| Step | Hyp | Ref
| Expression |
| 1 | | prmz 16712 |
. . . . . . 7
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℤ) |
| 2 | | gcddvds 16540 |
. . . . . . 7
⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → ((𝑃 gcd 𝑁) ∥ 𝑃 ∧ (𝑃 gcd 𝑁) ∥ 𝑁)) |
| 3 | 1, 2 | sylan 580 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((𝑃 gcd 𝑁) ∥ 𝑃 ∧ (𝑃 gcd 𝑁) ∥ 𝑁)) |
| 4 | 3 | simprd 495 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 gcd 𝑁) ∥ 𝑁) |
| 5 | | breq1 5146 |
. . . . 5
⊢ ((𝑃 gcd 𝑁) = 𝑃 → ((𝑃 gcd 𝑁) ∥ 𝑁 ↔ 𝑃 ∥ 𝑁)) |
| 6 | 4, 5 | syl5ibcom 245 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((𝑃 gcd 𝑁) = 𝑃 → 𝑃 ∥ 𝑁)) |
| 7 | 6 | con3d 152 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
𝑃 ∥ 𝑁 → ¬ (𝑃 gcd 𝑁) = 𝑃)) |
| 8 | | 0nnn 12302 |
. . . . . . . . 9
⊢ ¬ 0
∈ ℕ |
| 9 | | prmnn 16711 |
. . . . . . . . . 10
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℕ) |
| 10 | | eleq1 2829 |
. . . . . . . . . 10
⊢ (𝑃 = 0 → (𝑃 ∈ ℕ ↔ 0 ∈
ℕ)) |
| 11 | 9, 10 | syl5ibcom 245 |
. . . . . . . . 9
⊢ (𝑃 ∈ ℙ → (𝑃 = 0 → 0 ∈
ℕ)) |
| 12 | 8, 11 | mtoi 199 |
. . . . . . . 8
⊢ (𝑃 ∈ ℙ → ¬
𝑃 = 0) |
| 13 | 12 | intnanrd 489 |
. . . . . . 7
⊢ (𝑃 ∈ ℙ → ¬
(𝑃 = 0 ∧ 𝑁 = 0)) |
| 14 | 13 | adantr 480 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ¬
(𝑃 = 0 ∧ 𝑁 = 0)) |
| 15 | | gcdn0cl 16539 |
. . . . . . . 8
⊢ (((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑃 = 0 ∧ 𝑁 = 0)) → (𝑃 gcd 𝑁) ∈ ℕ) |
| 16 | 15 | ex 412 |
. . . . . . 7
⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬
(𝑃 = 0 ∧ 𝑁 = 0) → (𝑃 gcd 𝑁) ∈ ℕ)) |
| 17 | 1, 16 | sylan 580 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
(𝑃 = 0 ∧ 𝑁 = 0) → (𝑃 gcd 𝑁) ∈ ℕ)) |
| 18 | 14, 17 | mpd 15 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 gcd 𝑁) ∈ ℕ) |
| 19 | 3 | simpld 494 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 gcd 𝑁) ∥ 𝑃) |
| 20 | | isprm2 16719 |
. . . . . . . 8
⊢ (𝑃 ∈ ℙ ↔ (𝑃 ∈
(ℤ≥‘2) ∧ ∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃)))) |
| 21 | 20 | simprbi 496 |
. . . . . . 7
⊢ (𝑃 ∈ ℙ →
∀𝑧 ∈ ℕ
(𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃))) |
| 22 | | breq1 5146 |
. . . . . . . . 9
⊢ (𝑧 = (𝑃 gcd 𝑁) → (𝑧 ∥ 𝑃 ↔ (𝑃 gcd 𝑁) ∥ 𝑃)) |
| 23 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑃 gcd 𝑁) → (𝑧 = 1 ↔ (𝑃 gcd 𝑁) = 1)) |
| 24 | | eqeq1 2741 |
. . . . . . . . . 10
⊢ (𝑧 = (𝑃 gcd 𝑁) → (𝑧 = 𝑃 ↔ (𝑃 gcd 𝑁) = 𝑃)) |
| 25 | 23, 24 | orbi12d 919 |
. . . . . . . . 9
⊢ (𝑧 = (𝑃 gcd 𝑁) → ((𝑧 = 1 ∨ 𝑧 = 𝑃) ↔ ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃))) |
| 26 | 22, 25 | imbi12d 344 |
. . . . . . . 8
⊢ (𝑧 = (𝑃 gcd 𝑁) → ((𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃)) ↔ ((𝑃 gcd 𝑁) ∥ 𝑃 → ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃)))) |
| 27 | 26 | rspcv 3618 |
. . . . . . 7
⊢ ((𝑃 gcd 𝑁) ∈ ℕ → (∀𝑧 ∈ ℕ (𝑧 ∥ 𝑃 → (𝑧 = 1 ∨ 𝑧 = 𝑃)) → ((𝑃 gcd 𝑁) ∥ 𝑃 → ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃)))) |
| 28 | 21, 27 | syl5com 31 |
. . . . . 6
⊢ (𝑃 ∈ ℙ → ((𝑃 gcd 𝑁) ∈ ℕ → ((𝑃 gcd 𝑁) ∥ 𝑃 → ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃)))) |
