Theorem remulcli 7500

Description: Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)

Hypotheses

Ref	Expression
recni.1	|- A e. RR
axri.2	|- B e. RR

Assertion

Ref	Expression
remulcli	|- ( A x. B ) e. RR

Proof of Theorem remulcli

Step	Hyp	Ref	Expression
1		recni.1	. 2 |- A e. RR
2		axri.2	. 2 |- B e. RR
3		remulcl 7468	. 2 |- ( ( A e. RR /\ B e. RR ) -> ( A x. B ) e. RR )
4	1, 2, 3	mp2an 417	1 |- ( A x. B ) e. RR

Syntax hints:   e. wcel 1438  (class class class)co 5652   RRcr 7347   x. cmul 7353

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 106  ax-mulrcl 7442

This theorem is referenced by:  addltmul  8650  nn0lele2xi  8722  numltc  8900  ef01bndlem  11043  cos2bnd  11047  sin4lt0  11053