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Theorem List for Intuitionistic Logic Explorer - 7601-7700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-lt 7601* Define 'less than' on the real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
 |- 
 <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  = 
 <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
 
Theoremopelcn 7602 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.)
 |-  ( <. A ,  B >.  e.  CC  <->  ( A  e.  R. 
 /\  B  e.  R. ) )
 
Theoremopelreal 7603 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.)
 |-  ( <. A ,  0R >.  e.  RR  <->  A  e.  R. )
 
Theoremelreal 7604* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
 |-  ( A  e.  RR  <->  E. x  e.  R.  <. x ,  0R >.  =  A )
 
Theoremelrealeu 7605* The real number mapping in elreal 7604 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
 |-  ( A  e.  RR  <->  E! x  e.  R.  <. x ,  0R >.  =  A )
 
Theoremelreal2 7606 Ordered pair membership in the class of complex numbers. (Contributed by Mario Carneiro, 15-Jun-2013.)
 |-  ( A  e.  RR  <->  (
 ( 1st `  A )  e.  R.  /\  A  =  <. ( 1st `  A ) ,  0R >. ) )
 
Theorem0ncn 7607 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. See also cnm 7608 which is a related property. (Contributed by NM, 2-May-1996.)
 |- 
 -.  (/)  e.  CC
 
Theoremcnm 7608* A complex number is an inhabited set. Note: do not use this after the real number axioms are developed, since it is a construction-dependent property. (Contributed by Jim Kingdon, 23-Oct-2023.) (New usage is discouraged.)
 |-  ( A  e.  CC  ->  E. x  x  e.  A )
 
Theoremltrelre 7609 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
 |- 
 <RR  C_  ( RR  X.  RR )
 
Theoremaddcnsr 7610 Addition of complex numbers in terms of signed reals. (Contributed by NM, 28-May-1995.)
 |-  ( ( ( A  e.  R.  /\  B  e.  R. )  /\  ( C  e.  R.  /\  D  e.  R. ) )  ->  ( <. A ,  B >.  +  <. C ,  D >. )  =  <. ( A  +R  C ) ,  ( B  +R  D ) >. )
 
Theoremmulcnsr 7611 Multiplication of complex numbers in terms of signed reals. (Contributed by NM, 9-Aug-1995.)
 |-  ( ( ( 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 7612 Equality of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
 |-  A  e.  _V   =>    |-  ( <. A ,  0R >.  =  <. B ,  0R >. 
 <->  A  =  B )
 
Theoremaddresr 7613 Addition of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( <. A ,  0R >.  +  <. B ,  0R >. )  =  <. ( A  +R  B ) ,  0R >. )
 
Theoremmulresr 7614 Multiplication of real numbers in terms of intermediate signed reals. (Contributed by NM, 10-May-1996.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( <. A ,  0R >.  x.  <. B ,  0R >. )  =  <. ( A  .R  B ) ,  0R >. )
 
Theoremltresr 7615 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
 |-  ( <. A ,  0R >.  <RR 
 <. B ,  0R >.  <->  A  <R  B )
 
Theoremltresr2 7616 Ordering of real subset of complex numbers in terms of signed reals. (Contributed by NM, 22-Feb-1996.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <RR  B  <-> 
 ( 1st `  A )  <R  ( 1st `  B ) ) )
 
Theoremdfcnqs 7617 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6462, 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 7594), 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.)
 |- 
 CC  =  ( ( R.  X.  R. ) /. `'  _E  )
 
Theoremaddcnsrec 7618 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7617 and mulcnsrec 7619. (Contributed by NM, 13-Aug-1995.)
 |-  ( ( ( 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 7619 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6461, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7617. (Contributed by NM, 13-Aug-1995.)
 |-  ( ( ( 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  )
 
Theoremaddvalex 7620 Existence of a sum. This is dependent on how we define  + so once we proceed to real number axioms we will replace it with theorems such as addcl 7713. (Contributed by Jim Kingdon, 14-Jul-2021.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +  B )  e.  _V )
 
Theorempitonnlem1 7621* Lemma for pitonn 7624. Two ways to write the number one. (Contributed by Jim Kingdon, 24-Apr-2020.)
 |- 
 <. [ <. ( <. { l  |  l  <Q  [ <. 1o ,  1o >. ]  ~Q  } ,  { u  |  [ <. 1o ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  1
 
