Axiom ax-mulrcl 6782

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 6753. Proofs should normally use remulcl 6807 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 6710	. . . 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 6716	. . . 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  6807