Axiom ax-mulf 11252

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 11531 or eff 16215. 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 11256. Note that uses of ax-mulf 11252 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11267.

This axiom is justified by Theorem axmulf 11203. (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 11170	. . 3 class ℂ
2	1, 1	cxp 5645	. 2 class (ℂ × ℂ)
3		cmul 11177	. 2 class ·
4	2, 1, 3	wf 6523	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is used by:  mulnzcnf  11932  mulex  13089  cnfldmul  21648  mulcn  25149  dvdsmulf1o  27487  cncvcOLD  31119  xrge0pluscn  34506