Axiom ax-mulrcl 8130

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 8086. Proofs should normally use remulcl 8159 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 8030	. . . 4 class ℝ
3	1, 2	wcel 2202	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2202	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cmul 8036	. . . 4 class ·
8	1, 4, 7	co 6017	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2202	. 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  8159