Axiom ax-mulcl 8278

Description: Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 8234. Proofs should normally use mulcl 8307 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 8178	. . . 4 class ℂ
3	1, 2	wcel 2209	. . 3 wff 𝐴 ∈ ℂ
4		cB	. . . 4 class 𝐵
5	4, 2	wcel 2209	. . 3 wff 𝐵 ∈ ℂ
6	3, 5	wa 104	. 2 wff (𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ)
7		cmul 8185	. . . 4 class ·
8	1, 4, 7	co 6085	. . 3 class (𝐴 · 𝐵)
9	8, 2	wcel 2209	. 2 wff (𝐴 · 𝐵) ∈ ℂ
10	6, 9	wi 4	1 wff ((𝐴 ∈ ℂ ∧ 𝐵 ∈ ℂ) → (𝐴 · 𝐵) ∈ ℂ)

This axiom is used by:  mulcl  8307