Step |
Hyp |
Ref |
Expression |
1 |
|
subcl |
|- ( ( A e. CC /\ B e. CC ) -> ( A - B ) e. CC ) |
2 |
1
|
3adant3 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( A - B ) e. CC ) |
3 |
|
addsub |
|- ( ( ( A - B ) e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + B ) - C ) = ( ( ( A - B ) - C ) + B ) ) |
4 |
3
|
eqcomd |
|- ( ( ( A - B ) e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) + B ) = ( ( ( A - B ) + B ) - C ) ) |
5 |
2 4
|
syld3an1 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) + B ) = ( ( ( A - B ) + B ) - C ) ) |
6 |
|
npcan |
|- ( ( A e. CC /\ B e. CC ) -> ( ( A - B ) + B ) = A ) |
7 |
6
|
3adant3 |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( A - B ) + B ) = A ) |
8 |
7
|
oveq1d |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) + B ) - C ) = ( A - C ) ) |
9 |
5 8
|
eqtrd |
|- ( ( A e. CC /\ B e. CC /\ C e. CC ) -> ( ( ( A - B ) - C ) + B ) = ( A - C ) ) |