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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremaxmulf 8701 Multiplication is an operation on the complex numbers. This theorem can be used as an alternate axiom for complex numbers in place of the less specific axmulcl 8708. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8750. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxcnex 8702 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10282), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus we can avoid ax-rep 4071 in later theorems by invoking the axiom ax-cnex 8726 instead of cnexALT 10282. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 8703 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 8727. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 8704 1 is a complex number. Axiom 2 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 8728. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremaxicn 8705  _i is a complex number. Axiom 3 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 8729. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 8706 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-addcl 8730. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 8707 Closure law for addition in the real subfield of complex numbers. Axiom 5 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-addrcl 8731. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 8708 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulcl 8732. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 8709 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulrcl 8733. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxmulcom 8710 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulcom 8734. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 8711 Addition of complex numbers is associative. This theorem transfers the associative laws for the real and imaginary signed real components of complex number pairs, to complex number addition itself. Axiom 9 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-addass 8735. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremaxmulass 8712 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 8736. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Theoremaxdistr 8713 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-distr 8737. (Contributed by NM, 2-Sep-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C )
 )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Theoremaxi2m1 8714 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 8738. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax1ne0 8715 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1ne0 8739. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
 |-  1  =/=  0
 
Theoremax1rid 8716  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, derived from ZF set theory. Weakened from the original axiom in the form of statement in mulid1 8767, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8740. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremaxrnegex 8717* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 8741. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxrrecex 8718* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rrecex 8742. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Theoremaxcnre 8719* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, derived from ZF set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 8743. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theoremaxpre-lttri 8720 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttri 8827. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8744. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Theoremaxpre-lttrn 8721 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axlttrn 8828. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8745. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <RR  B 
 /\  B  <RR  C ) 
 ->  A  <RR  C ) )
 
Theoremaxpre-ltadd 8722 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axltadd 8829. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8746. (Contributed by NM, 11-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A ) 
 <RR  ( C  +  B ) ) )
 
Theoremaxpre-mulgt0 8723 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version for extended reals is axmulgt0 8830. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8747. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
 
Theoremaxpre-sup 8724* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, derived from ZF set theory. Note: The more general version with ordering on extended reals is axsup 8831. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8748. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 16-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x ) 
 ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
 y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
 
Theoremwuncn 8725 A weak universe containing  om contains the complex number construction. This theorem is construction-dependent in the literal sense, but will also be satisfied by any other reasonable implementation of the complex numbers. (Contributed by Mario Carneiro, 2-Jan-2017.)
 |-  ( ph  ->  U  e. WUni )   &    |-  ( ph  ->  om  e.  U )   =>    |-  ( ph  ->  CC  e.  U )
 
5.1.3  Real and complex number postulates restated as axioms
 
Axiomax-cnex 8726 The complex numbers form a set. This axiom is redundant - see cnexALT 10282- but we provide this axiom because the justification theorem axcnex 8702 does not use ax-rep 4071 even though the redundancy proof does. (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 8727 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8703. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 8728 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8704. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-icn 8729  _i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8705. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 8730 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8706. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Axiomax-addrcl 8731 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8707. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Axiomax-mulcl 8732 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8708. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 8733 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8709. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-mulcom 8734 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 8710. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Axiomax-addass 8735 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 8711. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Axiomax-mulass 8736 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8712. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Axiomax-distr 8737 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8713. (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C )
 )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Axiomax-i2m1 8738 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom 12 of 22 for real and complex numbers, justified by theorem axi2m1 8714. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-1ne0 8739 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 8715. (Contributed by NM, 29-Jan-1995.)
 |-  1  =/=  0
 
