Axiom ax-mulf 11118

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 11395 or eff 16046. 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 11122. Note that uses of ax-mulf 11118 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11133.

This axiom is justified by Theorem axmulf 11069. (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 11036	. . 3 class ℂ
2	1, 1	cxp 5629	. 2 class (ℂ × ℂ)
3		cmul 11043	. 2 class ·
4	2, 1, 3	wf 6495	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11796  mulex  12941  cnfldmul  21360  mulcn  24833  dvdsmulf1o  27159  cncvcOLD  30654  xrge0pluscn  34084