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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
StepHypRef Expression
1 recni.1 . 2 𝐴 ∈ ℝ
2 axri.2 . 2 𝐵 ∈ ℝ
3 remulcl 7468 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
41, 2, 3mp2an 417 1 (𝐴 · 𝐵) ∈ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1438  (class class class)co 5652  cr 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
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