Step |
Hyp |
Ref |
Expression |
1 |
|
evenz |
|- ( Z e. Even -> Z e. ZZ ) |
2 |
|
peano2zm |
|- ( Z e. ZZ -> ( Z - 1 ) e. ZZ ) |
3 |
1 2
|
syl |
|- ( Z e. Even -> ( Z - 1 ) e. ZZ ) |
4 |
|
iseven |
|- ( Z e. Even <-> ( Z e. ZZ /\ ( Z / 2 ) e. ZZ ) ) |
5 |
|
zcn |
|- ( Z e. ZZ -> Z e. CC ) |
6 |
|
npcan1 |
|- ( Z e. CC -> ( ( Z - 1 ) + 1 ) = Z ) |
7 |
5 6
|
syl |
|- ( Z e. ZZ -> ( ( Z - 1 ) + 1 ) = Z ) |
8 |
7
|
eqcomd |
|- ( Z e. ZZ -> Z = ( ( Z - 1 ) + 1 ) ) |
9 |
8
|
oveq1d |
|- ( Z e. ZZ -> ( Z / 2 ) = ( ( ( Z - 1 ) + 1 ) / 2 ) ) |
10 |
9
|
eleq1d |
|- ( Z e. ZZ -> ( ( Z / 2 ) e. ZZ <-> ( ( ( Z - 1 ) + 1 ) / 2 ) e. ZZ ) ) |
11 |
10
|
biimpa |
|- ( ( Z e. ZZ /\ ( Z / 2 ) e. ZZ ) -> ( ( ( Z - 1 ) + 1 ) / 2 ) e. ZZ ) |
12 |
4 11
|
sylbi |
|- ( Z e. Even -> ( ( ( Z - 1 ) + 1 ) / 2 ) e. ZZ ) |
13 |
|
isodd |
|- ( ( Z - 1 ) e. Odd <-> ( ( Z - 1 ) e. ZZ /\ ( ( ( Z - 1 ) + 1 ) / 2 ) e. ZZ ) ) |
14 |
3 12 13
|
sylanbrc |
|- ( Z e. Even -> ( Z - 1 ) e. Odd ) |