Step |
Hyp |
Ref |
Expression |
1 |
|
breq1 |
⊢ ( 𝑃 = 2 → ( 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ↔ 2 ∥ ( FermatNo ‘ 𝑁 ) ) ) |
2 |
1
|
adantr |
⊢ ( ( 𝑃 = 2 ∧ 𝑁 ∈ ℕ ) → ( 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ↔ 2 ∥ ( FermatNo ‘ 𝑁 ) ) ) |
3 |
|
nnnn0 |
⊢ ( 𝑁 ∈ ℕ → 𝑁 ∈ ℕ0 ) |
4 |
|
fmtnoodd |
⊢ ( 𝑁 ∈ ℕ0 → ¬ 2 ∥ ( FermatNo ‘ 𝑁 ) ) |
5 |
3 4
|
syl |
⊢ ( 𝑁 ∈ ℕ → ¬ 2 ∥ ( FermatNo ‘ 𝑁 ) ) |
6 |
5
|
adantl |
⊢ ( ( 𝑃 = 2 ∧ 𝑁 ∈ ℕ ) → ¬ 2 ∥ ( FermatNo ‘ 𝑁 ) ) |
7 |
6
|
pm2.21d |
⊢ ( ( 𝑃 = 2 ∧ 𝑁 ∈ ℕ ) → ( 2 ∥ ( FermatNo ‘ 𝑁 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
8 |
2 7
|
sylbid |
⊢ ( ( 𝑃 = 2 ∧ 𝑁 ∈ ℕ ) → ( 𝑃 ∥ ( FermatNo ‘ 𝑁 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
9 |
8
|
a1d |
⊢ ( ( 𝑃 = 2 ∧ 𝑁 ∈ ℕ ) → ( 𝑃 ∈ ℙ → ( 𝑃 ∥ ( FermatNo ‘ 𝑁 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) ) |
10 |
9
|
ex |
⊢ ( 𝑃 = 2 → ( 𝑁 ∈ ℕ → ( 𝑃 ∈ ℙ → ( 𝑃 ∥ ( FermatNo ‘ 𝑁 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) ) ) |
11 |
10
|
3impd |
⊢ ( 𝑃 = 2 → ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
12 |
|
simpr1 |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) ) → 𝑁 ∈ ℕ ) |
13 |
|
neqne |
⊢ ( ¬ 𝑃 = 2 → 𝑃 ≠ 2 ) |
14 |
13
|
anim2i |
⊢ ( ( 𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2 ) → ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 2 ) ) |
15 |
|
eldifsn |
⊢ ( 𝑃 ∈ ( ℙ ∖ { 2 } ) ↔ ( 𝑃 ∈ ℙ ∧ 𝑃 ≠ 2 ) ) |
16 |
14 15
|
sylibr |
⊢ ( ( 𝑃 ∈ ℙ ∧ ¬ 𝑃 = 2 ) → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
17 |
16
|
ex |
⊢ ( 𝑃 ∈ ℙ → ( ¬ 𝑃 = 2 → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) ) |
18 |
17
|
3ad2ant2 |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) → ( ¬ 𝑃 = 2 → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) ) |
19 |
18
|
impcom |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) ) → 𝑃 ∈ ( ℙ ∖ { 2 } ) ) |
20 |
|
simpr3 |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) ) → 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) |
21 |
|
fmtnoprmfac1lem |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ( ℙ ∖ { 2 } ) ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) → ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) ) |
22 |
12 19 20 21
|
syl3anc |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) ) → ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) ) |
23 |
|
prmnn |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ ) |
24 |
23
|
ad2antll |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → 𝑃 ∈ ℕ ) |
25 |
|
2z |
⊢ 2 ∈ ℤ |
26 |
25
|
a1i |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → 2 ∈ ℤ ) |
27 |
13
|
necomd |
⊢ ( ¬ 𝑃 = 2 → 2 ≠ 𝑃 ) |
28 |
27
|
adantr |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → 2 ≠ 𝑃 ) |
29 |
|
2prm |
⊢ 2 ∈ ℙ |
30 |
29
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℙ ) |
31 |
30
|
anim1i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) → ( 2 ∈ ℙ ∧ 𝑃 ∈ ℙ ) ) |
32 |
31
|
adantl |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( 2 ∈ ℙ ∧ 𝑃 ∈ ℙ ) ) |
33 |
|
prmrp |
⊢ ( ( 2 ∈ ℙ ∧ 𝑃 ∈ ℙ ) → ( ( 2 gcd 𝑃 ) = 1 ↔ 2 ≠ 𝑃 ) ) |
34 |
32 33
|
syl |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( 2 gcd 𝑃 ) = 1 ↔ 2 ≠ 𝑃 ) ) |
