Axiom ax-mulf 11186

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 11465 or eff 16141. 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 11190. Note that uses of ax-mulf 11186 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11201.

This axiom is justified by Theorem axmulf 11137. (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 11104	. . 3 class ℂ
2	1, 1	cxp 5658	. 2 class (ℂ × ℂ)
3		cmul 11111	. 2 class ·
4	2, 1, 3	wf 6532	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is used by:  mulnzcnf  11866  mulex  13021  cnfldmul  21541  mulcn  25036  dvdsmulf1o  27371  cncvcOLD  30946  xrge0pluscn  34339