Axiom ax-mulrcl 7638

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7596. Proofs should normally use remulcl 7666 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 7540	. . . 4 class ℝ
3	1, 2	wcel 1461	. . 3 wff 𝐴 ∈ ℝ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 1461	. . 3 wff 𝐵 ∈ ℝ
6	3, 5	wa 103	. 2 wff (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ)
7		cmul 7546	. . . 4 class ·
8	1, 4, 7	co 5726	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 1461	. 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  7666