Proof of Theorem perfectlem1
| Step | Hyp | Ref
| Expression |
| 1 | | 2nn 12339 |
. . 3
⊢ 2 ∈
ℕ |
| 2 | | perfectlem.1 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℕ) |
| 3 | 2 | nnnn0d 12587 |
. . . 4
⊢ (𝜑 → 𝐴 ∈
ℕ0) |
| 4 | | peano2nn0 12566 |
. . . 4
⊢ (𝐴 ∈ ℕ0
→ (𝐴 + 1) ∈
ℕ0) |
| 5 | 3, 4 | syl 17 |
. . 3
⊢ (𝜑 → (𝐴 + 1) ∈
ℕ0) |
| 6 | | nnexpcl 14115 |
. . 3
⊢ ((2
∈ ℕ ∧ (𝐴 +
1) ∈ ℕ0) → (2↑(𝐴 + 1)) ∈ ℕ) |
| 7 | 1, 5, 6 | sylancr 587 |
. 2
⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ) |
| 8 | | 2re 12340 |
. . . 4
⊢ 2 ∈
ℝ |
| 9 | 2 | peano2nnd 12283 |
. . . 4
⊢ (𝜑 → (𝐴 + 1) ∈ ℕ) |
| 10 | | 1lt2 12437 |
. . . . 5
⊢ 1 <
2 |
| 11 | 10 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 < 2) |
| 12 | | expgt1 14141 |
. . . 4
⊢ ((2
∈ ℝ ∧ (𝐴 +
1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1))) |
| 13 | 8, 9, 11, 12 | mp3an2i 1468 |
. . 3
⊢ (𝜑 → 1 < (2↑(𝐴 + 1))) |
| 14 | | 1nn 12277 |
. . . 4
⊢ 1 ∈
ℕ |
| 15 | | nnsub 12310 |
. . . 4
⊢ ((1
∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 <
(2↑(𝐴 + 1)) ↔
((2↑(𝐴 + 1)) −
1) ∈ ℕ)) |
| 16 | 14, 7, 15 | sylancr 587 |
. . 3
⊢ (𝜑 → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈
ℕ)) |
| 17 | 13, 16 | mpbid 232 |
. 2
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈
ℕ) |
| 18 | 7 | nnzd 12640 |
. . . . . . 7
⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ) |
| 19 | | peano2zm 12660 |
. . . . . . 7
⊢
((2↑(𝐴 + 1))
∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈
ℤ) |
| 20 | 18, 19 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈
ℤ) |
| 21 | | 1nn0 12542 |
. . . . . . . 8
⊢ 1 ∈
ℕ0 |
| 22 | | perfectlem.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℕ) |
| 23 | | sgmnncl 27190 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈
ℕ) |
| 24 | 21, 22, 23 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (1 σ 𝐵) ∈
ℕ) |
| 25 | 24 | nnzd 12640 |
. . . . . 6
⊢ (𝜑 → (1 σ 𝐵) ∈
ℤ) |
| 26 | | dvdsmul1 16315 |
. . . . . 6
⊢
((((2↑(𝐴 + 1))
− 1) ∈ ℤ ∧ (1 σ 𝐵) ∈ ℤ) → ((2↑(𝐴 + 1)) − 1) ∥
(((2↑(𝐴 + 1)) −
1) · (1 σ 𝐵))) |
| 27 | 20, 25, 26 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥
(((2↑(𝐴 + 1)) −
1) · (1 σ 𝐵))) |
| 28 | | 2cn 12341 |
. . . . . . . . 9
⊢ 2 ∈
ℂ |
| 29 | | expp1 14109 |
. . . . . . . . 9
⊢ ((2
∈ ℂ ∧ 𝐴
∈ ℕ0) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2)) |
| 30 | 28, 3, 29 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2)) |
| 31 | | nnexpcl 14115 |
. . . . . . . . . . 11
⊢ ((2
∈ ℕ ∧ 𝐴
∈ ℕ0) → (2↑𝐴) ∈ ℕ) |
| 32 | 1, 3, 31 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → (2↑𝐴) ∈ ℕ) |
| 33 | 32 | nncnd 12282 |
. . . . . . . . 9
⊢ (𝜑 → (2↑𝐴) ∈ ℂ) |
| 34 | | mulcom 11241 |
. . . . . . . . 9
⊢
(((2↑𝐴) ∈
ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴))) |
| 35 | 33, 28, 34 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴))) |
| 36 | 30, 35 | eqtrd 2777 |
. . . . . . 7
⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴))) |
| 37 | 36 | oveq1d 7446 |
. . . . . 6
⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵)) |
