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Axiom ax-mulrcl 7391
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7351. Proofs should normally use remulcl 7417 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 7296 . . . 4 class
31, 2wcel 1436 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1436 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 102 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cmul 7302 . . . 4 class ·
81, 4, 7co 5615 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1436 . 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  7417
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