Theorempitonnlem1p1 7622 Lemma for pitonn 7624. Simplifying an expression involving signed reals. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  P.  ->  [ <. ( A  +P.  ( 1P  +P.  1P )
 ) ,  ( 1P 
 +P.  1P ) >. ]  ~R  =  [ <. ( A  +P.  1P ) ,  1P >. ] 
 ~R  )
 
Theorempitonnlem2 7623* Lemma for pitonn 7624. Two ways to add one to a number. (Contributed by Jim Kingdon, 24-Apr-2020.)
 |-  ( K  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [
 <. K ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. K ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  +  1 )  =  <. [ <. ( <. { l  |  l  <Q  [
 <. ( K  +N  1o ) ,  1o >. ]  ~Q  } ,  { u  |  [ <. ( K  +N  1o ) ,  1o >. ] 
 ~Q  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ,  0R >. )
 
Theorempitonn 7624* Mapping from  N. to  NN. (Contributed by Jim Kingdon, 22-Apr-2020.)
 |-  ( N  e.  N.  -> 
 <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) } )
 
Theorempitoregt0 7625* Embedding from  N. to  RR yields a number greater than zero. (Contributed by Jim Kingdon, 15-Jul-2021.)
 |-  ( N  e.  N.  ->  0  <RR  <. [ <. ( <. { l  |  l  <Q  [
 <. N ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
 
Theorempitore 7626* Embedding from  N. to  RR. Similar to pitonn 7624 but separate in the sense that we have not proved nnssre 8688 yet. (Contributed by Jim Kingdon, 15-Jul-2021.)
 |-  ( N  e.  N.  -> 
 <. [ <. ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR )
 
Theoremrecnnre 7627* Embedding the reciprocal of a natural number into  RR. (Contributed by Jim Kingdon, 15-Jul-2021.)
 |-  ( N  e.  N.  -> 
 <. [ <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  e.  RR )
 
Theorempeano1nnnn 7628* One is an element of  NN. This is a counterpart to 1nn 8695 designed for real number axioms which involve natural numbers (notably, axcaucvg 7676). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   =>    |-  1  e.  N
 
Theorempeano2nnnn 7629* A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8696 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7676). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   =>    |-  ( A  e.  N  ->  ( A  +  1 )  e.  N )
 
Theoremltrennb 7630* Ordering of natural numbers with 
<N or  <RR. (Contributed by Jim Kingdon, 13-Jul-2021.)
 |-  ( ( J  e.  N. 
 /\  K  e.  N. )  ->  ( J  <N  K  <->  <. [ <. ( <. { l  |  l  <Q  [ <. J ,  1o >. ]  ~Q  } ,  { u  |  [ <. J ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l  <Q  [
 <. K ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. K ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. ) )
 
Theoremltrenn 7631* Ordering of natural numbers with 
<N or  <RR. (Contributed by Jim Kingdon, 12-Jul-2021.)
 |-  ( J  <N  K  ->  <. [ <. ( <. { l  |  l  <Q  [ <. J ,  1o >. ]  ~Q  } ,  { u  |  [ <. J ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  <RR  <. [ <. ( <. { l  |  l  <Q  [
 <. K ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. K ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
 
Theoremrecidpipr 7632* Another way of saying that a number times its reciprocal is one. (Contributed by Jim Kingdon, 17-Jul-2021.)
 |-  ( N  e.  N.  ->  ( <. { l  |  l  <Q  [ <. N ,  1o >. ]  ~Q  } ,  { u  |  [ <. N ,  1o >. ] 
 ~Q  <Q  u } >.  .P.  <. { l  |  l 
 <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >. )  =  1P )
 
Theoremrecidpirqlemcalc 7633 Lemma for recidpirq 7634. Rearranging some of the expressions. (Contributed by Jim Kingdon, 17-Jul-2021.)
 |-  ( ph  ->  A  e.  P. )   &    |-  ( ph  ->  B  e.  P. )   &    |-  ( ph  ->  ( A  .P.  B )  =  1P )   =>    |-  ( ph  ->  ( ( ( ( A  +P.  1P )  .P.  ( B  +P.  1P ) )  +P.  ( 1P  .P.  1P ) ) 
 +P.  1P )  =  ( ( ( ( A 
 +P.  1P )  .P.  1P )  +P.  ( 1P  .P.  ( B  +P.  1P )
 ) )  +P.  ( 1P  +P.  1P ) ) )
 
