Step |
Hyp |
Ref |
Expression |
1 |
|
fmtno |
⊢ ( 𝑁 ∈ ℕ0 → ( FermatNo ‘ 𝑁 ) = ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ) |
2 |
|
3z |
⊢ 3 ∈ ℤ |
3 |
2
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 3 ∈ ℤ ) |
4 |
|
2nn0 |
⊢ 2 ∈ ℕ0 |
5 |
4
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 2 ∈ ℕ0 ) |
6 |
|
id |
⊢ ( 𝑁 ∈ ℕ0 → 𝑁 ∈ ℕ0 ) |
7 |
5 6
|
nn0expcld |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 ↑ 𝑁 ) ∈ ℕ0 ) |
8 |
5 7
|
nn0expcld |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 ↑ ( 2 ↑ 𝑁 ) ) ∈ ℕ0 ) |
9 |
|
peano2nn0 |
⊢ ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) ∈ ℕ0 → ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ∈ ℕ0 ) |
10 |
8 9
|
syl |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ∈ ℕ0 ) |
11 |
10
|
nn0zd |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ∈ ℤ ) |
12 |
|
3m1e2 |
⊢ ( 3 − 1 ) = 2 |
13 |
|
2cn |
⊢ 2 ∈ ℂ |
14 |
|
exp1 |
⊢ ( 2 ∈ ℂ → ( 2 ↑ 1 ) = 2 ) |
15 |
13 14
|
ax-mp |
⊢ ( 2 ↑ 1 ) = 2 |
16 |
|
2re |
⊢ 2 ∈ ℝ |
17 |
16
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 2 ∈ ℝ ) |
18 |
|
1le2 |
⊢ 1 ≤ 2 |
19 |
18
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 1 ≤ 2 ) |
20 |
17 6 19
|
expge1d |
⊢ ( 𝑁 ∈ ℕ0 → 1 ≤ ( 2 ↑ 𝑁 ) ) |
21 |
|
1zzd |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℤ ) |
22 |
7
|
nn0zd |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 ↑ 𝑁 ) ∈ ℤ ) |
23 |
|
1lt2 |
⊢ 1 < 2 |
24 |
23
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 1 < 2 ) |
25 |
17 21 22 24
|
leexp2d |
⊢ ( 𝑁 ∈ ℕ0 → ( 1 ≤ ( 2 ↑ 𝑁 ) ↔ ( 2 ↑ 1 ) ≤ ( 2 ↑ ( 2 ↑ 𝑁 ) ) ) ) |
26 |
20 25
|
mpbid |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 ↑ 1 ) ≤ ( 2 ↑ ( 2 ↑ 𝑁 ) ) ) |
27 |
15 26
|
eqbrtrrid |
⊢ ( 𝑁 ∈ ℕ0 → 2 ≤ ( 2 ↑ ( 2 ↑ 𝑁 ) ) ) |
28 |
12 27
|
eqbrtrid |
⊢ ( 𝑁 ∈ ℕ0 → ( 3 − 1 ) ≤ ( 2 ↑ ( 2 ↑ 𝑁 ) ) ) |
29 |
|
3re |
⊢ 3 ∈ ℝ |
30 |
29
|
a1i |
⊢ ( 𝑁 ∈ ℕ0 → 3 ∈ ℝ ) |
31 |
|
1red |
⊢ ( 𝑁 ∈ ℕ0 → 1 ∈ ℝ ) |
32 |
8
|
nn0red |
⊢ ( 𝑁 ∈ ℕ0 → ( 2 ↑ ( 2 ↑ 𝑁 ) ) ∈ ℝ ) |
33 |
30 31 32
|
lesubaddd |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 3 − 1 ) ≤ ( 2 ↑ ( 2 ↑ 𝑁 ) ) ↔ 3 ≤ ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ) ) |
34 |
28 33
|
mpbid |
⊢ ( 𝑁 ∈ ℕ0 → 3 ≤ ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ) |
35 |
|
eluz2 |
⊢ ( ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ∈ ( ℤ≥ ‘ 3 ) ↔ ( 3 ∈ ℤ ∧ ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ∈ ℤ ∧ 3 ≤ ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ) ) |
36 |
3 11 34 35
|
syl3anbrc |
⊢ ( 𝑁 ∈ ℕ0 → ( ( 2 ↑ ( 2 ↑ 𝑁 ) ) + 1 ) ∈ ( ℤ≥ ‘ 3 ) ) |
37 |
1 36
|
eqeltrd |
⊢ ( 𝑁 ∈ ℕ0 → ( FermatNo ‘ 𝑁 ) ∈ ( ℤ≥ ‘ 3 ) ) |