Step |
Hyp |
Ref |
Expression |
1 |
|
perfectlem.1 |
⊢ ( 𝜑 → 𝐴 ∈ ℕ ) |
2 |
|
perfectlem.2 |
⊢ ( 𝜑 → 𝐵 ∈ ℕ ) |
3 |
|
perfectlem.3 |
⊢ ( 𝜑 → ¬ 2 ∥ 𝐵 ) |
4 |
|
perfectlem.4 |
⊢ ( 𝜑 → ( 1 σ ( ( 2 ↑ 𝐴 ) · 𝐵 ) ) = ( 2 · ( ( 2 ↑ 𝐴 ) · 𝐵 ) ) ) |
5 |
|
2nn |
⊢ 2 ∈ ℕ |
6 |
1
|
nnnn0d |
⊢ ( 𝜑 → 𝐴 ∈ ℕ0 ) |
7 |
|
peano2nn0 |
⊢ ( 𝐴 ∈ ℕ0 → ( 𝐴 + 1 ) ∈ ℕ0 ) |
8 |
6 7
|
syl |
⊢ ( 𝜑 → ( 𝐴 + 1 ) ∈ ℕ0 ) |
9 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ ( 𝐴 + 1 ) ∈ ℕ0 ) → ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℕ ) |
10 |
5 8 9
|
sylancr |
⊢ ( 𝜑 → ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℕ ) |
11 |
|
2re |
⊢ 2 ∈ ℝ |
12 |
1
|
peano2nnd |
⊢ ( 𝜑 → ( 𝐴 + 1 ) ∈ ℕ ) |
13 |
|
1lt2 |
⊢ 1 < 2 |
14 |
13
|
a1i |
⊢ ( 𝜑 → 1 < 2 ) |
15 |
|
expgt1 |
⊢ ( ( 2 ∈ ℝ ∧ ( 𝐴 + 1 ) ∈ ℕ ∧ 1 < 2 ) → 1 < ( 2 ↑ ( 𝐴 + 1 ) ) ) |
16 |
11 12 14 15
|
mp3an2i |
⊢ ( 𝜑 → 1 < ( 2 ↑ ( 𝐴 + 1 ) ) ) |
17 |
|
1nn |
⊢ 1 ∈ ℕ |
18 |
|
nnsub |
⊢ ( ( 1 ∈ ℕ ∧ ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℕ ) → ( 1 < ( 2 ↑ ( 𝐴 + 1 ) ) ↔ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℕ ) ) |
19 |
17 10 18
|
sylancr |
⊢ ( 𝜑 → ( 1 < ( 2 ↑ ( 𝐴 + 1 ) ) ↔ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℕ ) ) |
20 |
16 19
|
mpbid |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℕ ) |
21 |
10
|
nnzd |
⊢ ( 𝜑 → ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℤ ) |
22 |
|
peano2zm |
⊢ ( ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℤ → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℤ ) |
23 |
21 22
|
syl |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℤ ) |
24 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
25 |
|
sgmnncl |
⊢ ( ( 1 ∈ ℕ0 ∧ 𝐵 ∈ ℕ ) → ( 1 σ 𝐵 ) ∈ ℕ ) |
26 |
24 2 25
|
sylancr |
⊢ ( 𝜑 → ( 1 σ 𝐵 ) ∈ ℕ ) |
27 |
26
|
nnzd |
⊢ ( 𝜑 → ( 1 σ 𝐵 ) ∈ ℤ ) |
28 |
|
dvdsmul1 |
⊢ ( ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℤ ∧ ( 1 σ 𝐵 ) ∈ ℤ ) → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∥ ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) · ( 1 σ 𝐵 ) ) ) |
29 |
23 27 28
|
syl2anc |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∥ ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) · ( 1 σ 𝐵 ) ) ) |
30 |
|
2cn |
⊢ 2 ∈ ℂ |
31 |
|
expp1 |
⊢ ( ( 2 ∈ ℂ ∧ 𝐴 ∈ ℕ0 ) → ( 2 ↑ ( 𝐴 + 1 ) ) = ( ( 2 ↑ 𝐴 ) · 2 ) ) |
32 |
30 6 31
|
sylancr |
⊢ ( 𝜑 → ( 2 ↑ ( 𝐴 + 1 ) ) = ( ( 2 ↑ 𝐴 ) · 2 ) ) |
33 |
|
nnexpcl |
⊢ ( ( 2 ∈ ℕ ∧ 𝐴 ∈ ℕ0 ) → ( 2 ↑ 𝐴 ) ∈ ℕ ) |
34 |
5 6 33
|
sylancr |
⊢ ( 𝜑 → ( 2 ↑ 𝐴 ) ∈ ℕ ) |
35 |
34
|
nncnd |
⊢ ( 𝜑 → ( 2 ↑ 𝐴 ) ∈ ℂ ) |
36 |
|
mulcom |
⊢ ( ( ( 2 ↑ 𝐴 ) ∈ ℂ ∧ 2 ∈ ℂ ) → ( ( 2 ↑ 𝐴 ) · 2 ) = ( 2 · ( 2 ↑ 𝐴 ) ) ) |
37 |
35 30 36
|
sylancl |
⊢ ( 𝜑 → ( ( 2 ↑ 𝐴 ) · 2 ) = ( 2 · ( 2 ↑ 𝐴 ) ) ) |
38 |
32 37
|
eqtrd |
⊢ ( 𝜑 → ( 2 ↑ ( 𝐴 + 1 ) ) = ( 2 · ( 2 ↑ 𝐴 ) ) ) |
39 |
