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

Axiom ax-mulrcl 7041
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7001. Proofs should normally use remulcl 7067 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
Assertion
Ref Expression
ax-mulrcl ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)

Detailed syntax breakdown of Axiom ax-mulrcl
StepHypRef Expression
1 cA . . . 4 class 𝐴
2 cr 6946 . . . 4 class
31, 2wcel 1409 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1409 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 101 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cmul 6952 . . . 4 class ·
81, 4, 7co 5540 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1409 . 2 wff (𝐴 · 𝐵) ∈ ℝ
106, 9wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Colors of variables: wff set class
This axiom is referenced by:  remulcl  7067
  Copyright terms: Public domain W3C validator