| 38 | 28 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℂ) |
| 39 | 22 | nncnd 12282 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) |
| 40 | 38, 33, 39 | mulassd 11284 |
. . . . . 6
⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵))) |
| 41 | | ax-1cn 11213 |
. . . . . . . . 9
⊢ 1 ∈
ℂ |
| 42 | 41 | a1i 11 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℂ) |
| 43 | | perfectlem.3 |
. . . . . . . . . 10
⊢ (𝜑 → ¬ 2 ∥ 𝐵) |
| 44 | | 2prm 16729 |
. . . . . . . . . . 11
⊢ 2 ∈
ℙ |
| 45 | 22 | nnzd 12640 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐵 ∈ ℤ) |
| 46 | | coprm 16748 |
. . . . . . . . . . 11
⊢ ((2
∈ ℙ ∧ 𝐵
∈ ℤ) → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1)) |
| 47 | 44, 45, 46 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → (¬ 2 ∥ 𝐵 ↔ (2 gcd 𝐵) = 1)) |
| 48 | 43, 47 | mpbid 232 |
. . . . . . . . 9
⊢ (𝜑 → (2 gcd 𝐵) = 1) |
| 49 | | 2z 12649 |
. . . . . . . . . 10
⊢ 2 ∈
ℤ |
| 50 | | rpexp1i 16760 |
. . . . . . . . . 10
⊢ ((2
∈ ℤ ∧ 𝐵
∈ ℤ ∧ 𝐴
∈ ℕ0) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1)) |
| 51 | 49, 45, 3, 50 | mp3an2i 1468 |
. . . . . . . . 9
⊢ (𝜑 → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1)) |
| 52 | 48, 51 | mpd 15 |
. . . . . . . 8
⊢ (𝜑 → ((2↑𝐴) gcd 𝐵) = 1) |
| 53 | | sgmmul 27245 |
. . . . . . . 8
⊢ ((1
∈ ℂ ∧ ((2↑𝐴) ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ((2↑𝐴) gcd 𝐵) = 1)) → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵))) |
| 54 | 42, 32, 22, 52, 53 | syl13anc 1374 |
. . . . . . 7
⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵))) |
| 55 | | perfectlem.4 |
. . . . . . 7
⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵))) |
| 56 | 2 | nncnd 12282 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 57 | | pncan 11514 |
. . . . . . . . . . . 12
⊢ ((𝐴 ∈ ℂ ∧ 1 ∈
ℂ) → ((𝐴 + 1)
− 1) = 𝐴) |
| 58 | 56, 41, 57 | sylancl 586 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴) |
| 59 | 58 | oveq2d 7447 |
. . . . . . . . . 10
⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) =
(2↑𝐴)) |
| 60 | 59 | oveq2d 7447 |
. . . . . . . . 9
⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ
(2↑𝐴))) |
| 61 | | 1sgm2ppw 27244 |
. . . . . . . . . 10
⊢ ((𝐴 + 1) ∈ ℕ → (1
σ (2↑((𝐴 + 1)
− 1))) = ((2↑(𝐴
+ 1)) − 1)) |
| 62 | 9, 61 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) =
((2↑(𝐴 + 1)) −
1)) |
| 63 | 60, 62 | eqtr3d 2779 |
. . . . . . . 8
⊢ (𝜑 → (1 σ (2↑𝐴)) = ((2↑(𝐴 + 1)) − 1)) |
| 64 | 63 | oveq1d 7446 |
. . . . . . 7
⊢ (𝜑 → ((1 σ (2↑𝐴)) · (1 σ 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1
σ 𝐵))) |
| 65 | 54, 55, 64 | 3eqtr3d 2785 |
. . . . . 6
⊢ (𝜑 → (2 · ((2↑𝐴) · 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵))) |
| 66 | 37, 40, 65 | 3eqtrd 2781 |
. . . . 5
⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵))) |
| 67 | 27, 66 | breqtrrd 5171 |
. . . 4
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥
((2↑(𝐴 + 1)) ·
𝐵)) |
| 68 | 20, 18 | gcdcomd 16551 |
. . . . 5
⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd
(2↑(𝐴 + 1))) =
((2↑(𝐴 + 1)) gcd
((2↑(𝐴 + 1)) −