Axiomax-1rid 8740  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 8716. Weakened from the original axiom in the form of statement in mulid1 8767, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-rnegex 8741* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 8717. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-rrecex 8742* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 8718. (Contributed by Eric Schmidt, 11-Apr-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Axiomax-cnre 8743* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom 17 of 22 for real and complex numbers, justified by theorem axcnre 8719. (Contributed by NM, 9-May-1999.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Axiomax-pre-lttri 8744 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 8720. Note: The more general version for extended reals is axlttri 8827. (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Axiomax-pre-lttrn 8745 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 8721. Note: The more general version for extended reals is axlttrn 8828. (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  <RR  B 
 /\  B  <RR  C ) 
 ->  A  <RR  C ) )
 
Axiomax-pre-ltadd 8746 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 8722. Note: The more general version for extended reals is axltadd 8829. (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( C  +  A ) 
 <RR  ( C  +  B ) ) )
 
Axiomax-pre-mulgt0 8747 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 8723. Note: The more general version for extended reals is axmulgt0 8830. (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( 0 
 <RR  A  /\  0  <RR  B )  ->  0  <RR  ( A  x.  B ) ) )
 
Axiomax-pre-sup 8748* A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 8724. Note: The more general version with ordering on extended reals is axsup 8831. (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x ) 
 ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
 y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
 
Axiomax-addf 8749 Addition 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-addcl 8730 should be used. Note that uses of ax-addf 8749 can be eliminated by using the defined operation  ( x  e.  CC ,  y  e.  CC  |->  ( 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 axaddf 8700. (Contributed by NM, 19-Oct-2004.)

 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Axiomax-mulf 8750 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 8732 should be used. Note that uses of ax-mulf 8750 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 8701. (Contributed by NM, 19-Oct-2004.)

 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
5.2  Derive the basic properties from the field axioms
 
5.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 8751 Alias for ax-cnex 8726. See also cnexALT 10282. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 CC  e.  _V
 
Theoremaddcl 8752 Alias for ax-addcl 8730, for naming consistency with addcli 8774. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremreaddcl 8753 Alias for ax-addrcl 8731, for naming consistency with readdcli 8783. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremmulcl 8754 Alias for ax-mulcl 8732, for naming consistency with mulcli 8775. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremremulcl 8755 Alias for ax-mulrcl 8733, for naming consistency with remulcli 8784. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremmulcom 8756 Alias for ax-mulcom 8734, for naming consistency with mulcomi 8776. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaddass 8757 Alias for ax-addass 8735, for naming consistency with addassi 8778. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremmulass 8758 Alias for ax-mulass 8736, for naming consistency with mulassi 8779. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Theoremadddi 8759 Alias for ax-distr 8737, for naming consistency with adddii 8780. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C )
 )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Theoremrecn 8760 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  RR  ->  A  e.  CC )
 
Theoremreex 8761 The real numbers form a set. See also reexALT 10280. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 RR  e.  _V
 
Theoremelimne0 8762 Hypothesis for weak deduction theorem to eliminate  A  =/=  0. (Contributed by NM, 15-May-1999.)
 |- 
 if ( A  =/=  0 ,  A , 
 1 )  =/=  0
 
Theoremadddir 8763 Distributive law for complex numbers. (Contributed by NM, 10-Oct-2004.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )
 
Theorem0cn 8764 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
 |-  0  e.  CC
 
Theoremc0ex 8765 0 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  e.  _V
 
Theorem1ex 8766 1 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  1  e.  _V
 
Theoremmulid1 8767  1 is an identity element for multiplication. Based on ideas by Eric Schmidt. (Contributed by Scott Fenton, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  ( A  x.  1
 )  =  A )
 
Theoremmulid2 8768 Identity law for multiplication. Note: see mulid1 8767 for commuted version. (Contributed by NM, 8-Oct-1999.)
 |-  ( A  e.  CC  ->  ( 1  x.  A )  =  A )
 
Theoremcnre 8769* Alias for ax-cnre 8743, for naming consistency. (Contributed by Mario Carneiro, 3-Jan-2013.)
 |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
 
Theorem1re 8770  1 is a real number. This used to be one of our postulates for complex numbers, but Eric Schmidt discovered that it could be derived from a weaker postulate, ax-1cn 8728, by exploiting properties of the imaginary unit  _i. (Contributed by Eric Schmidt, 11-Apr-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  1  e.  RR
 
