Axiom ax-mulrcl 6983

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 6943. Proofs should normally use remulcl 7009 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 6888	. . . 4 class ℝ
3	1, 2	wcel 1393	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1393	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 97	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cmul 6894	. . . 4 class ·
8	1, 4, 7	co 5512	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 1393	. 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  7009