ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  remulcli GIF version

Theorem remulcli 7041
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 7009 . 2 ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
41, 2, 3mp2an 402 1 (𝐴 · 𝐵) ∈ ℝ
Colors of variables: wff set class
Syntax hints:  wcel 1393  (class class class)co 5512  cr 6888   · cmul 6894
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia3 101  ax-mulrcl 6983
This theorem is referenced by:  addltmul  8161  nn0lele2xi  8233  numltc  8387
  Copyright terms: Public domain W3C validator