Step |
Hyp |
Ref |
Expression |
1 |
|
8nn |
⊢ 8 ∈ ℕ |
2 |
|
4z |
⊢ 4 ∈ ℤ |
3 |
|
9nn |
⊢ 9 ∈ ℕ |
4 |
3
|
nnzi |
⊢ 9 ∈ ℤ |
5 |
|
4re |
⊢ 4 ∈ ℝ |
6 |
|
9re |
⊢ 9 ∈ ℝ |
7 |
|
4lt9 |
⊢ 4 < 9 |
8 |
5 6 7
|
ltleii |
⊢ 4 ≤ 9 |
9 |
|
eluz2 |
⊢ ( 9 ∈ ( ℤ≥ ‘ 4 ) ↔ ( 4 ∈ ℤ ∧ 9 ∈ ℤ ∧ 4 ≤ 9 ) ) |
10 |
2 4 8 9
|
mpbir3an |
⊢ 9 ∈ ( ℤ≥ ‘ 4 ) |
11 |
|
2z |
⊢ 2 ∈ ℤ |
12 |
|
3z |
⊢ 3 ∈ ℤ |
13 |
|
2re |
⊢ 2 ∈ ℝ |
14 |
|
3re |
⊢ 3 ∈ ℝ |
15 |
|
2lt3 |
⊢ 2 < 3 |
16 |
13 14 15
|
ltleii |
⊢ 2 ≤ 3 |
17 |
|
eluz2 |
⊢ ( 3 ∈ ( ℤ≥ ‘ 2 ) ↔ ( 2 ∈ ℤ ∧ 3 ∈ ℤ ∧ 2 ≤ 3 ) ) |
18 |
11 12 16 17
|
mpbir3an |
⊢ 3 ∈ ( ℤ≥ ‘ 2 ) |
19 |
|
nprm |
⊢ ( ( 3 ∈ ( ℤ≥ ‘ 2 ) ∧ 3 ∈ ( ℤ≥ ‘ 2 ) ) → ¬ ( 3 · 3 ) ∈ ℙ ) |
20 |
18 18 19
|
mp2an |
⊢ ¬ ( 3 · 3 ) ∈ ℙ |
21 |
|
df-nel |
⊢ ( 9 ∉ ℙ ↔ ¬ 9 ∈ ℙ ) |
22 |
|
3t3e9 |
⊢ ( 3 · 3 ) = 9 |
23 |
22
|
eqcomi |
⊢ 9 = ( 3 · 3 ) |
24 |
23
|
eleq1i |
⊢ ( 9 ∈ ℙ ↔ ( 3 · 3 ) ∈ ℙ ) |
25 |
21 24
|
xchbinx |
⊢ ( 9 ∉ ℙ ↔ ¬ ( 3 · 3 ) ∈ ℙ ) |
26 |
20 25
|
mpbir |
⊢ 9 ∉ ℙ |
27 |
|
9m1e8 |
⊢ ( 9 − 1 ) = 8 |
28 |
27
|
oveq2i |
⊢ ( 8 ↑ ( 9 − 1 ) ) = ( 8 ↑ 8 ) |
29 |
28
|
oveq1i |
⊢ ( ( 8 ↑ ( 9 − 1 ) ) mod 9 ) = ( ( 8 ↑ 8 ) mod 9 ) |
30 |
|
8exp8mod9 |
⊢ ( ( 8 ↑ 8 ) mod 9 ) = 1 |
31 |
29 30
|
eqtri |
⊢ ( ( 8 ↑ ( 9 − 1 ) ) mod 9 ) = 1 |
32 |
10 26 31
|
3pm3.2i |
⊢ ( 9 ∈ ( ℤ≥ ‘ 4 ) ∧ 9 ∉ ℙ ∧ ( ( 8 ↑ ( 9 − 1 ) ) mod 9 ) = 1 ) |
33 |
|
fpprel |
⊢ ( 8 ∈ ℕ → ( 9 ∈ ( FPPr ‘ 8 ) ↔ ( 9 ∈ ( ℤ≥ ‘ 4 ) ∧ 9 ∉ ℙ ∧ ( ( 8 ↑ ( 9 − 1 ) ) mod 9 ) = 1 ) ) ) |
34 |
32 33
|
mpbiri |
⊢ ( 8 ∈ ℕ → 9 ∈ ( FPPr ‘ 8 ) ) |
35 |
1 34
|
ax-mp |
⊢ 9 ∈ ( FPPr ‘ 8 ) |