Step |
Hyp |
Ref |
Expression |
1 |
|
evenz |
⊢ ( 𝐴 ∈ Even → 𝐴 ∈ ℤ ) |
2 |
1
|
zcnd |
⊢ ( 𝐴 ∈ Even → 𝐴 ∈ ℂ ) |
3 |
|
evenz |
⊢ ( 𝐵 ∈ Even → 𝐵 ∈ ℤ ) |
4 |
3
|
zcnd |
⊢ ( 𝐵 ∈ Even → 𝐵 ∈ ℂ ) |
5 |
|
negsub |
⊢ ( ( 𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) ) |
6 |
2 4 5
|
syl2an |
⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → ( 𝐴 + - 𝐵 ) = ( 𝐴 − 𝐵 ) ) |
7 |
|
enege |
⊢ ( 𝐵 ∈ Even → - 𝐵 ∈ Even ) |
8 |
|
epee |
⊢ ( ( 𝐴 ∈ Even ∧ - 𝐵 ∈ Even ) → ( 𝐴 + - 𝐵 ) ∈ Even ) |
9 |
7 8
|
sylan2 |
⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → ( 𝐴 + - 𝐵 ) ∈ Even ) |
10 |
6 9
|
eqeltrrd |
⊢ ( ( 𝐴 ∈ Even ∧ 𝐵 ∈ Even ) → ( 𝐴 − 𝐵 ) ∈ Even ) |