Proof of Theorem perfectALTVlem1
Step | Hyp | Ref
| Expression |
1 | | 2nn 12056 |
. . 3
⊢ 2 ∈
ℕ |
2 | | perfectALTVlem.1 |
. . . . 5
⊢ (𝜑 → 𝐴 ∈ ℕ) |
3 | 2 | nnnn0d 12303 |
. . . 4
⊢ (𝜑 → 𝐴 ∈
ℕ0) |
4 | | peano2nn0 12283 |
. . . 4
⊢ (𝐴 ∈ ℕ0
→ (𝐴 + 1) ∈
ℕ0) |
5 | 3, 4 | syl 17 |
. . 3
⊢ (𝜑 → (𝐴 + 1) ∈
ℕ0) |
6 | | nnexpcl 13805 |
. . 3
⊢ ((2
∈ ℕ ∧ (𝐴 +
1) ∈ ℕ0) → (2↑(𝐴 + 1)) ∈ ℕ) |
7 | 1, 5, 6 | sylancr 587 |
. 2
⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℕ) |
8 | | 2re 12057 |
. . . . 5
⊢ 2 ∈
ℝ |
9 | 8 | a1i 11 |
. . . 4
⊢ (𝜑 → 2 ∈
ℝ) |
10 | 2 | peano2nnd 12000 |
. . . 4
⊢ (𝜑 → (𝐴 + 1) ∈ ℕ) |
11 | | 1lt2 12154 |
. . . . 5
⊢ 1 <
2 |
12 | 11 | a1i 11 |
. . . 4
⊢ (𝜑 → 1 < 2) |
13 | | expgt1 13831 |
. . . 4
⊢ ((2
∈ ℝ ∧ (𝐴 +
1) ∈ ℕ ∧ 1 < 2) → 1 < (2↑(𝐴 + 1))) |
14 | 9, 10, 12, 13 | syl3anc 1370 |
. . 3
⊢ (𝜑 → 1 < (2↑(𝐴 + 1))) |
15 | | 1nn 11994 |
. . . 4
⊢ 1 ∈
ℕ |
16 | | nnsub 12027 |
. . . 4
⊢ ((1
∈ ℕ ∧ (2↑(𝐴 + 1)) ∈ ℕ) → (1 <
(2↑(𝐴 + 1)) ↔
((2↑(𝐴 + 1)) −
1) ∈ ℕ)) |
17 | 15, 7, 16 | sylancr 587 |
. . 3
⊢ (𝜑 → (1 < (2↑(𝐴 + 1)) ↔ ((2↑(𝐴 + 1)) − 1) ∈
ℕ)) |
18 | 14, 17 | mpbid 231 |
. 2
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈
ℕ) |
19 | 7 | nnzd 12435 |
. . . . . . 7
⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ ℤ) |
20 | | peano2zm 12373 |
. . . . . . 7
⊢
((2↑(𝐴 + 1))
∈ ℤ → ((2↑(𝐴 + 1)) − 1) ∈
ℤ) |
21 | 19, 20 | syl 17 |
. . . . . 6
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈
ℤ) |
22 | | 1nn0 12259 |
. . . . . . . 8
⊢ 1 ∈
ℕ0 |
23 | | perfectALTVlem.2 |
. . . . . . . 8
⊢ (𝜑 → 𝐵 ∈ ℕ) |
24 | | sgmnncl 26306 |
. . . . . . . 8
⊢ ((1
∈ ℕ0 ∧ 𝐵 ∈ ℕ) → (1 σ 𝐵) ∈
ℕ) |
25 | 22, 23, 24 | sylancr 587 |
. . . . . . 7
⊢ (𝜑 → (1 σ 𝐵) ∈
ℕ) |
26 | 25 | nnzd 12435 |
. . . . . 6
⊢ (𝜑 → (1 σ 𝐵) ∈
ℤ) |
27 | | dvdsmul1 15997 |
. . . . . 6
⊢
((((2↑(𝐴 + 1))
− 1) ∈ ℤ ∧ (1 σ 𝐵) ∈ ℤ) → ((2↑(𝐴 + 1)) − 1) ∥
(((2↑(𝐴 + 1)) −
1) · (1 σ 𝐵))) |
28 | 21, 26, 27 | syl2anc 584 |
. . . . 5
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥
(((2↑(𝐴 + 1)) −
1) · (1 σ 𝐵))) |
29 | | 2cn 12058 |
. . . . . . . . 9
⊢ 2 ∈
ℂ |
30 | | expp1 13799 |
. . . . . . . . 9
⊢ ((2
∈ ℂ ∧ 𝐴
∈ ℕ0) → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2)) |
31 | 29, 3, 30 | sylancr 587 |
. . . . . . . 8
⊢ (𝜑 → (2↑(𝐴 + 1)) = ((2↑𝐴) · 2)) |
32 | | nnexpcl 13805 |
. . . . . . . . . . 11
⊢ ((2
∈ ℕ ∧ 𝐴
∈ ℕ0) → (2↑𝐴) ∈ ℕ) |
33 | 1, 3, 32 | sylancr 587 |
. . . . . . . . . 10
⊢ (𝜑 → (2↑𝐴) ∈ ℕ) |
34 | 33 | nncnd 11999 |
. . . . . . . . 9
⊢ (𝜑 → (2↑𝐴) ∈ ℂ) |
35 | | mulcom 10967 |