Theoremrecidpirq 7634* A real number times its reciprocal is one, where reciprocal is expressed with  *Q. (Contributed by Jim Kingdon, 15-Jul-2021.)
 |-  ( N  e.  N.  ->  ( <. [ <. ( <. { l  |  l  <Q  [
 <. N ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  <. [ <. (
 <. { l  |  l 
 <Q  ( *Q `  [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ,  0R >. )  =  1 )
 
4.1.2  Final derivation of real and complex number postulates
 
Theoremaxcnex 7635 The complex numbers form a set. Use cnex 7712 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 7636 The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-resscn 7680. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 7637 1 is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1cn 7681. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremax1re 7638 1 is a real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1re 7682.

In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7681 and the other axioms. It is not known whether we can do so here, but the Metamath Proof Explorer proof (accessed 13-Jan-2020) uses excluded middle. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)

 |-  1  e.  RR
 
Theoremaxicn 7639  _i is a complex number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-icn 7683. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 7640 Closure law for addition of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcl 7684 be used later. Instead, in most cases use addcl 7713. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 7641 Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addrcl 7685 be used later. Instead, in most cases use readdcl 7714. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 7642 Closure law for multiplication of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcl 7686 be used later. Instead, in most cases use mulcl 7715. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 7643 Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulrcl 7687 be used later. Instead, in most cases use remulcl 7716. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxaddf 7644 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 7640. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 7710. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Theoremaxmulf 7645 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 7642. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 7711. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxaddcom 7646 Addition commutes. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addcom 7688 be used later. Instead, use addcom 7867.

In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on real number trichotomy and it is not known whether it is possible to prove this from the other axioms without it. (Contributed by Jim Kingdon, 17-Jan-2020.) (New usage is discouraged.)

 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Theoremaxmulcom 7647 Multiplication of complex numbers is commutative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-mulcom 7689 be used later. Instead, use mulcom 7717. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 7648 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 for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-addass 7690 be used later. Instead, use addass 7718. (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 7649 Multiplication of complex numbers is associative. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-mulass 7691. (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 7650 Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly, nor should the proven axiom ax-distr 7692 be used later. Instead, use adddi 7720. (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 7651 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-i2m1 7693. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax0lt1 7652 0 is less than 1. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0lt1 7694.

The version of this axiom in the Metamath Proof Explorer reads  1  =/=  0; here we change it to  0  <RR  1. The proof of  0  <RR  1 from  1  =/=  0 in the Metamath Proof Explorer (accessed 12-Jan-2020) relies on real number trichotomy. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)

 |-  0  <RR  1
 
Theoremax1rid 7653  1 is an identity element for real multiplication. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-1rid 7695. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremax0id 7654  0 is an identity element for real addition. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-0id 7696.

In the Metamath Proof Explorer this is not a complex number axiom but is instead proved from other axioms. That proof relies on excluded middle and it is not known whether it is possible to prove this from the other axioms without excluded middle. (Contributed by Jim Kingdon, 16-Jan-2020.) (New usage is discouraged.)

 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Theoremaxrnegex 7655* Existence of negative of real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-rnegex 7697. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxprecex 7656* Existence of positive reciprocal of positive real number. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-precex 7698.

In treatments which assume excluded middle, the  0 
<RR  A condition is generally replaced by  A  =/=  0, and it may not be necessary to state that the reciproacal is positive. (Contributed by Jim Kingdon, 6-Feb-2020.) (New usage is discouraged.)

 |-  ( ( A  e.  RR  /\  0  <RR  A ) 
 ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x )  =  1 )
 )
 
Theoremaxcnre 7657* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-cnre 7699. (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-ltirr 7658 Real number less-than is irreflexive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltirr 7700. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  -.  A  <RR  A )
 
Theoremaxpre-ltwlin 7659 Real number less-than is weakly linear. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltwlin 7701. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )
 
Theoremaxpre-lttrn 7660 Ordering on reals is transitive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-lttrn 7702. (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-apti 7661 Apartness of reals is tight. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-apti 7703.

