Metamath Proof Explorer


Theorem subled

Description: Swap subtrahends in an inequality. (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 )
subled.4
|- ( ph -> ( A - B ) <_ C )
Assertion subled
|- ( ph -> ( A - C ) <_ B )

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 subled.4
 |-  ( ph -> ( A - B ) <_ C )
5 suble
 |-  ( ( A e. RR /\ B e. RR /\ C e. RR ) -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) )
6 1 2 3 5 syl3anc
 |-  ( ph -> ( ( A - B ) <_ C <-> ( A - C ) <_ B ) )
7 4 6 mpbid
 |-  ( ph -> ( A - C ) <_ B )