Axiom ax-mulrcl 7981

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7937. Proofs should normally use remulcl 8010 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

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cr 7881	. . . 4 class ℝ
3	1, 2	wcel 2167	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2167	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cmul 7887	. . . 4 class ·
8	1, 4, 7	co 5923	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2167	. 2 wff (𝐴 · 𝐵) ∈ ℝ
10	6, 9	wi 4	1 wff ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 · 𝐵) ∈ ℝ)

This axiom is referenced by:  remulcl  8010