Metamath Proof Explorer


Theorem lttrd

Description: Transitive law deduction for 'less than'. (Contributed by NM, 9-Jan-2006)

Ref Expression
Hypotheses ltd.1
|- ( ph -> A e. RR )
ltd.2
|- ( ph -> B e. RR )
letrd.3
|- ( ph -> C e. RR )
lttrd.4
|- ( ph -> A < B )
lttrd.5
|- ( ph -> B < C )
Assertion lttrd
|- ( ph -> A < C )

Proof

Step Hyp Ref Expression
1 ltd.1
 |-  ( ph -> A e. RR )
2 ltd.2
 |-  ( ph -> B e. RR )
3 letrd.3
 |-  ( ph -> C e. RR )
4 lttrd.4
 |-  ( ph -> A < B )
5 lttrd.5
 |-  ( ph -> B < C )
6 lttr
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A < B /\ B < C ) -> A < C ) )
7 1 2 3 6 syl3anc
 |-  ( ph -> ( ( A < B /\ B < C ) -> A < C ) )
8 4 5 7 mp2and
 |-  ( ph -> A < C )