(Contributed by Jim Kingdon, 29-Jan-2020.) (New usage is discouraged.)

 |-  ( ( A  e.  RR  /\  B  e.  RR  /\ 
 -.  ( A  <RR  B  \/  B  <RR  A ) )  ->  A  =  B )
 
Theoremaxpre-ltadd 7662 Ordering property of addition on reals. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-ltadd 7704. (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 7663 The product of two positive reals is positive. Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulgt0 7705. (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-mulext 7664 Strong extensionality of multiplication (expressed in terms of  <RR). Axiom for real and complex numbers, derived from set theory. This construction-dependent theorem should not be referenced directly; instead, use ax-pre-mulext 7706.

(Contributed by Jim Kingdon, 18-Feb-2020.) (New usage is discouraged.)

 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  x.  C )  <RR  ( B  x.  C )  ->  ( A  <RR  B  \/  B  <RR  A ) ) )
 
Theoremrereceu 7665* The reciprocal from axprecex 7656 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
 |-  ( ( A  e.  RR  /\  0  <RR  A ) 
 ->  E! x  e.  RR  ( A  x.  x )  =  1 )
 
Theoremrecriota 7666* Two ways to express the reciprocal of a natural number. (Contributed by Jim Kingdon, 11-Jul-2021.)
 |-  ( N  e.  N.  ->  ( iota_ r  e.  RR  ( <. [ <. ( <. { l  |  l  <Q  [
 <. N ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. N ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  x.  r )  =  1 )  = 
 <. [ <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. N ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. N ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )
 
Theoremaxarch 7667* Archimedean axiom. The Archimedean property is more naturally stated once we have defined  NN. Unless we find another way to state it, we'll just use the right hand side of dfnn2 8686 in stating what we mean by "natural number" in the context of this axiom.

This construction-dependent theorem should not be referenced directly; instead, use ax-arch 7707. (Contributed by Jim Kingdon, 22-Apr-2020.) (New usage is discouraged.)

 |-  ( A  e.  RR  ->  E. n  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  (
 y  +  1 )  e.  x ) } A  <RR  n )
 
Theorempeano5nnnn 7668* Peano's inductive postulate. This is a counterpart to peano5nni 8687 designed for real number axioms which involve natural numbers (notably, axcaucvg 7676). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   =>    |-  ( ( 1  e.  A  /\  A. z  e.  A  ( z  +  1 )  e.  A )  ->  N  C_  A )
 
Theoremnnindnn 7669* Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8700 designed for real number axioms which involve natural numbers (notably, axcaucvg 7676). (Contributed by Jim Kingdon, 14-Jul-2021.) (New usage is discouraged.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( z  =  1  ->  ( ph  <->  ps ) )   &    |-  ( z  =  k  ->  ( ph  <->  ch ) )   &    |-  ( z  =  ( k  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( z  =  A  ->  ( ph  <->  ta ) )   &    |-  ps   &    |-  ( k  e.  N  ->  ( ch  ->  th ) )   =>    |-  ( A  e.  N  ->  ta )
 
Theoremnntopi 7670* Mapping from  NN to  N.. (Contributed by Jim Kingdon, 13-Jul-2021.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   =>    |-  ( A  e.  N  ->  E. z  e.  N.  <. [ <. ( <. { l  |  l  <Q  [ <. z ,  1o >. ]  ~Q  } ,  { u  |  [ <. z ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >.  =  A )
 
Theoremaxcaucvglemcl 7671* Lemma for axcaucvg 7676. Mapping to  N. and  R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( ph  ->  F : N --> RR )   =>    |-  (
 ( ph  /\  J  e.  N. )  ->  ( iota_ z  e. 
 R.  ( F `  <. [ <. ( <. { l  |  l  <Q  [ <. J ,  1o >. ]  ~Q  } ,  { u  |  [ <. J ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. )  e. 
 R. )
 
Theoremaxcaucvglemf 7672* Lemma for axcaucvg 7676. Mapping to  N. and  R. yields a sequence. (Contributed by Jim Kingdon, 9-Jul-2021.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( ph  ->  F : N --> RR )   &    |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )   &    |-  G  =  ( j  e.  N.  |->  (
 iota_ z  e.  R.  ( F `  <. [ <. (
 <. { l  |  l 
 <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. ) )   =>    |-  ( ph  ->  G : N.
 --> R. )
 
