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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremsupsrlem 8701* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  B  =  { w  |  ( C  +R  [ <. w ,  1P >. ] 
 ~R  )  e.  A }   &    |-  C  e.  R.   =>    |-  ( ( C  e.  A  /\  E. x  e.  R.  A. y  e.  A  y  <R  x ) 
 ->  E. x  e.  R.  ( A. y  e.  A  -.  x  <R  y  /\  A. y  e.  R.  (
 y  <R  x  ->  E. z  e.  A  y  <R  z ) ) )
 
Theoremsupsr 8702* A non-empty, bounded set of signed reals has a supremum. (Cotributed by Mario Carneiro, 15-Jun-2013.) (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  R.  A. y  e.  A  y 
 <R  x )  ->  E. x  e.  R.  ( A. y  e.  A  -.  x  <R  y 
 /\  A. y  e.  R.  ( y  <R  x  ->  E. z  e.  A  y  <R  z ) ) )
 
Syntaxcc 8703 Class of complex numbers.
 class  CC
 
Syntaxcr 8704 Class of real numbers.
 class  RR
 
Syntaxcc0 8705 Extend class notation to include the complex number 0.
 class 
 0
 
Syntaxc1 8706 Extend class notation to include the complex number 1.
 class 
 1
 
Syntaxci 8707 Extend class notation to include the complex number i.
 class  _i
 
Syntaxcaddc 8708 Addition on complex numbers.
 class  +
 
Syntaxcltrr 8709 'Less than' predicate (defined over real subset of complex numbers).
 class  <RR
 
Syntaxcmul 8710 Multiplication on complex numbers. The token  x. is a center dot.
 class  x.
 
Definitiondf-c 8711 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8738. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 CC  =  ( R. 
 X.  R. )
 
Definitiondf-0 8712 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  0  =  <. 0R ,  0R >.
 
Definitiondf-1 8713 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  1  =  <. 1R ,  0R >.
 
Definitiondf-i 8714 Define the complex number  _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  _i  =  <. 0R ,  1R >.
 
Definitiondf-r 8715 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 RR  =  ( R. 
 X.  { 0R } )
 
Definitiondf-plus 8716* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
 |- 
 +  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( w  +R  u ) ,  ( v  +R  f ) >. ) ) }
 
Definitiondf-mul 8717* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 x.  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) ) ,  ( ( v 
 .R  u )  +R  ( w  .R  f ) ) >. ) ) }
 
Definitiondf-lt 8718* Define 'less than' on the real subset of complex numbers. Proofs should typically use  < instead; see df-ltxr 8840. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  = 
 <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
 
Theoremopelcn 8719 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( <. A ,  B >.  e.  CC  <->  ( A  e.  R. 
 /\  B  e.  R. ) )
 
Theoremopelreal 8720 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( <. A ,  0R >.  e.  RR  <->  A  e.  R. )
 
Theoremelreal 8721* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  <->  E. x  e.  R.  <. x ,  0R >.  =  A )
 
Theoremelreal2 8722 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  <->  (
 ( 1st `  A )  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
 
Theorem0ncn 8723 The empty set is not a complex number. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  CC
 
Theoremltrelre 8724 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  C_  ( RR  X.  RR )
 
Theoremaddcnsr 8725 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( <. A ,  B >.  +  <. C ,  D >. )  =  <. ( A  +R  C ) ,  ( B  +R  D ) >. )
 
Theoremmulcnsr 8726 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( <. A ,  B >.  x.  <. C ,  D >. )  =  <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  (
 ( B  .R  C )  +R  ( A  .R  D ) ) >. )
 
Theoremeqresr 8727 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  A  e.  _V   =>    |-  ( <. A ,  0R >.  =  <. B ,  0R >. 
 <->  A  =  B )
 
Theoremaddresr 8728 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( <. A ,  0R >.  +  <. B ,  0R >. )  =  <. ( A  +R  B ) ,  0R >. )
 
Theoremmulresr 8729 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( <. A ,  0R >.  x.  <. B ,  0R >. )  =  <. ( A  .R  B ) ,  0R >. )
 
Theoremltresr 8730 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( <. A ,  0R >.  <RR 
 <. B ,  0R >.  <->  A  <R  B )
 
Theoremltresr2 8731 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <-> 
 ( 1st `  A )  <R  ( 1st `  B ) ) )
 
Theoremdfcnqs 8732 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6693, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) acts as an identity divisor for the quotient set operation. This lets us "pretend" that  CC is a quotient set, even though it is not (compare df-c 8711), and allows us to reuse some of the equivalence class lemmas we developed for the transition from positive reals to signed reals, etc. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
 |- 
 CC  =  ( ( R.  X.  R. ) /. `'  _E  )
 
Theoremaddcnsrec 8733 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8732 and mulcnsrec 8734. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( [ <. A ,  B >. ] `'  _E  +  [ <. C ,  D >. ] `'  _E  )  =  [ <. ( A  +R  C ) ,  ( B  +R  D ) >. ] `'  _E  )
 
Theoremmulcnsrec 8734 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6692, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8732.

