Metamath Proof Explorer


Theorem elznn0

Description: Integer property expressed in terms of nonnegative integers. (Contributed by NM, 9-May-2004)

Ref Expression
Assertion elznn0 ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) )

Proof

Step Hyp Ref Expression
1 elz ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ) )
2 elnn0 ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) )
3 2 a1i ( 𝑁 ∈ ℝ → ( 𝑁 ∈ ℕ0 ↔ ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) )
4 elnn0 ( - 𝑁 ∈ ℕ0 ↔ ( - 𝑁 ∈ ℕ ∨ - 𝑁 = 0 ) )
5 recn ( 𝑁 ∈ ℝ → 𝑁 ∈ ℂ )
6 0cn 0 ∈ ℂ
7 negcon1 ( ( 𝑁 ∈ ℂ ∧ 0 ∈ ℂ ) → ( - 𝑁 = 0 ↔ - 0 = 𝑁 ) )
8 5 6 7 sylancl ( 𝑁 ∈ ℝ → ( - 𝑁 = 0 ↔ - 0 = 𝑁 ) )
9 neg0 - 0 = 0
10 9 eqeq1i ( - 0 = 𝑁 ↔ 0 = 𝑁 )
11 eqcom ( 0 = 𝑁𝑁 = 0 )
12 10 11 bitri ( - 0 = 𝑁𝑁 = 0 )
13 8 12 bitrdi ( 𝑁 ∈ ℝ → ( - 𝑁 = 0 ↔ 𝑁 = 0 ) )
14 13 orbi2d ( 𝑁 ∈ ℝ → ( ( - 𝑁 ∈ ℕ ∨ - 𝑁 = 0 ) ↔ ( - 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) )
15 4 14 syl5bb ( 𝑁 ∈ ℝ → ( - 𝑁 ∈ ℕ0 ↔ ( - 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) )
16 3 15 orbi12d ( 𝑁 ∈ ℝ → ( ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ↔ ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ∨ ( - 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) ) )
17 3orass ( ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ↔ ( 𝑁 = 0 ∨ ( 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ) )
18 orcom ( ( 𝑁 = 0 ∨ ( 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ) ↔ ( ( 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ∨ 𝑁 = 0 ) )
19 orordir ( ( ( 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ∨ 𝑁 = 0 ) ↔ ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ∨ ( - 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) )
20 17 18 19 3bitrri ( ( ( 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ∨ ( - 𝑁 ∈ ℕ ∨ 𝑁 = 0 ) ) ↔ ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) )
21 16 20 bitr2di ( 𝑁 ∈ ℝ → ( ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ↔ ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) )
22 21 pm5.32i ( ( 𝑁 ∈ ℝ ∧ ( 𝑁 = 0 ∨ 𝑁 ∈ ℕ ∨ - 𝑁 ∈ ℕ ) ) ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) )
23 1 22 bitri ( 𝑁 ∈ ℤ ↔ ( 𝑁 ∈ ℝ ∧ ( 𝑁 ∈ ℕ0 ∨ - 𝑁 ∈ ℕ0 ) ) )