Step |
Hyp |
Ref |
Expression |
1 |
|
dfodd2 |
⊢ Odd = { 𝑧 ∈ ℤ ∣ ( ( 𝑧 − 1 ) / 2 ) ∈ ℤ } |
2 |
|
peano2zm |
⊢ ( 𝑧 ∈ ℤ → ( 𝑧 − 1 ) ∈ ℤ ) |
3 |
2
|
zred |
⊢ ( 𝑧 ∈ ℤ → ( 𝑧 − 1 ) ∈ ℝ ) |
4 |
|
2rp |
⊢ 2 ∈ ℝ+ |
5 |
|
mod0 |
⊢ ( ( ( 𝑧 − 1 ) ∈ ℝ ∧ 2 ∈ ℝ+ ) → ( ( ( 𝑧 − 1 ) mod 2 ) = 0 ↔ ( ( 𝑧 − 1 ) / 2 ) ∈ ℤ ) ) |
6 |
3 4 5
|
sylancl |
⊢ ( 𝑧 ∈ ℤ → ( ( ( 𝑧 − 1 ) mod 2 ) = 0 ↔ ( ( 𝑧 − 1 ) / 2 ) ∈ ℤ ) ) |
7 |
|
zre |
⊢ ( 𝑧 ∈ ℤ → 𝑧 ∈ ℝ ) |
8 |
|
2re |
⊢ 2 ∈ ℝ |
9 |
8
|
a1i |
⊢ ( 𝑧 ∈ ℤ → 2 ∈ ℝ ) |
10 |
|
1lt2 |
⊢ 1 < 2 |
11 |
10
|
a1i |
⊢ ( 𝑧 ∈ ℤ → 1 < 2 ) |
12 |
|
m1mod0mod1 |
⊢ ( ( 𝑧 ∈ ℝ ∧ 2 ∈ ℝ ∧ 1 < 2 ) → ( ( ( 𝑧 − 1 ) mod 2 ) = 0 ↔ ( 𝑧 mod 2 ) = 1 ) ) |
13 |
7 9 11 12
|
syl3anc |
⊢ ( 𝑧 ∈ ℤ → ( ( ( 𝑧 − 1 ) mod 2 ) = 0 ↔ ( 𝑧 mod 2 ) = 1 ) ) |
14 |
6 13
|
bitr3d |
⊢ ( 𝑧 ∈ ℤ → ( ( ( 𝑧 − 1 ) / 2 ) ∈ ℤ ↔ ( 𝑧 mod 2 ) = 1 ) ) |
15 |
14
|
rabbiia |
⊢ { 𝑧 ∈ ℤ ∣ ( ( 𝑧 − 1 ) / 2 ) ∈ ℤ } = { 𝑧 ∈ ℤ ∣ ( 𝑧 mod 2 ) = 1 } |
16 |
1 15
|
eqtri |
⊢ Odd = { 𝑧 ∈ ℤ ∣ ( 𝑧 mod 2 ) = 1 } |