Step |
Hyp |
Ref |
Expression |
1 |
|
nnaord |
|- ( ( A e. _om /\ B e. _om /\ C e. _om ) -> ( A e. B <-> ( C +o A ) e. ( C +o B ) ) ) |
2 |
|
nnacom |
|- ( ( C e. _om /\ A e. _om ) -> ( C +o A ) = ( A +o C ) ) |
3 |
2
|
ancoms |
|- ( ( A e. _om /\ C e. _om ) -> ( C +o A ) = ( A +o C ) ) |
4 |
3
|
3adant2 |
|- ( ( A e. _om /\ B e. _om /\ C e. _om ) -> ( C +o A ) = ( A +o C ) ) |
5 |
|
nnacom |
|- ( ( C e. _om /\ B e. _om ) -> ( C +o B ) = ( B +o C ) ) |
6 |
5
|
ancoms |
|- ( ( B e. _om /\ C e. _om ) -> ( C +o B ) = ( B +o C ) ) |
7 |
6
|
3adant1 |
|- ( ( A e. _om /\ B e. _om /\ C e. _om ) -> ( C +o B ) = ( B +o C ) ) |
8 |
4 7
|
eleq12d |
|- ( ( A e. _om /\ B e. _om /\ C e. _om ) -> ( ( C +o A ) e. ( C +o B ) <-> ( A +o C ) e. ( B +o C ) ) ) |
9 |
1 8
|
bitrd |
|- ( ( A e. _om /\ B e. _om /\ C e. _om ) -> ( A e. B <-> ( A +o C ) e. ( B +o C ) ) ) |