Axiom ax-mulrcl 8268

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 8224. Proofs should normally use remulcl 8297 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 8168	. . . 4 class ℝ
3	1, 2	wcel 2209	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2209	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cmul 8174	. . . 4 class ·
8	1, 4, 7	co 6075	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2209	. 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  8297