35 |
28 34
|
mpbird |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( 2 gcd 𝑃 ) = 1 ) |
36 |
|
odzphi |
⊢ ( ( 𝑃 ∈ ℕ ∧ 2 ∈ ℤ ∧ ( 2 gcd 𝑃 ) = 1 ) → ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( ϕ ‘ 𝑃 ) ) |
37 |
24 26 35 36
|
syl3anc |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( ϕ ‘ 𝑃 ) ) |
38 |
|
phiprm |
⊢ ( 𝑃 ∈ ℙ → ( ϕ ‘ 𝑃 ) = ( 𝑃 − 1 ) ) |
39 |
38
|
ad2antll |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ϕ ‘ 𝑃 ) = ( 𝑃 − 1 ) ) |
40 |
39
|
breq2d |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( ϕ ‘ 𝑃 ) ↔ ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( 𝑃 − 1 ) ) ) |
41 |
|
breq1 |
⊢ ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( 𝑃 − 1 ) ↔ ( 2 ↑ ( 𝑁 + 1 ) ) ∥ ( 𝑃 − 1 ) ) ) |
42 |
41
|
adantl |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( 𝑃 − 1 ) ↔ ( 2 ↑ ( 𝑁 + 1 ) ) ∥ ( 𝑃 − 1 ) ) ) |
43 |
|
2nn |
⊢ 2 ∈ ℕ |
44 |
43
|
a1i |
⊢ ( 𝑁 ∈ ℕ → 2 ∈ ℕ ) |
45 |
|
peano2nn |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 + 1 ) ∈ ℕ ) |
46 |
45
|
nnnn0d |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 + 1 ) ∈ ℕ0 ) |
47 |
44 46
|
nnexpcld |
⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℕ ) |
48 |
23
|
nnnn0d |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℕ0 ) |
49 |
|
prmuz2 |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ( ℤ≥ ‘ 2 ) ) |
50 |
|
eluzle |
⊢ ( 𝑃 ∈ ( ℤ≥ ‘ 2 ) → 2 ≤ 𝑃 ) |
51 |
49 50
|
syl |
⊢ ( 𝑃 ∈ ℙ → 2 ≤ 𝑃 ) |
52 |
|
nn0ge2m1nn |
⊢ ( ( 𝑃 ∈ ℕ0 ∧ 2 ≤ 𝑃 ) → ( 𝑃 − 1 ) ∈ ℕ ) |
53 |
48 51 52
|
syl2anc |
⊢ ( 𝑃 ∈ ℙ → ( 𝑃 − 1 ) ∈ ℕ ) |
54 |
47 53
|
anim12i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) → ( ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℕ ∧ ( 𝑃 − 1 ) ∈ ℕ ) ) |
55 |
54
|
adantl |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℕ ∧ ( 𝑃 − 1 ) ∈ ℕ ) ) |
56 |
|
nndivides |
⊢ ( ( ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℕ ∧ ( 𝑃 − 1 ) ∈ ℕ ) → ( ( 2 ↑ ( 𝑁 + 1 ) ) ∥ ( 𝑃 − 1 ) ↔ ∃ 𝑘 ∈ ℕ ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) ) ) |
57 |
55 56
|
syl |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( 2 ↑ ( 𝑁 + 1 ) ) ∥ ( 𝑃 − 1 ) ↔ ∃ 𝑘 ∈ ℕ ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) ) ) |
58 |
|
eqcom |
⊢ ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) ↔ ( 𝑃 − 1 ) = ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) ) |
59 |
58
|
a1i |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) ↔ ( 𝑃 − 1 ) = ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) ) ) |
60 |
23
|
nncnd |
⊢ ( 𝑃 ∈ ℙ → 𝑃 ∈ ℂ ) |
61 |
60
|
adantl |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) → 𝑃 ∈ ℂ ) |
62 |
61
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ∧ 𝑘 ∈ ℕ ) → 𝑃 ∈ ℂ ) |
63 |
|
1cnd |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ∧ 𝑘 ∈ ℕ ) → 1 ∈ ℂ ) |
64 |
|
nncn |
⊢ ( 𝑘 ∈ ℕ → 𝑘 ∈ ℂ ) |
65 |
64
|
adantl |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ∧ 𝑘 ∈ ℕ ) → 𝑘 ∈ ℂ ) |
66 |
|
peano2nn0 |
⊢ ( 𝑁 ∈ ℕ0 → ( 𝑁 + 1 ) ∈ ℕ0 ) |
67 |
3 66
|
syl |
⊢ ( 𝑁 ∈ ℕ → ( 𝑁 + 1 ) ∈ ℕ0 ) |
68 |
44 67
|
nnexpcld |
⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℕ ) |
69 |
68
|
nncnd |
⊢ ( 𝑁 ∈ ℕ → ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℂ ) |
