Step |
Hyp |
Ref |
Expression |
1 |
|
znegcl |
⊢ ( 𝐴 ∈ ℤ → - 𝐴 ∈ ℤ ) |
2 |
1
|
adantr |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) → - 𝐴 ∈ ℤ ) |
3 |
|
znegcl |
⊢ ( ( 𝐴 / 2 ) ∈ ℤ → - ( 𝐴 / 2 ) ∈ ℤ ) |
4 |
3
|
adantl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) → - ( 𝐴 / 2 ) ∈ ℤ ) |
5 |
|
zcn |
⊢ ( 𝐴 ∈ ℤ → 𝐴 ∈ ℂ ) |
6 |
|
2cnd |
⊢ ( 𝐴 ∈ ℤ → 2 ∈ ℂ ) |
7 |
|
2ne0 |
⊢ 2 ≠ 0 |
8 |
7
|
a1i |
⊢ ( 𝐴 ∈ ℤ → 2 ≠ 0 ) |
9 |
5 6 8
|
3jca |
⊢ ( 𝐴 ∈ ℤ → ( 𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
10 |
9
|
adantr |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) → ( 𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
11 |
|
divneg |
⊢ ( ( 𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → - ( 𝐴 / 2 ) = ( - 𝐴 / 2 ) ) |
12 |
11
|
eleq1d |
⊢ ( ( 𝐴 ∈ ℂ ∧ 2 ∈ ℂ ∧ 2 ≠ 0 ) → ( - ( 𝐴 / 2 ) ∈ ℤ ↔ ( - 𝐴 / 2 ) ∈ ℤ ) ) |
13 |
10 12
|
syl |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) → ( - ( 𝐴 / 2 ) ∈ ℤ ↔ ( - 𝐴 / 2 ) ∈ ℤ ) ) |
14 |
4 13
|
mpbid |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) → ( - 𝐴 / 2 ) ∈ ℤ ) |
15 |
2 14
|
jca |
⊢ ( ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) → ( - 𝐴 ∈ ℤ ∧ ( - 𝐴 / 2 ) ∈ ℤ ) ) |
16 |
|
iseven |
⊢ ( 𝐴 ∈ Even ↔ ( 𝐴 ∈ ℤ ∧ ( 𝐴 / 2 ) ∈ ℤ ) ) |
17 |
|
iseven |
⊢ ( - 𝐴 ∈ Even ↔ ( - 𝐴 ∈ ℤ ∧ ( - 𝐴 / 2 ) ∈ ℤ ) ) |
18 |
15 16 17
|
3imtr4i |
⊢ ( 𝐴 ∈ Even → - 𝐴 ∈ Even ) |