| 29 | 28 | adantr 480 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((𝑃 gcd 𝑁) ∈ ℕ → ((𝑃 gcd 𝑁) ∥ 𝑃 → ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃)))) |
| 30 | 18, 19, 29 | mp2d 49 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃)) |
| 31 | | biorf 937 |
. . . . 5
⊢ (¬
(𝑃 gcd 𝑁) = 𝑃 → ((𝑃 gcd 𝑁) = 1 ↔ ((𝑃 gcd 𝑁) = 𝑃 ∨ (𝑃 gcd 𝑁) = 1))) |
| 32 | | orcom 871 |
. . . . 5
⊢ (((𝑃 gcd 𝑁) = 𝑃 ∨ (𝑃 gcd 𝑁) = 1) ↔ ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃)) |
| 33 | 31, 32 | bitrdi 287 |
. . . 4
⊢ (¬
(𝑃 gcd 𝑁) = 𝑃 → ((𝑃 gcd 𝑁) = 1 ↔ ((𝑃 gcd 𝑁) = 1 ∨ (𝑃 gcd 𝑁) = 𝑃))) |
| 34 | 30, 33 | syl5ibrcom 247 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
(𝑃 gcd 𝑁) = 𝑃 → (𝑃 gcd 𝑁) = 1)) |
| 35 | 7, 34 | syld 47 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
𝑃 ∥ 𝑁 → (𝑃 gcd 𝑁) = 1)) |
| 36 | | iddvds 16307 |
. . . . . . 7
⊢ (𝑃 ∈ ℤ → 𝑃 ∥ 𝑃) |
| 37 | 1, 36 | syl 17 |
. . . . . 6
⊢ (𝑃 ∈ ℙ → 𝑃 ∥ 𝑃) |
| 38 | 37 | adantr 480 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → 𝑃 ∥ 𝑃) |
| 39 | | dvdslegcd 16541 |
. . . . . . . . 9
⊢ (((𝑃 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ ¬
(𝑃 = 0 ∧ 𝑁 = 0)) → ((𝑃 ∥ 𝑃 ∧ 𝑃 ∥ 𝑁) → 𝑃 ≤ (𝑃 gcd 𝑁))) |
| 40 | 39 | ex 412 |
. . . . . . . 8
⊢ ((𝑃 ∈ ℤ ∧ 𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬
(𝑃 = 0 ∧ 𝑁 = 0) → ((𝑃 ∥ 𝑃 ∧ 𝑃 ∥ 𝑁) → 𝑃 ≤ (𝑃 gcd 𝑁)))) |
| 41 | 40 | 3anidm12 1421 |
. . . . . . 7
⊢ ((𝑃 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (¬
(𝑃 = 0 ∧ 𝑁 = 0) → ((𝑃 ∥ 𝑃 ∧ 𝑃 ∥ 𝑁) → 𝑃 ≤ (𝑃 gcd 𝑁)))) |
| 42 | 1, 41 | sylan 580 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
(𝑃 = 0 ∧ 𝑁 = 0) → ((𝑃 ∥ 𝑃 ∧ 𝑃 ∥ 𝑁) → 𝑃 ≤ (𝑃 gcd 𝑁)))) |
| 43 | 14, 42 | mpd 15 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((𝑃 ∥ 𝑃 ∧ 𝑃 ∥ 𝑁) → 𝑃 ≤ (𝑃 gcd 𝑁))) |
| 44 | 38, 43 | mpand 695 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 → 𝑃 ≤ (𝑃 gcd 𝑁))) |
| 45 | | prmgt1 16734 |
. . . . . 6
⊢ (𝑃 ∈ ℙ → 1 <
𝑃) |
| 46 | 45 | adantr 480 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → 1 <
𝑃) |
| 47 | | 1re 11261 |
. . . . . 6
⊢ 1 ∈
ℝ |
| 48 | 1 | zred 12722 |
. . . . . 6
⊢ (𝑃 ∈ ℙ → 𝑃 ∈
ℝ) |
| 49 | 18 | nnred 12281 |
. . . . . 6
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 gcd 𝑁) ∈ ℝ) |
| 50 | | ltletr 11353 |
. . . . . 6
⊢ ((1
∈ ℝ ∧ 𝑃
∈ ℝ ∧ (𝑃 gcd
𝑁) ∈ ℝ) →
((1 < 𝑃 ∧ 𝑃 ≤ (𝑃 gcd 𝑁)) → 1 < (𝑃 gcd 𝑁))) |
| 51 | 47, 48, 49, 50 | mp3an2ani 1470 |
. . . . 5
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((1 <
𝑃 ∧ 𝑃 ≤ (𝑃 gcd 𝑁)) → 1 < (𝑃 gcd 𝑁))) |
| 52 | 46, 51 | mpand 695 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 ≤ (𝑃 gcd 𝑁) → 1 < (𝑃 gcd 𝑁))) |
| 53 | | ltne 11358 |
. . . . . 6
⊢ ((1
∈ ℝ ∧ 1 < (𝑃 gcd 𝑁)) → (𝑃 gcd 𝑁) ≠ 1) |
| 54 | 47, 53 | mpan 690 |
. . . . 5
⊢ (1 <
(𝑃 gcd 𝑁) → (𝑃 gcd 𝑁) ≠ 1) |
| 55 | 54 | a1i 11 |
. . . 4
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (1 <
(𝑃 gcd 𝑁) → (𝑃 gcd 𝑁) ≠ 1)) |
| 56 | 44, 52, 55 | 3syld 60 |
. . 3
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (𝑃 ∥ 𝑁 → (𝑃 gcd 𝑁) ≠ 1)) |
| 57 | 56 | necon2bd 2956 |
. 2
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → ((𝑃 gcd 𝑁) = 1 → ¬ 𝑃 ∥ 𝑁)) |
| 58 | 35, 57 | impbid 212 |
1
⊢ ((𝑃 ∈ ℙ ∧ 𝑁 ∈ ℤ) → (¬
𝑃 ∥ 𝑁 ↔ (𝑃 gcd 𝑁) = 1)) |