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Axiom ax-mulrcl 10287
Description: Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 10263. Proofs should normally use remulcl 10309 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 10223 . . . 4 class
31, 2wcel 2157 . . 3 wff 𝐴 ∈ ℝ
4 cB . . . 4 class 𝐵
54, 2wcel 2157 . . 3 wff 𝐵 ∈ ℝ
63, 5wa 384 . 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7 cmul 10229 . . . 4 class ·
81, 4, 7co 6877 . . 3 class (𝐴 · 𝐵)
98, 2wcel 2157 . 2 wff (𝐴 · 𝐵) ∈ ℝ
106, 9wi 4 1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)
Colors of variables: wff setvar class
This axiom is referenced by:  remulcl  10309
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