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Axiom ax-mulrcl 7888
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7844. Proofs should normally use remulcl 7917 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 7788 . . . 4 class
31, 2wcel 2148 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 2148 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 104 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cmul 7794 . . . 4 class ·
81, 4, 7co 5868 . . 3 class (𝐴 · 𝐵)
98, 2wcel 2148 . 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  7917
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