. . . . . . . . 9
⊢
(((2↑𝐴) ∈
ℂ ∧ 2 ∈ ℂ) → ((2↑𝐴) · 2) = (2 · (2↑𝐴))) |
36 | 34, 29, 35 | sylancl 586 |
. . . . . . . 8
⊢ (𝜑 → ((2↑𝐴) · 2) = (2 · (2↑𝐴))) |
37 | 31, 36 | eqtrd 2778 |
. . . . . . 7
⊢ (𝜑 → (2↑(𝐴 + 1)) = (2 · (2↑𝐴))) |
38 | 37 | oveq1d 7282 |
. . . . . 6
⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = ((2 · (2↑𝐴)) · 𝐵)) |
39 | 29 | a1i 11 |
. . . . . . 7
⊢ (𝜑 → 2 ∈
ℂ) |
40 | 23 | nncnd 11999 |
. . . . . . 7
⊢ (𝜑 → 𝐵 ∈ ℂ) |
41 | 39, 34, 40 | mulassd 11008 |
. . . . . 6
⊢ (𝜑 → ((2 · (2↑𝐴)) · 𝐵) = (2 · ((2↑𝐴) · 𝐵))) |
42 | | 1cnd 10980 |
. . . . . . . 8
⊢ (𝜑 → 1 ∈
ℂ) |
43 | | perfectALTVlem.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ Odd ) |
44 | | isodd7 45095 |
. . . . . . . . . . 11
⊢ (𝐵 ∈ Odd ↔ (𝐵 ∈ ℤ ∧ (2 gcd
𝐵) = 1)) |
45 | 44 | simprbi 497 |
. . . . . . . . . 10
⊢ (𝐵 ∈ Odd → (2 gcd 𝐵) = 1) |
46 | 43, 45 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (2 gcd 𝐵) = 1) |
47 | | 2z 12362 |
. . . . . . . . . . 11
⊢ 2 ∈
ℤ |
48 | 47 | a1i 11 |
. . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℤ) |
49 | 23 | nnzd 12435 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℤ) |
50 | | rpexp1i 16438 |
. . . . . . . . . 10
⊢ ((2
∈ ℤ ∧ 𝐵
∈ ℤ ∧ 𝐴
∈ ℕ0) → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1)) |
51 | 48, 49, 3, 50 | syl3anc 1370 |
. . . . . . . . 9
⊢ (𝜑 → ((2 gcd 𝐵) = 1 → ((2↑𝐴) gcd 𝐵) = 1)) |
52 | 46, 51 | mpd 15 |
. . . . . . . 8
⊢ (𝜑 → ((2↑𝐴) gcd 𝐵) = 1) |
53 | | sgmmul 26359 |
. . . . . . . 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, 33, 23, 52, 53 | syl13anc 1371 |
. . . . . . 7
⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = ((1 σ (2↑𝐴)) · (1 σ 𝐵))) |
55 | | perfectALTVlem.4 |
. . . . . . 7
⊢ (𝜑 → (1 σ ((2↑𝐴) · 𝐵)) = (2 · ((2↑𝐴) · 𝐵))) |
56 | 2 | nncnd 11999 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ ℂ) |
57 | | pncan1 11409 |
. . . . . . . . . . . 12
⊢ (𝐴 ∈ ℂ → ((𝐴 + 1) − 1) = 𝐴) |
58 | 56, 57 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → ((𝐴 + 1) − 1) = 𝐴) |
59 | 58 | oveq2d 7283 |
. . . . . . . . . 10
⊢ (𝜑 → (2↑((𝐴 + 1) − 1)) =
(2↑𝐴)) |
60 | 59 | oveq2d 7283 |
. . . . . . . . 9
⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) = (1 σ
(2↑𝐴))) |
61 | | 1sgm2ppw 26358 |
. . . . . . . . . 10
⊢ ((𝐴 + 1) ∈ ℕ → (1
σ (2↑((𝐴 + 1)
− 1))) = ((2↑(𝐴
+ 1)) − 1)) |
62 | 10, 61 | syl 17 |
. . . . . . . . 9
⊢ (𝜑 → (1 σ (2↑((𝐴 + 1) − 1))) =
((2↑(𝐴 + 1)) −
1)) |
63 | 60, 62 | eqtr3d 2780 |
. . . . . . . 8
⊢ (𝜑 → (1 σ (2↑𝐴)) = ((2↑(𝐴 + 1)) − 1)) |
