Step |
Hyp |
Ref |
Expression |
1 |
|
zcn |
|- ( C e. ZZ -> C e. CC ) |
2 |
1
|
3ad2ant3 |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> C e. CC ) |
3 |
2
|
negnegd |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> -u -u C = C ) |
4 |
3
|
oveq2d |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( B - -u -u C ) = ( B - C ) ) |
5 |
4
|
breq2d |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A || ( B - -u -u C ) <-> A || ( B - C ) ) ) |
6 |
5
|
biimpd |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( A || ( B - -u -u C ) -> A || ( B - C ) ) ) |
7 |
6
|
orim2d |
|- ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) -> ( ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) -> ( A || ( B - -u C ) \/ A || ( B - C ) ) ) ) |
8 |
7
|
imp |
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) ) -> ( A || ( B - -u C ) \/ A || ( B - C ) ) ) |
9 |
8
|
orcomd |
|- ( ( ( A e. ZZ /\ B e. ZZ /\ C e. ZZ ) /\ ( A || ( B - -u C ) \/ A || ( B - -u -u C ) ) ) -> ( A || ( B - C ) \/ A || ( B - -u C ) ) ) |