Step |
Hyp |
Ref |
Expression |
1 |
|
repnpcan |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + B ) -R ( A + C ) ) = ( B -R C ) ) |
2 |
|
readdcl |
|- ( ( A e. RR /\ B e. RR ) -> ( A + B ) e. RR ) |
3 |
2
|
3adant3 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + B ) e. RR ) |
4 |
|
readdcl |
|- ( ( A e. RR /\ C e. RR ) -> ( A + C ) e. RR ) |
5 |
4
|
3adant2 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A + C ) e. RR ) |
6 |
|
rersubcl |
|- ( ( B e. RR /\ C e. RR ) -> ( B -R C ) e. RR ) |
7 |
6
|
3adant1 |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( B -R C ) e. RR ) |
8 |
3 5 7
|
resubaddd |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( ( A + B ) -R ( A + C ) ) = ( B -R C ) <-> ( ( A + C ) + ( B -R C ) ) = ( A + B ) ) ) |
9 |
1 8
|
mpbid |
|- ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A + C ) + ( B -R C ) ) = ( A + B ) ) |