Metamath Proof Explorer

Theorem xrlelttr

Description: Transitive law for ordering on extended reals. (Contributed by NM, 19-Jan-2006)

Ref Expression
Assertion xrlelttr 
|- ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B < C ) -> A < C ) )

Proof

Step Hyp Ref Expression
1 xrleloe 
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )
2 1 3adant3 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ B <-> ( A < B \/ A = B ) ) )
3 xrlttr 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B /\ B < C ) -> A < C ) )
4 3 expd 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A < B -> ( B < C -> A < C ) ) )
5 breq1 
 |-  ( A = B -> ( A < C <-> B < C ) )
6 5 biimprd 
 |-  ( A = B -> ( B < C -> A < C ) )
7 6 a1i 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A = B -> ( B < C -> A < C ) ) )
8 4 7 jaod 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A < B \/ A = B ) -> ( B < C -> A < C ) ) )
9 2 8 sylbid 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( A <_ B -> ( B < C -> A < C ) ) )
10 9 impd 
 |-  ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) -> ( ( A <_ B /\ B < C ) -> A < C ) )