Axiom ax-mulrcl 6742

Description: Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 6713. Proofs should normally use remulcl 6767 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 6670	. . . 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 6676	. . . 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  6767