Axiom ax-mulcl 7847

Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7803. Proofs should normally use mulcl 7876 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 7747	. . . 4 class ℂ
3	1, 2	wcel 2136	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2136	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 103	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 7754	. . . 4 class ·
8	1, 4, 7	co 5841	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2136	. 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  7876