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

Axiom ax-mulf 8833
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 http://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 8815 should be used. Note that uses of ax-mulf 8833 can be eliminated by using the defined operation  ( x  e.  CC ,  y  e.  CC  |->  ( x  x.  y ) ) in place of  x., from which this axiom (with the defined operation in place of  x.) follows as a theorem.

This axiom is justified by theorem axmulf 8784. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

Assertion
Ref Expression
ax-mulf  |-  x.  :
( CC  X.  CC )
--> CC

Detailed syntax breakdown of Axiom ax-mulf
StepHypRef Expression
1 cc 8751 . . 3  class  CC
21, 1cxp 4703 . 2  class  ( CC 
X.  CC )
3 cmul 8758 . 2  class  x.
42, 1, 3wf 5267 1  wff  x.  :
( CC  X.  CC )
--> CC
Colors of variables: wff set class
This axiom is referenced by:  mulnzcnopr  9430  mulex  10369  rlimmul  12134  mulcn  18387  iimulcn  18452  dvdsmulf1o  20450  fsumdvdsmul  20451  efghgrp  21056  cnrngo  21086  cncvc  21155  rmulccn  23316  xrge0pluscn  23337  nZdef  25283
  Copyright terms: Public domain W3C validator