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Theorem List for Metamath Proof Explorer - 8701-8800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltsrpr 8701 Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
 |-  ( [ <. A ,  B >. ]  ~R  <R  [
 <. C ,  D >. ] 
 ~R 
 <->  ( A  +P.  D )  <P  ( B  +P.  C ) )
 
Theoremgt0srpr 8702 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( 0R  <R  [ <. A ,  B >. ]  ~R  <->  B  <P  A )
 
Theorem0nsr 8703 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  R.
 
Theorem0r 8704 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 0R  e.  R.
 
Theorem1sr 8705 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 1R  e.  R.
 
Theoremm1r 8706 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 -1R  e.  R.
 
Theoremaddclsr 8707 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  +R  B )  e.  R. )
 
Theoremmulclsr 8708 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  .R  B )  e.  R. )
 
Theoremdmaddsr 8709 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  +R  =  ( R.  X.  R. )
 
Theoremdmmulsr 8710 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  .R  =  ( R. 
 X.  R. )
 
Theoremaddcomsr 8711 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  +R  B )  =  ( B  +R  A )
 
Theoremaddasssr 8712 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) )
 
Theoremmulcomsr 8713 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  .R  B )  =  ( B  .R  A )
 
Theoremmulasssr 8714 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) )
 
Theoremdistrsr 8715 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  .R  ( B  +R  C ) )  =  ( ( A 
 .R  B )  +R  ( A  .R  C ) )
 
Theoremm1p1sr 8716 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 8717 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( -1R  .R  -1R )  =  1R
 
Theoremltsosr 8718 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 8719 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 8720 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 8721 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
 
Theorem1idsr 8722 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
 
Theorem00sr 8723 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
 
Theoremltasr 8724 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( C  e.  R.  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
 
Theorempn0sr 8725 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  +R  ( A  .R  -1R ) )  =  0R )
 
Theoremnegexsr 8726* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  E. x  e.  R.  ( A  +R  x )  =  0R )
 
Theoremrecexsrlem 8727* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremaddgt0sr 8728 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  +R  B ) )
 
Theoremmulgt0sr 8729 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  .R  B ) )
 
Theoremsqgt0sr 8730 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  A  =/=  0R )  ->  0R  <R  ( A 
 .R  A ) )
 
Theoremrecexsr 8731* The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  A  =/=  0R )  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremmappsrpr 8732 Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  ) 
 <->  A  e.  P. )
 
Theoremltpsrpr 8733 Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  [ <. A ,  1P >. ]  ~R  )  <R  ( C  +R  [ <. B ,  1P >. ] 
 ~R  )  <->  A  <P  B )
 
Theoremmap2psrpr 8734* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  -1R )  <R  A  <->  E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
 
Theoremsupsrlem 8735* 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 8736* 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 8737 Class of complex numbers.
 class  CC
 
Syntaxcr 8738 Class of real numbers.
 class  RR
 
Syntaxcc0 8739 Extend class notation to include the complex number 0.
 class 
 0
 
Syntaxc1 8740 Extend class notation to include the complex number 1.
 class 
 1
 
Syntaxci 8741 Extend class notation to include the complex number i.
 class  _i
 
Syntaxcaddc 8742 Addition on complex numbers.
 class  +
 
Syntaxcltrr 8743 'Less than' predicate (defined over real subset of complex numbers).
 class  <RR
 
Syntaxcmul 8744 Multiplication on complex numbers. The token  x. is a center dot.
 class  x.
 
Definitiondf-c 8745 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 8772. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 CC  =  ( R. 
 X.  R. )
 
Definitiondf-0 8746 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  0  =  <. 0R ,  0R >.
 
Definitiondf-1 8747 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  1  =  <. 1R ,  0R >.
 
Definitiondf-i 8748 Define the complex number  _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  _i  =  <. 0R ,  1R >.
 
Definitiondf-r 8749 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 RR  =  ( R. 
 X.  { 0R } )
 
Definitiondf-add 8750* 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 8751* 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 8752* Define 'less than' on the real subset of complex numbers. Proofs should typically use  < instead; see df-ltxr 8874. (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 8753 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 8754 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 8755* 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 8756 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 8757 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 8758 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  C_  ( RR  X.  RR )
 
Theoremaddcnsr 8759 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 8760 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 8761 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 8762 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 8763 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 8764 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 8765 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 8766 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6727, 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 8745), 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 8767 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 8766 and mulcnsrec 8768. (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 8768 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecid 6726, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 8766.

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 8468. (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 8769 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 8775. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 8818. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Theoremaxmulf 8770 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 8777. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 8819. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxcnex 8771 The complex numbers form a set. This axiom is redundant in the presence of the other axioms (see cnexALT 10352), but the proof requires the axiom of replacement, while the derivation from the construction here does not. Thus, we can avoid ax-rep 4133 in later theorems by invoking the axiom ax-cnex 8795 instead of cnexALT 10352. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 8772 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 8796. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 8773 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 8797. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremaxicn 8774  _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 8798. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 8775 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 8799 be used later. Instead, in most cases use addcl 8821. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 8776 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 8800 be used later. Instead, in most cases use readdcl 8822. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 8777 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 8801 be used later. Instead, in most cases use mulcl 8823. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 8778 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 8802 be used later. Instead, in most cases use remulcl 8824. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxmulcom 8779 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 8803 be used later. Instead, use mulcom 8825. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 8780 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 8804 be used later. Instead, use addass 8826. (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 8781 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 8805. (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 8782 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 8806 be used later. Instead, use adddi 8828. (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 8783 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 8807. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax1ne0 8784 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 8808. (Contributed by NM, 19-Mar-1996.) (New usage is discouraged.)
 |-  1  =/=  0
 
Theoremax1rid 8785  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 8837, based on ideas by Eric Schmidt. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 8809. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremaxrnegex 8786* 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 8810. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxrrecex 8787* 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 8811. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  A  =/=  0
 )  ->  E. x  e.  RR  ( A  x.  x )  =  1
 )
 
Theoremaxcnre 8788* 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 8812. (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 8789 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 8896. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttri 8813. (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 8790 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 8897. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 8814. (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 8791 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 8898. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 8815. (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 8792 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 8899. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 8816. (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 8793* 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 8900. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-sup 8817. (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 8794 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 8795 The complex numbers form a set. This axiom is redundant - see cnexALT 10352- but we provide this axiom because the justification theorem axcnex 8771 does not use ax-rep 4133 even though the redundancy proof does. Proofs should normally use cnex 8820 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 8796 The real numbers are a subset of the complex numbers. Axiom 1 of 22 for real and complex numbers, justified by theorem axresscn 8772. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 8797 1 is a complex number. Axiom 2 of 22 for real and complex numbers, justified by theorem ax1cn 8773. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-icn 8798  _i is a complex number. Axiom 3 of 22 for real and complex numbers, justified by theorem axicn 8774. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 8799 Closure law for addition of complex numbers. Axiom 4 of 22 for real and complex numbers, justified by theorem axaddcl 8775. Proofs should normally use addcl 8821 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 8800 Closure law for addition in the real subfield of complex numbers. Axiom 6 of 23 for real and complex numbers, justified by theorem axaddrcl 8776. Proofs should normally use readdcl 8822 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
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