70 |
69
|
adantr |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) → ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℂ ) |
71 |
70
|
adantr |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ∧ 𝑘 ∈ ℕ ) → ( 2 ↑ ( 𝑁 + 1 ) ) ∈ ℂ ) |
72 |
65 71
|
mulcld |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ∧ 𝑘 ∈ ℕ ) → ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) ∈ ℂ ) |
73 |
62 63 72
|
subadd2d |
⊢ ( ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑃 − 1 ) = ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) ↔ ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) = 𝑃 ) ) |
74 |
73
|
adantll |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑃 − 1 ) = ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) ↔ ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) = 𝑃 ) ) |
75 |
|
eqcom |
⊢ ( ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) = 𝑃 ↔ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) |
76 |
75
|
a1i |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) = 𝑃 ↔ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
77 |
59 74 76
|
3bitrd |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ 𝑘 ∈ ℕ ) → ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) ↔ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
78 |
77
|
rexbidva |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ∃ 𝑘 ∈ ℕ ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) ↔ ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
79 |
78
|
biimpd |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ∃ 𝑘 ∈ ℕ ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) = ( 𝑃 − 1 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
80 |
57 79
|
sylbid |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( 2 ↑ ( 𝑁 + 1 ) ) ∥ ( 𝑃 − 1 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
81 |
80
|
adantr |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) ) → ( ( 2 ↑ ( 𝑁 + 1 ) ) ∥ ( 𝑃 − 1 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
82 |
42 81
|
sylbid |
⊢ ( ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) ∧ ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( 𝑃 − 1 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
83 |
82
|
ex |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( 𝑃 − 1 ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) ) |
84 |
83
|
com23 |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( 𝑃 − 1 ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) ) |
85 |
40 84
|
sylbid |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) ∥ ( ϕ ‘ 𝑃 ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) ) |
86 |
37 85
|
mpd |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
87 |
86
|
3adantr3 |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) ) → ( ( ( odℤ ‘ 𝑃 ) ‘ 2 ) = ( 2 ↑ ( 𝑁 + 1 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
88 |
22 87
|
mpd |
⊢ ( ( ¬ 𝑃 = 2 ∧ ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) |
89 |
88
|
ex |
⊢ ( ¬ 𝑃 = 2 → ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) ) |
90 |
11 89
|
pm2.61i |
⊢ ( ( 𝑁 ∈ ℕ ∧ 𝑃 ∈ ℙ ∧ 𝑃 ∥ ( FermatNo ‘ 𝑁 ) ) → ∃ 𝑘 ∈ ℕ 𝑃 = ( ( 𝑘 · ( 2 ↑ ( 𝑁 + 1 ) ) ) + 1 ) ) |