Metamath Proof Explorer


Theorem ltsubrpd

Description: Subtracting a positive real from another number decreases it. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses rpgecld.1
|- ( ph -> A e. RR )
rpgecld.2
|- ( ph -> B e. RR+ )
Assertion ltsubrpd
|- ( ph -> ( A - B ) < A )

Proof

Step Hyp Ref Expression
1 rpgecld.1
 |-  ( ph -> A e. RR )
2 rpgecld.2
 |-  ( ph -> B e. RR+ )
3 ltsubrp
 |-  ( ( A e. RR /\ B e. RR+ ) -> ( A - B ) < A )
4 1 2 3 syl2anc
 |-  ( ph -> ( A - B ) < A )