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Axiom ax-mulrcl 7638
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7596. Proofs should normally use remulcl 7666 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 7540 . . . 4 class
31, 2wcel 1461 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 1461 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 103 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cmul 7546 . . . 4 class ·
81, 4, 7co 5726 . . 3 class (𝐴 · 𝐵)
98, 2wcel 1461 . 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  7666
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