Metamath Proof Explorer


Theorem xleadd2d

Description: Addition of extended reals preserves the "less than or equal to" relation, in the right slot. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses xleadd2d.1
|- ( ph -> A e. RR* )
xleadd2d.2
|- ( ph -> B e. RR* )
xleadd2d.3
|- ( ph -> C e. RR* )
xleadd2d.4
|- ( ph -> A <_ B )
Assertion xleadd2d
|- ( ph -> ( C +e A ) <_ ( C +e B ) )

Proof

Step Hyp Ref Expression
1 xleadd2d.1
 |-  ( ph -> A e. RR* )
2 xleadd2d.2
 |-  ( ph -> B e. RR* )
3 xleadd2d.3
 |-  ( ph -> C e. RR* )
4 xleadd2d.4
 |-  ( ph -> A <_ B )
5 xleadd2a
 |-  ( ( ( A e. RR* /\ B e. RR* /\ C e. RR* ) /\ A <_ B ) -> ( C +e A ) <_ ( C +e B ) )
6 1 2 3 4 5 syl31anc
 |-  ( ph -> ( C +e A ) <_ ( C +e B ) )