Axiom ax-mulrcl 6762

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 6733. Proofs should normally use remulcl 6787 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 6690	. . . 4 class ℝ
3	1, 2	wcel 1390	. . 3 wff A ∈ ℝ
4		cB	. . . 4 class B
5	4, 2	wcel 1390	. . 3 wff B ∈ ℝ
6	3, 5	wa 97	. 2 wff (A ∈ ℝ ∧ B ∈ ℝ)
7		cmul 6696	. . . 4 class ·
8	1, 4, 7	co 5455	. . 3 class (A · B)
9	8, 2	wcel 1390	. 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  6787