Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ax-mulf Structured version   Visualization version   GIF version

Axiom ax-mulf 10609
 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- 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 10591 should be used. Note that uses of ax-mulf 10609 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 10560. (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
StepHypRef Expression
1 cc 10527 . . 3 class
21, 1cxp 5551 . 2 class (ℂ × ℂ)
3 cmul 10534 . 2 class ·
42, 1, 3wf 6347 1 wff · :(ℂ × ℂ)⟶ℂ
 Colors of variables: wff setvar class This axiom is referenced by:  mulnzcnopr  11278  mulex  12381  rlimmul  14994  mulcn  23390  iimulcn  23457  dvdsmulf1o  25685  fsumdvdsmul  25686  cncvcOLD  28275  rmulccn  31058  xrge0pluscn  31070
 Copyright terms: Public domain W3C validator