Metamath Proof Explorer


Theorem ltsub2dd

Description: Subtraction of both sides of 'less than'. (Contributed by Mario Carneiro, 30-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
ltadd1d.3
|- ( ph -> C e. RR )
ltadd1dd.4
|- ( ph -> A < B )
Assertion ltsub2dd
|- ( ph -> ( C - B ) < ( C - A ) )

Proof

Step Hyp Ref Expression
1 leidd.1
 |-  ( ph -> A e. RR )
2 ltnegd.2
 |-  ( ph -> B e. RR )
3 ltadd1d.3
 |-  ( ph -> C e. RR )
4 ltadd1dd.4
 |-  ( ph -> A < B )
5 1 2 3 ltsub2d
 |-  ( ph -> ( A < B <-> ( C - B ) < ( C - A ) ) )
6 4 5 mpbid
 |-  ( ph -> ( C - B ) < ( C - A ) )