Theoremaxcaucvglemval 7673* Lemma for axcaucvg 7676. Value of sequence when mapping to  N. and  R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( ph  ->  F : N --> RR )   &    |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )   &    |-  G  =  ( j  e.  N.  |->  (
 iota_ z  e.  R.  ( F `  <. [ <. (
 <. { l  |  l 
 <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. ) )   =>    |-  ( ( ph  /\  J  e.  N. )  ->  ( F `  <. [ <. ( <. { l  |  l  <Q  [
 <. J ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. J ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. ( G `  J ) ,  0R >. )
 
Theoremaxcaucvglemcau 7674* Lemma for axcaucvg 7676. The result of mapping to  N. and  R. satisfies the Cauchy condition. (Contributed by Jim Kingdon, 9-Jul-2021.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( ph  ->  F : N --> RR )   &    |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )   &    |-  G  =  ( j  e.  N.  |->  (
 iota_ z  e.  R.  ( F `  <. [ <. (
 <. { l  |  l 
 <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. ) )   =>    |-  ( ph  ->  A. n  e. 
 N.  A. k  e.  N.  ( n  <N  k  ->  ( ( G `  n )  <R  ( ( G `  k )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  )  /\  ( G `  k )  <R  ( ( G `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )
 
Theoremaxcaucvglemres 7675* Lemma for axcaucvg 7676. Mapping the limit from  N. and  R.. (Contributed by Jim Kingdon, 10-Jul-2021.)
 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( ph  ->  F : N --> RR )   &    |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )   &    |-  G  =  ( j  e.  N.  |->  (
 iota_ z  e.  R.  ( F `  <. [ <. (
 <. { l  |  l 
 <Q  [ <. j ,  1o >. ]  ~Q  } ,  { u  |  [ <. j ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  ,  0R >. )  =  <. z ,  0R >. ) )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e. 
 RR  ( 0  <RR  x 
 ->  E. j  e.  N  A. k  e.  N  ( j  <RR  k  ->  (
 ( F `  k
 )  <RR  ( y  +  x )  /\  y  <RR  ( ( F `  k
 )  +  x ) ) ) ) )
 
Theoremaxcaucvg 7676* Real number completeness axiom. A Cauchy sequence with a modulus of convergence converges. This is basically Corollary 11.2.13 of [HoTT], p. (varies). The HoTT book theorem has a modulus of convergence (that is, a rate of convergence) specified by (11.2.9) in HoTT whereas this theorem fixes the rate of convergence to say that all terms after the nth term must be within 
1  /  n of the nth term (it should later be able to prove versions of this theorem with a different fixed rate or a modulus of convergence supplied as a hypothesis).

Because we are stating this axiom before we have introduced notations for  NN or division, we use  N for the natural numbers and express a reciprocal in terms of  iota_.

This construction-dependent theorem should not be referenced directly; instead, use ax-caucvg 7708. (Contributed by Jim Kingdon, 8-Jul-2021.) (New usage is discouraged.)

 |-  N  =  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1
 )  e.  x ) }   &    |-  ( ph  ->  F : N --> RR )   &    |-  ( ph  ->  A. n  e.  N  A. k  e.  N  ( n  <RR  k  ->  (
 ( F `  n )  <RR  ( ( F `
  k )  +  ( iota_ r  e.  RR  ( n  x.  r
 )  =  1 ) )  /\  ( F `
  k )  <RR  ( ( F `  n )  +  ( iota_ r  e. 
 RR  ( n  x.  r )  =  1
 ) ) ) ) )   =>    |-  ( ph  ->  E. y  e.  RR  A. x  e. 
 RR  ( 0  <RR  x 
 ->  E. j  e.  N  A. k  e.  N  ( j  <RR  k  ->  (
 ( F `  k
 )  <RR  ( y  +  x )  /\  y  <RR  ( ( F `  k
 )  +  x ) ) ) ) )
 