38
|
oveq1d |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) · 𝐵 ) = ( ( 2 · ( 2 ↑ 𝐴 ) ) · 𝐵 ) ) |
40 |
30
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℂ ) |
41 |
2
|
nncnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
42 |
40 35 41
|
mulassd |
⊢ ( 𝜑 → ( ( 2 · ( 2 ↑ 𝐴 ) ) · 𝐵 ) = ( 2 · ( ( 2 ↑ 𝐴 ) · 𝐵 ) ) ) |
43 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
44 |
43
|
a1i |
⊢ ( 𝜑 → 1 ∈ ℂ ) |
45 |
|
2prm |
⊢ 2 ∈ ℙ |
46 |
2
|
nnzd |
⊢ ( 𝜑 → 𝐵 ∈ ℤ ) |
47 |
|
coprm |
⊢ ( ( 2 ∈ ℙ ∧ 𝐵 ∈ ℤ ) → ( ¬ 2 ∥ 𝐵 ↔ ( 2 gcd 𝐵 ) = 1 ) ) |
48 |
45 46 47
|
sylancr |
⊢ ( 𝜑 → ( ¬ 2 ∥ 𝐵 ↔ ( 2 gcd 𝐵 ) = 1 ) ) |
49 |
3 48
|
mpbid |
⊢ ( 𝜑 → ( 2 gcd 𝐵 ) = 1 ) |
50 |
|
2z |
⊢ 2 ∈ ℤ |
51 |
|
rpexp1i |
⊢ ( ( 2 ∈ ℤ ∧ 𝐵 ∈ ℤ ∧ 𝐴 ∈ ℕ0 ) → ( ( 2 gcd 𝐵 ) = 1 → ( ( 2 ↑ 𝐴 ) gcd 𝐵 ) = 1 ) ) |
52 |
50 46 6 51
|
mp3an2i |
⊢ ( 𝜑 → ( ( 2 gcd 𝐵 ) = 1 → ( ( 2 ↑ 𝐴 ) gcd 𝐵 ) = 1 ) ) |
53 |
49 52
|
mpd |
⊢ ( 𝜑 → ( ( 2 ↑ 𝐴 ) gcd 𝐵 ) = 1 ) |
54 |
|
sgmmul |
⊢ ( ( 1 ∈ ℂ ∧ ( ( 2 ↑ 𝐴 ) ∈ ℕ ∧ 𝐵 ∈ ℕ ∧ ( ( 2 ↑ 𝐴 ) gcd 𝐵 ) = 1 ) ) → ( 1 σ ( ( 2 ↑ 𝐴 ) · 𝐵 ) ) = ( ( 1 σ ( 2 ↑ 𝐴 ) ) · ( 1 σ 𝐵 ) ) ) |
55 |
44 34 2 53 54
|
syl13anc |
⊢ ( 𝜑 → ( 1 σ ( ( 2 ↑ 𝐴 ) · 𝐵 ) ) = ( ( 1 σ ( 2 ↑ 𝐴 ) ) · ( 1 σ 𝐵 ) ) ) |
56 |
1
|
nncnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
57 |
|
pncan |
⊢ ( ( 𝐴 ∈ ℂ ∧ 1 ∈ ℂ ) → ( ( 𝐴 + 1 ) − 1 ) = 𝐴 ) |
58 |
56 43 57
|
sylancl |
⊢ ( 𝜑 → ( ( 𝐴 + 1 ) − 1 ) = 𝐴 ) |
59 |
58
|
oveq2d |
⊢ ( 𝜑 → ( 2 ↑ ( ( 𝐴 + 1 ) − 1 ) ) = ( 2 ↑ 𝐴 ) ) |
60 |
59
|
oveq2d |
⊢ ( 𝜑 → ( 1 σ ( 2 ↑ ( ( 𝐴 + 1 ) − 1 ) ) ) = ( 1 σ ( 2 ↑ 𝐴 ) ) ) |
61 |
|
1sgm2ppw |
⊢ ( ( 𝐴 + 1 ) ∈ ℕ → ( 1 σ ( 2 ↑ ( ( 𝐴 + 1 ) − 1 ) ) ) = ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) |
62 |
12 61
|
syl |
⊢ ( 𝜑 → ( 1 σ ( 2 ↑ ( ( 𝐴 + 1 ) − 1 ) ) ) = ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) |
63 |
60 62
|
eqtr3d |
⊢ ( 𝜑 → ( 1 σ ( 2 ↑ 𝐴 ) ) = ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) |
64 |
63
|
oveq1d |
⊢ ( 𝜑 → ( ( 1 σ ( 2 ↑ 𝐴 ) ) · ( 1 σ 𝐵 ) ) = ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) · ( 1 σ 𝐵 ) ) ) |
65 |
55 4 64
|
3eqtr3d |
⊢ ( 𝜑 → ( 2 · ( ( 2 ↑ 𝐴 ) · 𝐵 ) ) = ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) · ( 1 σ 𝐵 ) ) ) |
66 |
39 42 65
|
3eqtrd |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) · 𝐵 ) = ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) · ( 1 σ 𝐵 ) ) ) |
67 |
29 66
|
breqtrrd |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) · 𝐵 ) ) |
68 |
23 21
|
gcdcomd |
⊢ ( 𝜑 → ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) gcd ( 2 ↑ ( 𝐴 + 1 ) ) ) = ( ( 2 ↑ ( 𝐴 + 1 ) ) gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) ) |
69 |
|
iddvdsexp |
⊢ ( ( 2 ∈ ℤ ∧ ( 𝐴 + 1 ) ∈ ℕ ) → 2 ∥ ( 2 ↑ ( 𝐴 + 1 ) ) ) |
70 |
50 12 69
|
sylancr |
⊢ ( 𝜑 → 2 ∥ ( 2 ↑ ( 𝐴 + 1 ) ) ) |
