Axiom ax-mulf 11281

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 11560 or eff 16248. 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 11285. Note that uses of ax-mulf 11281 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, as seen in mpomulf 11296.

This axiom is justified by Theorem axmulf 11232. (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 11199	. . 3 class ℂ
2	1, 1	cxp 5648	. 2 class (ℂ × ℂ)
3		cmul 11206	. 2 class ·
4	2, 1, 3	wf 6534	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is used by:  mulnzcnf  11963  mulex  13120  cnfldmul  21687  mulcn  25188  dvdsmulf1o  27524  cncvcOLD  31186  xrge0pluscn  34615