Axiom ax-mulcl 7859

Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7815. Proofs should normally use mulcl 7888 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)

Assertion

Ref	Expression
ax-mulcl	⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

Detailed syntax breakdown of Axiom ax-mulcl

Step	Hyp	Ref	Expression
1		cA	. . . 4 class 𝐴
2		cc 7759	. . . 4 class ℂ
3	1, 2	wcel 2141	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2141	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 7766	. . . 4 class ·
8	1, 4, 7	co 5850	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2141	. 2 wff (𝐴 · 𝐵) ∈ ℂ
10	6, 9	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

This axiom is referenced by:  mulcl  7888