ →
(∀𝑥 ∈
(1...𝑘)𝜑 → ∀𝑥 ∈ (1...(𝑘 + 1))𝜑)) | 
| 153 | 3, 5, 7, 9, 15, 152 | nnind 12284 | . 2
⊢ (𝐴 ∈ ℕ →
∀𝑥 ∈ (1...𝐴)𝜑) | 
| 154 |  | elfz1end 13594 | . . 3
⊢ (𝐴 ∈ ℕ ↔ 𝐴 ∈ (1...𝐴)) | 
| 155 | 154 | biimpi 216 | . 2
⊢ (𝐴 ∈ ℕ → 𝐴 ∈ (1...𝐴)) | 
| 156 | 1, 153, 155 | rspcdva 3623 | 1
⊢ (𝐴 ∈ ℕ → 𝜂) |