Axiom ax-mulf 11207

Description: Multiplication is an operation on the complex numbers. This axiom tells us that · is defined only on complex numbers which is analogous to the way that other operations are defined, for example see subf 11486 or eff 16171. However, while Metamath can handle this axiom, if we wish to work with weaker complex number axioms, we can avoid it by using the less specific mulcl 11211. Note that uses of ax-mulf 11207 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11222.

This axiom is justified by Theorem axmulf 11158. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

Assertion

Ref	Expression
ax-mulf	⊢ · :(ℂ × ℂ)⟶ℂ

Detailed syntax breakdown of Axiom ax-mulf

Step	Hyp	Ref	Expression
1		cc 11125	. . 3 class ℂ
2	1, 1	cxp 5657	. 2 class (ℂ × ℂ)
3		cmul 11132	. 2 class ·
4	2, 1, 3	wf 6533	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is used by:  mulnzcnf  11887  mulex  13043  cnfldmul  21597  mulcn  25098  dvdsmulf1o  27433  cncvcOLD  31065  xrge0pluscn  34452