Step |
Hyp |
Ref |
Expression |
1 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
2 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
3 |
|
eluzle |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 3 ≤ 𝑀 ) |
4 |
2 3
|
eqbrtrid |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → ( 2 + 1 ) ≤ 𝑀 ) |
5 |
|
2z |
⊢ 2 ∈ ℤ |
6 |
|
eluzelz |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 𝑀 ∈ ℤ ) |
7 |
|
zltp1le |
⊢ ( ( 2 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 2 < 𝑀 ↔ ( 2 + 1 ) ≤ 𝑀 ) ) |
8 |
5 6 7
|
sylancr |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → ( 2 < 𝑀 ↔ ( 2 + 1 ) ≤ 𝑀 ) ) |
9 |
4 8
|
mpbird |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 2 < 𝑀 ) |
10 |
1 9
|
eqbrtrid |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → ( 1 + 1 ) < 𝑀 ) |
11 |
|
1red |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 1 ∈ ℝ ) |
12 |
|
eluzelre |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 𝑀 ∈ ℝ ) |
13 |
11 11 12
|
ltaddsub2d |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → ( ( 1 + 1 ) < 𝑀 ↔ 1 < ( 𝑀 − 1 ) ) ) |
14 |
10 13
|
mpbid |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 1 < ( 𝑀 − 1 ) ) |
15 |
|
eluzge3nn |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 𝑀 ∈ ℕ ) |
16 |
|
m1modnnsub1 |
⊢ ( 𝑀 ∈ ℕ → ( - 1 mod 𝑀 ) = ( 𝑀 − 1 ) ) |
17 |
15 16
|
syl |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → ( - 1 mod 𝑀 ) = ( 𝑀 − 1 ) ) |
18 |
14 17
|
breqtrrd |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 3 ) → 1 < ( - 1 mod 𝑀 ) ) |