Theorem remulcli 7500

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

Hypotheses

Ref	Expression
recni.1	⊢ 𝐴 ∈ ℝ
axri.2	⊢ 𝐵 ∈ ℝ

Assertion

Ref	Expression
remulcli	⊢ (𝐴 · 𝐵) ∈ ℝ

Proof of Theorem remulcli

Step	Hyp	Ref	Expression
1		recni.1	. 2 ⊢ 𝐴 ∈ ℝ
2		axri.2	. 2 ⊢ 𝐵 ∈ ℝ
3		remulcl 7468	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
4	1, 2, 3	mp2an 417	1 ⊢ (𝐴 · 𝐵) ∈ ℝ

Syntax hints:   ∈ wcel 1438  (class class class)co 5652  ℝcr 7347   · 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