Metamath Proof Explorer


Theorem ltsub13d

Description: 'Less than' relationship between subtraction and addition. (Contributed by Mario Carneiro, 27-May-2016)

Ref Expression
Hypotheses leidd.1
|- ( ph -> A e. RR )
ltnegd.2
|- ( ph -> B e. RR )
ltadd1d.3
|- ( ph -> C e. RR )
ltsub13d.4
|- ( ph -> A < ( B - C ) )
Assertion ltsub13d
|- ( ph -> C < ( B - 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 ltsub13d.4
 |-  ( ph -> A < ( B - C ) )
5 ltsub13
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( A < ( B - C ) <-> C < ( B - A ) ) )
6 1 2 3 5 syl3anc
 |-  ( ph -> ( A < ( B - C ) <-> C < ( B - A ) ) )
7 4 6 mpbid
 |-  ( ph -> C < ( B - A ) )