Metamath Proof Explorer


Theorem dfodd4

Description: Alternate definition for odd numbers. (Contributed by AV, 18-Jun-2020)

Ref Expression
Assertion dfodd4 Odd = { 𝑧 ∈ ℤ ∣ ( 𝑧 mod 2 ) = 1 }

Proof

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 }