Theoremaxpre-suploclemres 7677* Lemma for axpre-suploc 7678. The result. The proof just needs to define  B as basically the same set as  A (but expressed as a subset of  R. rather than a subset of  RR), and apply suplocsr 7585. (Contributed by Jim Kingdon, 24-Jan-2024.)
 |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  C  e.  A )   &    |-  ( ph  ->  E. x  e.  RR  A. y  e.  A  y 
 <RR  x )   &    |-  ( ph  ->  A. x  e.  RR  A. y  e.  RR  ( x  <RR  y  ->  ( E. z  e.  A  x  <RR  z  \/  A. z  e.  A  z  <RR  y ) ) )   &    |-  B  =  { w  e.  R.  |  <. w ,  0R >.  e.  A }   =>    |-  ( ph  ->  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 ) ) )
 
Theoremaxpre-suploc 7678* An inhabited, bounded-above, located set of reals has a supremum.

Locatedness here means that given  x  <  y, either there is an element of the set greater than  x, or  y is an upper bound.

This construction-dependent theorem should not be referenced directly; instead, use ax-pre-suploc 7709. (Contributed by Jim Kingdon, 23-Jan-2024.) (New usage is discouraged.)

 |-  ( ( ( A 
 C_  RR  /\  E. x  x  e.  A )  /\  ( E. x  e. 
 RR  A. y  e.  A  y  <RR  x  /\  A. x  e.  RR  A. y  e.  RR  ( x  <RR  y 
 ->  ( E. z  e.  A  x  <RR  z  \/ 
 A. z  e.  A  z  <RR  y ) ) ) )  ->  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 ) ) )
 
4.1.3  Real and complex number postulates restated as axioms
 
Axiomax-cnex 7679 The complex numbers form a set. Proofs should normally use cnex 7712 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 7680 The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by theorem axresscn 7636. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 7681 1 is a complex number. Axiom for real and complex numbers, justified by theorem ax1cn 7637. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-1re 7682 1 is a real number. Axiom for real and complex numbers, justified by theorem ax1re 7638. Proofs should use 1re 7733 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
 |-  1  e.  RR
 
Axiomax-icn 7683  _i is a complex number. Axiom for real and complex numbers, justified by theorem axicn 7639. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 7684 Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by theorem axaddcl 7640. Proofs should normally use addcl 7713 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 7685 Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axaddrcl 7641. Proofs should normally use readdcl 7714 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Axiomax-mulcl 7686 Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by theorem axmulcl 7642. Proofs should normally use mulcl 7715 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 7687 Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by theorem axmulrcl 7643. Proofs should normally use remulcl 7716 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-addcom 7688 Addition commutes. Axiom for real and complex numbers, justified by theorem axaddcom 7646. Proofs should normally use addcom 7867 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Axiomax-mulcom 7689 Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by theorem axmulcom 7647. Proofs should normally use mulcom 7717 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 7690 Addition of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axaddass 7648. Proofs should normally use addass 7718 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 7691 Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by theorem axmulass 7649. Proofs should normally use mulass 7719 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 7692 Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by theorem axdistr 7650. Proofs should normally use adddi 7720 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 7693 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by theorem axi2m1 7651. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-0lt1 7694 0 is less than 1. Axiom for real and complex numbers, justified by theorem ax0lt1 7652. Proofs should normally use 0lt1 7857 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
 |-  0  <RR  1
 
Axiomax-1rid 7695  1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by theorem ax1rid 7653. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-0id 7696  0 is an identity element for real addition. Axiom for real and complex numbers, justified by theorem ax0id 7654.

Proofs should normally use addid1 7868 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)

 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Axiomax-rnegex 7697* Existence of negative of real number. Axiom for real and complex numbers, justified by theorem axrnegex 7655. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-precex 7698* Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by theorem axprecex 7656. (Contributed by Jim Kingdon, 6-Feb-2020.)
 |-  ( ( A  e.  RR  /\  0  <RR  A ) 
 ->  E. x  e.  RR  ( 0  <RR  x  /\  ( A  x.  x )  =  1 )
 )
 
Axiomax-cnre 7699* A complex number can be expressed in terms of two reals. Definition 10-1.1(v) of [Gleason] p. 130. Axiom for real and complex numbers, justified by theorem axcnre 7657. For naming consistency, use cnre 7730 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-ltirr 7700 Real number less-than is irreflexive. Axiom for real and complex numbers, justified by theorem ax-pre-ltirr 7700. (Contributed by Jim Kingdon, 12-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A  <RR  A )
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