Description: A complex number is an integer iff its negative is. (Contributed by Stefan O'Rear, 13-Sep-2014)
Ref | Expression | ||
---|---|---|---|
Assertion | znegclb | ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℤ ↔ - 𝐴 ∈ ℤ ) ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | znegcl | ⊢ ( 𝐴 ∈ ℤ → - 𝐴 ∈ ℤ ) | |
2 | znegcl | ⊢ ( - 𝐴 ∈ ℤ → - - 𝐴 ∈ ℤ ) | |
3 | negneg | ⊢ ( 𝐴 ∈ ℂ → - - 𝐴 = 𝐴 ) | |
4 | 3 | eleq1d | ⊢ ( 𝐴 ∈ ℂ → ( - - 𝐴 ∈ ℤ ↔ 𝐴 ∈ ℤ ) ) |
5 | 2 4 | syl5ib | ⊢ ( 𝐴 ∈ ℂ → ( - 𝐴 ∈ ℤ → 𝐴 ∈ ℤ ) ) |
6 | 1 5 | impbid2 | ⊢ ( 𝐴 ∈ ℂ → ( 𝐴 ∈ ℤ ↔ - 𝐴 ∈ ℤ ) ) |