Metamath Proof Explorer


Theorem fmtnoge3

Description: Each Fermat number is greater than or equal to 3. (Contributed by AV, 4-Aug-2021)

Ref Expression
Assertion fmtnoge3 ( 𝑁 ∈ ℕ0 → ( FermatNo ‘ 𝑁 ) ∈ ( ℤ ‘ 3 ) )

Proof

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 ∈ ℕ0𝑁 ∈ 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 ) )