Note: This is the last lemma (from which the axioms will be derived) in the construction of real and complex numbers. The construction starts at cnpi 8434. (Contributed by NM, 13-Aug-1995.) (New usage is discouraged.)

 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( [ <. A ,  B >. ] `'  _E  x.  [
 <. C ,  D >. ] `'  _E  )  =  [ <. ( ( A  .R  C )  +R  ( -1R  .R  ( B  .R  D ) ) ) ,  ( ( B 
 .R  C )  +R  ( A  .R  D ) ) >. ] `'  _E  )
 
5.1.2  Final derivation of real and complex number postulates
 
Theoremaxaddf 8735 Addition 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 axaddcl 8741. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8784. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Theoremaxmulf 8736 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 8743. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8785. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxcnex 8737 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10318), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus we can avoid ax-rep 4105 in later theorems by invoking the axiom ax-cnex 8761 instead of cnexALT 10318. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 8738 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 8762. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 8739 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 8763. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremaxicn 8740  _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 8764. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 8741 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, nor should the proven axiom ax-addcl 8765 be used later. Instead, in most cases use addcl 8787. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 8742 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, nor should the proven axiom ax-addrcl 8766 be used later. Instead, in most cases use readdcl 8788. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 8743 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, nor should the proven axiom ax-mulcl 8767 be used later. Instead, in most cases use mulcl 8789. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 8744 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, nor should the proven axiom ax-mulrcl 8768 be used later. Instead, in most cases use remulcl 8790. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxmulcom 8745 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, nor should the proven axiom ax-mulcom 8769 be used later. Instead, use mulcom 8791. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 8746 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, nor should the proven axiom ax-addass 8770 be used later. Instead, use addass 8792. (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 8747 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 8771. (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 8748 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, nor should the proven axiom ax-distr 8772 be used later. Instead, use adddi 8794. (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 8749 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 8773. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax1ne0 8750 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 8774. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
 |-  1  =/=  0
 
Theoremax1rid 8751  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 8803, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8775. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremaxrnegex 8752* 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 8776. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxrrecex 8753* 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 8777. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Theoremaxcnre 8754* 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 8778. (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 8755 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 8862. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8779. (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 8756 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 8863. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8780. (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 8757 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 8864. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8781. (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 8758 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 8865. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8782. (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 8759* 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 8866. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8783. (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 8760 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 8761 The complex numbers form a set. This axiom is redundant - see cnexALT 10318- but we provide this axiom because the justification theorem axcnex 8737 does not use ax-rep 4105 even though the redundancy proof does. Proofs should normally use cnex 8786 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 8762 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8738. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 8763 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8739. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-icn 8764  _i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8740. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 8765 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8741. Proofs should normally use addcl 8787 instead, which asserts the same thing but follows our naming conventions for closures. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Axiomax-addrcl 8766 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8742. Proofs should normally use readdcl 8788 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Axiomax-mulcl 8767 Closure law for multiplication of complex numbers. Axiom 6 of 22 for real and complex numbers, justified by theorem axmulcl 8743. Proofs should normally use mulcl 8789 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 8768 Closure law for multiplication in the real subfield of complex numbers. Axiom 7 of 22 for real and complex numbers, justified by theorem axmulrcl 8744. Proofs should normally use remulcl 8790 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-mulcom 8769 Multiplication of complex numbers is commutative. Axiom 8 of 22 for real and complex numbers, justified by theorem axmulcom 8745. Proofs should normally use mulcom 8791 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Axiomax-addass 8770 Addition of complex numbers is associative. Axiom 9 of 22 for real and complex numbers, justified by theorem axaddass 8746. Proofs should normally use addass 8792 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Axiomax-mulass 8771 Multiplication of complex numbers is associative. Axiom 10 of 22 for real and complex numbers, justified by theorem axmulass 8747. Proofs should normally use mulass 8793 instead. (New usage is discouraged.) (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 8772 Distributive law for complex numbers. Axiom 11 of 22 for real and complex numbers, justified by theorem axdistr 8748. Proofs should normally use adddi 8794 instead. (New usage is discouraged.) (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 8773 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 8749. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-1ne0 8774 1 and 0 are distinct. Axiom 13 of 22 for real and complex numbers, justified by theorem ax1ne0 8750. (Contributed by NM, 29-Jan-1995.)
 |-  1  =/=  0
 
