Step |
Hyp |
Ref |
Expression |
1 |
|
xleadd1a |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( A +e C ) <_ ( B +e C ) ) |
2 |
|
xaddcom |
|- ( ( A e. RR* /\ C e. RR* ) -> ( A +e C ) = ( C +e A ) ) |
3 |
2
|
3adant2 |
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A +e C ) = ( C +e A ) ) |
4 |
3
|
adantr |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( A +e C ) = ( C +e A ) ) |
5 |
|
xaddcom |
|- ( ( B e. RR* /\ C e. RR* ) -> ( B +e C ) = ( C +e B ) ) |
6 |
5
|
3adant1 |
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( B +e C ) = ( C +e B ) ) |
7 |
6
|
adantr |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( B +e C ) = ( C +e B ) ) |
8 |
1 4 7
|
3brtr3d |
|- ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( C +e A ) <_ ( C +e B ) ) |