Axiom ax-mulf 11189

Description: Multiplication is an operation on the complex numbers. This deprecated axiom is provided for historical compatibility but is not a bona fide axiom for complex numbers (independent of set theory) since it cannot be interpreted as a first-order or second-order statement (see https://us.metamath.org/downloads/schmidt-cnaxioms.pdf). It may be deleted in the future and should be avoided for new theorems. Instead, the less specific ax-mulcl 11171 should be used. Note that uses of ax-mulf 11189 can be eliminated by using the defined operation (𝑥 ∈ ℂ, 𝑦 ∈ ℂ ↦ (𝑥 · 𝑦)) in place of ·, from which this axiom (with the defined operation in place of ·) follows as a theorem.

This axiom is justified by Theorem axmulf 11140. (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 11107	. . 3 class ℂ
2	1, 1	cxp 5667	. 2 class (ℂ × ℂ)
3		cmul 11114	. 2 class ·
4	2, 1, 3	wf 6532	1 wff · :(ℂ × ℂ)⟶ℂ

This axiom is referenced by:  mulnzcnf  11861  mulex  12976  rlimmulOLD  15594  cnfldmul  21243  dfcnfldOLD  21251  mulcn  24733  iimulcnOLD  24812  dvdsmulf1o  27078  fsumdvdsmulOLD  27079  cncvcOLD  30340  xrge0pluscn  33449