1))) |
| 69 | | iddvdsexp 16317 |
. . . . . . . . 9
⊢ ((2
∈ ℤ ∧ (𝐴 +
1) ∈ ℕ) → 2 ∥ (2↑(𝐴 + 1))) |
| 70 | 49, 9, 69 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → 2 ∥ (2↑(𝐴 + 1))) |
| 71 | | n2dvds1 16405 |
. . . . . . . . . 10
⊢ ¬ 2
∥ 1 |
| 72 | 49 | a1i 11 |
. . . . . . . . . . . 12
⊢ (𝜑 → 2 ∈
ℤ) |
| 73 | | 1zzd 12648 |
. . . . . . . . . . . 12
⊢ (𝜑 → 1 ∈
ℤ) |
| 74 | 72, 18, 73 | 3jca 1129 |
. . . . . . . . . . 11
⊢ (𝜑 → (2 ∈ ℤ ∧
(2↑(𝐴 + 1)) ∈
ℤ ∧ 1 ∈ ℤ)) |
| 75 | | dvdssub2 16338 |
. . . . . . . . . . 11
⊢ (((2
∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 1 ∈
ℤ) ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2 ∥
(2↑(𝐴 + 1)) ↔ 2
∥ 1)) |
| 76 | 74, 75 | sylan 580 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → (2
∥ (2↑(𝐴 + 1))
↔ 2 ∥ 1)) |
| 77 | 71, 76 | mtbiri 327 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 2 ∥ ((2↑(𝐴 + 1)) − 1)) → ¬
2 ∥ (2↑(𝐴 +
1))) |
| 78 | 77 | ex 412 |
. . . . . . . 8
⊢ (𝜑 → (2 ∥ ((2↑(𝐴 + 1)) − 1) → ¬ 2
∥ (2↑(𝐴 +
1)))) |
| 79 | 70, 78 | mt2d 136 |
. . . . . . 7
⊢ (𝜑 → ¬ 2 ∥
((2↑(𝐴 + 1)) −
1)) |
| 80 | | coprm 16748 |
. . . . . . . 8
⊢ ((2
∈ ℙ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℤ) →
(¬ 2 ∥ ((2↑(𝐴 + 1)) − 1) ↔ (2 gcd
((2↑(𝐴 + 1)) −
1)) = 1)) |
| 81 | 44, 20, 80 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (¬ 2 ∥
((2↑(𝐴 + 1)) −
1) ↔ (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1)) |
| 82 | 79, 81 | mpbid 232 |
. . . . . 6
⊢ (𝜑 → (2 gcd ((2↑(𝐴 + 1)) − 1)) =
1) |
| 83 | | rpexp1i 16760 |
. . . . . . 7
⊢ ((2
∈ ℤ ∧ ((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧
(𝐴 + 1) ∈
ℕ0) → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 →
((2↑(𝐴 + 1)) gcd
((2↑(𝐴 + 1)) −
1)) = 1)) |
| 84 | 49, 20, 5, 83 | mp3an2i 1468 |
. . . . . 6
⊢ (𝜑 → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 →
((2↑(𝐴 + 1)) gcd
((2↑(𝐴 + 1)) −
1)) = 1)) |
| 85 | 82, 84 | mpd 15 |
. . . . 5
⊢ (𝜑 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) =
1) |
| 86 | 68, 85 | eqtrd 2777 |
. . . 4
⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd
(2↑(𝐴 + 1))) =
1) |
| 87 | | coprmdvds 16690 |
. . . . 5
⊢
((((2↑(𝐴 + 1))
− 1) ∈ ℤ ∧ (2↑(𝐴 + 1)) ∈ ℤ ∧ 𝐵 ∈ ℤ) →
((((2↑(𝐴 + 1)) −
1) ∥ ((2↑(𝐴 +
1)) · 𝐵) ∧
(((2↑(𝐴 + 1)) −
1) gcd (2↑(𝐴 + 1))) =
1) → ((2↑(𝐴 + 1))
− 1) ∥ 𝐵)) |
| 88 | 20, 18, 45, 87 | syl3anc 1373 |
. . . 4
⊢ (𝜑 → ((((2↑(𝐴 + 1)) − 1) ∥
((2↑(𝐴 + 1)) ·
𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd
(2↑(𝐴 + 1))) = 1)
→ ((2↑(𝐴 + 1))
− 1) ∥ 𝐵)) |
| 89 | 67, 86, 88 | mp2and 699 |
. . 3
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵) |
| 90 | | nndivdvds 16299 |
. . . 4
⊢ ((𝐵 ∈ ℕ ∧
((2↑(𝐴 + 1)) −
1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈
ℕ)) |
| 91 | 22, 17, 90 | syl2anc 584 |
. . 3
⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈
ℕ)) |
| 92 | 89, 91 | mpbid 232 |
. 2
⊢ (𝜑 → (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈
ℕ) |
| 93 | 7, 17, 92 | 3jca 1129 |
1
⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧
((2↑(𝐴 + 1)) −
1) ∈ ℕ ∧ (𝐵
/ ((2↑(𝐴 + 1)) −
1)) ∈ ℕ)) |