Metamath Proof Explorer


Theorem xmulcld

Description: Closure of extended real multiplication. (Contributed by Mario Carneiro, 28-May-2016)

Ref Expression
Hypotheses xnegcld.1
|- ( ph -> A e. RR* )
xaddcld.2
|- ( ph -> B e. RR* )
Assertion xmulcld
|- ( ph -> ( A *e B ) e. RR* )

Proof

Step Hyp Ref Expression
1 xnegcld.1
 |-  ( ph -> A e. RR* )
2 xaddcld.2
 |-  ( ph -> B e. RR* )
3 xmulcl
 |-  ( ( A e. RR* /\ B e. RR* ) -> ( A *e B ) e. RR* )
4 1 2 3 syl2anc
 |-  ( ph -> ( A *e B ) e. RR* )