71 |
|
n2dvds1 |
⊢ ¬ 2 ∥ 1 |
72 |
50
|
a1i |
⊢ ( 𝜑 → 2 ∈ ℤ ) |
73 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
74 |
72 21 73
|
3jca |
⊢ ( 𝜑 → ( 2 ∈ ℤ ∧ ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℤ ∧ 1 ∈ ℤ ) ) |
75 |
|
dvdssub2 |
⊢ ( ( ( 2 ∈ ℤ ∧ ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℤ ∧ 1 ∈ ℤ ) ∧ 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) → ( 2 ∥ ( 2 ↑ ( 𝐴 + 1 ) ) ↔ 2 ∥ 1 ) ) |
76 |
74 75
|
sylan |
⊢ ( ( 𝜑 ∧ 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) → ( 2 ∥ ( 2 ↑ ( 𝐴 + 1 ) ) ↔ 2 ∥ 1 ) ) |
77 |
71 76
|
mtbiri |
⊢ ( ( 𝜑 ∧ 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) → ¬ 2 ∥ ( 2 ↑ ( 𝐴 + 1 ) ) ) |
78 |
77
|
ex |
⊢ ( 𝜑 → ( 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) → ¬ 2 ∥ ( 2 ↑ ( 𝐴 + 1 ) ) ) ) |
79 |
70 78
|
mt2d |
⊢ ( 𝜑 → ¬ 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) |
80 |
|
coprm |
⊢ ( ( 2 ∈ ℙ ∧ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℤ ) → ( ¬ 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ↔ ( 2 gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) = 1 ) ) |
81 |
45 23 80
|
sylancr |
⊢ ( 𝜑 → ( ¬ 2 ∥ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ↔ ( 2 gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) = 1 ) ) |
82 |
79 81
|
mpbid |
⊢ ( 𝜑 → ( 2 gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) = 1 ) |
83 |
|
rpexp1i |
⊢ ( ( 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 |
50 23 8 83
|
mp3an2i |
⊢ ( 𝜑 → ( ( 2 gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) = 1 → ( ( 2 ↑ ( 𝐴 + 1 ) ) gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) = 1 ) ) |
85 |
82 84
|
mpd |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) gcd ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) = 1 ) |
86 |
68 85
|
eqtrd |
⊢ ( 𝜑 → ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) gcd ( 2 ↑ ( 𝐴 + 1 ) ) ) = 1 ) |
87 |
|
coprmdvds |
⊢ ( ( ( ( 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 |
23 21 46 87
|
syl3anc |
⊢ ( 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 |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∥ 𝐵 ) |
90 |
|
nndivdvds |
⊢ ( ( 𝐵 ∈ ℕ ∧ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℕ ) → ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∥ 𝐵 ↔ ( 𝐵 / ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) ∈ ℕ ) ) |
91 |
2 20 90
|
syl2anc |
⊢ ( 𝜑 → ( ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∥ 𝐵 ↔ ( 𝐵 / ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) ∈ ℕ ) ) |
92 |
89 91
|
mpbid |
⊢ ( 𝜑 → ( 𝐵 / ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) ∈ ℕ ) |
93 |
10 20 92
|
3jca |
⊢ ( 𝜑 → ( ( 2 ↑ ( 𝐴 + 1 ) ) ∈ ℕ ∧ ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ∈ ℕ ∧ ( 𝐵 / ( ( 2 ↑ ( 𝐴 + 1 ) ) − 1 ) ) ∈ ℕ ) ) |