Theorem0re 8771  0 is a real number. (Contributed by Eric Schmidt, 21-May-2007.) (Revised by Scott Fenton, 3-Jan-2013.)
 |-  0  e.  RR
 
Theoremmulid1i 8772 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 |-  A  e.  CC   =>    |-  ( A  x.  1 )  =  A
 
Theoremmulid2i 8773 Identity law for multiplication. (Contributed by NM, 14-Feb-1995.)
 |-  A  e.  CC   =>    |-  ( 1  x.  A )  =  A
 
Theoremaddcli 8774 Closure law for addition. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  +  B )  e.  CC
 
Theoremmulcli 8775 Closure law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  B )  e.  CC
 
Theoremmulcomi 8776 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   =>    |-  ( A  x.  B )  =  ( B  x.  A )
 
Theoremmulcomli 8777 Commutative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  ( A  x.  B )  =  C   =>    |-  ( B  x.  A )  =  C
 
Theoremaddassi 8778 Associative law for addition. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  +  C )  =  ( A  +  ( B  +  C )
 )
 
Theoremmulassi 8779 Associative law for multiplication. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) )
 
Theoremadddii 8780 Distributive law. (Contributed by NM, 23-Nov-1994.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) )
 
Theoremadddiri 8781 Distributive law. (Contributed by NM, 16-Feb-1995.)
 |-  A  e.  CC   &    |-  B  e.  CC   &    |-  C  e.  CC   =>    |-  (
 ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) )
 
Theoremrecni 8782 A real number is a complex number. (Contributed by NM, 1-Mar-1995.)
 |-  A  e.  RR   =>    |-  A  e.  CC
 
Theoremreaddcli 8783 Closure law for addition of reals. (Contributed by NM, 17-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  +  B )  e.  RR
 
Theoremremulcli 8784 Closure law for multiplication of reals. (Contributed by NM, 17-Jan-1997.)
 |-  A  e.  RR   &    |-  B  e.  RR   =>    |-  ( A  x.  B )  e.  RR
 
Theoremmulid1d 8785 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  ( A  x.  1 )  =  A )
 
Theoremmulid2d 8786 Identity law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   =>    |-  ( ph  ->  (
 1  x.  A )  =  A )
 
Theoremaddcld 8787 Closure law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  +  B )  e.  CC )
 
Theoremmulcld 8788 Closure law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  B )  e.  CC )
 
Theoremmulcomd 8789 Commutative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaddassd 8790 Associative law for addition. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  +  C )  =  ( A  +  ( B  +  C )
 ) )
 
Theoremmulassd 8791 Associative law for multiplication. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  x.  B )  x.  C )  =  ( A  x.  ( B  x.  C ) ) )
 
Theoremadddid 8792 Distributive law. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
 
Theoremadddird 8793 Distributive law. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  (
 ( A  +  B )  x.  C )  =  ( ( A  x.  C )  +  ( B  x.  C ) ) )
 
Theoremrecnd 8794 Deduction from real number to complex number. (Contributed by NM, 26-Oct-1999.)
 |-  ( ph  ->  A  e.  RR )   =>    |-  ( ph  ->  A  e.  CC )
 
Theoremreaddcld 8795 Closure law for addition of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  +  B )  e.  RR )
 
Theoremremulcld 8796 Closure law for multiplication of reals. (Contributed by Mario Carneiro, 27-May-2016.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  ( A  x.  B )  e.  RR )
 
5.2.2  Infinity and the extended real number system
 
Syntaxcpnf 8797 Plus infinity.
 class  +oo
 
Syntaxcmnf 8798 Minus infinity.
 class  -oo
 
Syntaxcxr 8799 The set of extended reals (includes plus and minus infinity).
 class  RR*
 
Syntaxclt 8800 'Less than' predicate (extended to include the extended reals).
 class  <
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