64 | 63 | oveq1d 7282 |
. . . . . . 7
⊢ (𝜑 → ((1 σ (2↑𝐴)) · (1 σ 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1
σ 𝐵))) |
65 | 54, 55, 64 | 3eqtr3d 2786 |
. . . . . 6
⊢ (𝜑 → (2 · ((2↑𝐴) · 𝐵)) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵))) |
66 | 38, 41, 65 | 3eqtrd 2782 |
. . . . 5
⊢ (𝜑 → ((2↑(𝐴 + 1)) · 𝐵) = (((2↑(𝐴 + 1)) − 1) · (1 σ 𝐵))) |
67 | 28, 66 | breqtrrd 5101 |
. . . 4
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥
((2↑(𝐴 + 1)) ·
𝐵)) |
68 | 21, 19 | gcdcomd 16231 |
. . . . 5
⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd
(2↑(𝐴 + 1))) =
((2↑(𝐴 + 1)) gcd
((2↑(𝐴 + 1)) −
1))) |
69 | | nnpw2evenALTV 45132 |
. . . . . . . . 9
⊢ ((𝐴 + 1) ∈ ℕ →
(2↑(𝐴 + 1)) ∈
Even ) |
70 | 10, 69 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → (2↑(𝐴 + 1)) ∈ Even ) |
71 | | evenm1odd 45069 |
. . . . . . . 8
⊢
((2↑(𝐴 + 1))
∈ Even → ((2↑(𝐴 + 1)) − 1) ∈ Odd
) |
72 | 70, 71 | syl 17 |
. . . . . . 7
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∈ Odd
) |
73 | | isodd7 45095 |
. . . . . . . 8
⊢
(((2↑(𝐴 + 1))
− 1) ∈ Odd ↔ (((2↑(𝐴 + 1)) − 1) ∈ ℤ ∧ (2
gcd ((2↑(𝐴 + 1))
− 1)) = 1)) |
74 | 73 | simprbi 497 |
. . . . . . 7
⊢
(((2↑(𝐴 + 1))
− 1) ∈ Odd → (2 gcd ((2↑(𝐴 + 1)) − 1)) = 1) |
75 | 72, 74 | syl 17 |
. . . . . 6
⊢ (𝜑 → (2 gcd ((2↑(𝐴 + 1)) − 1)) =
1) |
76 | | rpexp1i 16438 |
. . . . . . 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)) |
77 | 48, 21, 5, 76 | syl3anc 1370 |
. . . . . 6
⊢ (𝜑 → ((2 gcd ((2↑(𝐴 + 1)) − 1)) = 1 →
((2↑(𝐴 + 1)) gcd
((2↑(𝐴 + 1)) −
1)) = 1)) |
78 | 75, 77 | mpd 15 |
. . . . 5
⊢ (𝜑 → ((2↑(𝐴 + 1)) gcd ((2↑(𝐴 + 1)) − 1)) =
1) |
79 | 68, 78 | eqtrd 2778 |
. . . 4
⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) gcd
(2↑(𝐴 + 1))) =
1) |
80 | | coprmdvds 16368 |
. . . . 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)) |
81 | 21, 19, 49, 80 | syl3anc 1370 |
. . . 4
⊢ (𝜑 → ((((2↑(𝐴 + 1)) − 1) ∥
((2↑(𝐴 + 1)) ·
𝐵) ∧ (((2↑(𝐴 + 1)) − 1) gcd
(2↑(𝐴 + 1))) = 1)
→ ((2↑(𝐴 + 1))
− 1) ∥ 𝐵)) |
82 | 67, 79, 81 | mp2and 696 |
. . 3
⊢ (𝜑 → ((2↑(𝐴 + 1)) − 1) ∥ 𝐵) |
83 | | nndivdvds 15982 |
. . . 4
⊢ ((𝐵 ∈ ℕ ∧
((2↑(𝐴 + 1)) −
1) ∈ ℕ) → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈
ℕ)) |
84 | 23, 18, 83 | syl2anc 584 |
. . 3
⊢ (𝜑 → (((2↑(𝐴 + 1)) − 1) ∥ 𝐵 ↔ (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈
ℕ)) |
85 | 82, 84 | mpbid 231 |
. 2
⊢ (𝜑 → (𝐵 / ((2↑(𝐴 + 1)) − 1)) ∈
ℕ) |
86 | 7, 18, 85 | 3jca 1127 |
1
⊢ (𝜑 → ((2↑(𝐴 + 1)) ∈ ℕ ∧
((2↑(𝐴 + 1)) −
1) ∈ ℕ ∧ (𝐵
/ ((2↑(𝐴 + 1)) −
1)) ∈ ℕ)) |