Axiomax-1rid 8775  1 is an identity element for real multiplication. Axiom 14 of 22 for real and complex numbers, justified by theorem ax1rid 8751. Weakened from the original axiom in the form of statement in mulid1 8803, based on ideas by Eric Schmidt. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-rnegex 8776* Existence of negative of real number. Axiom 15 of 22 for real and complex numbers, justified by theorem axrnegex 8752. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-rrecex 8777* Existence of reciprocal of nonzero real number. Axiom 16 of 22 for real and complex numbers, justified by theorem axrrecex 8753. (Contributed by Eric Schmidt, 11-Apr-2007.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Axiomax-cnre 8778* 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 8754. For naming consistency, use cnre 8802 for new proofs. (New usage is discouraged.) (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 8779 Ordering on reals satisfies strict trichotomy. Axiom 18 of 22 for real and complex numbers, justified by theorem axpre-lttri 8755. Note: The more general version for extended reals is axlttri 8862. Normally new proofs would use xrlttri 10441. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <->  -.  ( A  =  B  \/  B  <RR  A ) ) )
 
Axiomax-pre-lttrn 8780 Ordering on reals is transitive. Axiom 19 of 22 for real and complex numbers, justified by theorem axpre-lttrn 8756. Note: The more general version for extended reals is axlttrn 8863. Normally new proofs would use lttr 8867. (New usage is discouraged.) (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 8781 Ordering property of addition on reals. Axiom 20 of 22 for real and complex numbers, justified by theorem axpre-ltadd 8757. Normally new proofs would use axltadd 8864. (New usage is discouraged.) (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 8782 The product of two positive reals is positive. Axiom 21 of 22 for real and complex numbers, justified by theorem axpre-mulgt0 8758. Normally new proofs would use axmulgt0 8865. (New usage is discouraged.) (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 8783* 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 8759. Note: Normally new proofs would use axsup 8866. (New usage is discouraged.) (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 8784 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 addcl 8787 should be used. Note that uses of ax-addf 8784 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 8735. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Axiomax-mulf 8785 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 8767 should be used. Note that uses of ax-mulf 8785 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 8736. (New usage is discouraged.) (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 8786 Alias for ax-cnex 8761. See also cnexALT 10318. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 CC  e.  _V
 
Theoremaddcl 8787 Alias for ax-addcl 8765, for naming consistency with addcli 8809. Use this theorem instead of ax-addcl 8765 or axaddcl 8741. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremreaddcl 8788 Alias for ax-addrcl 8766, for naming consistency with readdcli 8818. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremmulcl 8789 Alias for ax-mulcl 8767, for naming consistency with mulcli 8810. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremremulcl 8790 Alias for ax-mulrcl 8768, for naming consistency with remulcli 8819. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremmulcom 8791 Alias for ax-mulcom 8769, for naming consistency with mulcomi 8811. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaddass 8792 Alias for ax-addass 8770, for naming consistency with addassi 8813. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( ( A  +  B )  +  C )  =  ( A  +  ( B  +  C ) ) )
 
Theoremmulass 8793 Alias for ax-mulass 8771, for naming consistency with mulassi 8814. (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 8794 Alias for ax-distr 8772, for naming consistency with adddii 8815. (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 8795 A real number is a complex number. (Contributed by NM, 10-Aug-1999.)
 |-  ( A  e.  RR  ->  A  e.  CC )
 
Theoremreex 8796 The real numbers form a set. See also reexALT 10316. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 RR  e.  _V
 
Theoremelimne0 8797 Hypothesis for weak deduction theorem to eliminate  A  =/=  0. (Contributed by NM, 15-May-1999.)
 |- 
 if ( A  =/=  0 ,  A , 
 1 )  =/=  0
 
Theoremadddir 8798 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 8799 0 is a complex number. (Contributed by NM, 19-Feb-2005.)
 |-  0  e.  CC
 
Theoremc0ex 8800 0 is a set. Common special case. (Contributed by David A. Wheeler, 7-Jul-2016.)
 |-  0  e.  _V
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