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Theorem List for Intuitionistic Logic Explorer - 7801-7900   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltresr 7801 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 7802 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 7803 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6578, 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 7780), 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 7804 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 7803 and mulcnsrec 7805. (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 7805 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6577, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 7803. (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 7806 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 7899. (Contributed by Jim Kingdon, 14-Jul-2021.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +  B )  e.  _V )
 
Theorempitonnlem1 7807* Lemma for pitonn 7810. 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 7808 Lemma for pitonn 7810. 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 7809* Lemma for pitonn 7810. 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 7810* 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 7811* 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 7812* Embedding from  N. to  RR. Similar to pitonn 7810 but separate in the sense that we have not proved nnssre 8882 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 7813* 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 7814* One is an element of  NN. This is a counterpart to 1nn 8889 designed for real number axioms which involve natural numbers (notably, axcaucvg 7862). (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 7815* A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8890 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7862). (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 7816* 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 7817* 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 7818* 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 7819 Lemma for recidpirq 7820. 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 7820* 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 7821 The complex numbers form a set. Use cnex 7898 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 7822 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 7866. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 7823 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 7867. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremax1re 7824 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 7868.

In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7867 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 7825  _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 7869. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 7826 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 7870 be used later. Instead, in most cases use addcl 7899. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 7827 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 7871 be used later. Instead, in most cases use readdcl 7900. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 7828 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 7872 be used later. Instead, in most cases use mulcl 7901. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Theoremaxmulrcl 7829 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 7873 be used later. Instead, in most cases use remulcl 7902. (New usage is discouraged.) (Contributed by NM, 31-Mar-1996.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Theoremaxaddf 7830 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 7826. This construction-dependent theorem should not be referenced directly; instead, use ax-addf 7896. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Theoremaxmulf 7831 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 7828. This construction-dependent theorem should not be referenced directly; instead, use ax-mulf 7897. (Contributed by NM, 8-Feb-2005.) (New usage is discouraged.)
 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
Theoremaxaddcom 7832 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 7874 be used later. Instead, use addcom 8056.

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 7833 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 7875 be used later. Instead, use mulcom 7903. (Contributed by NM, 31-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  =  ( B  x.  A ) )
 
Theoremaxaddass 7834 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 7876 be used later. Instead, use addass 7904. (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 7835 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 7877. (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 7836 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 7878 be used later. Instead, use adddi 7906. (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 7837 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 7879. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Theoremax0lt1 7838 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 7880.

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 7839  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 7881. (Contributed by Scott Fenton, 3-Jan-2013.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Theoremax0id 7840  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 7882.

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 7841* 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 7883. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Theoremaxprecex 7842* 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 7884.

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 7843* 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 7885. (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 7844 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 7886. (Contributed by Jim Kingdon, 12-Jan-2020.) (New usage is discouraged.)
 |-  ( A  e.  RR  ->  -.  A  <RR  A )
 
Theoremaxpre-ltwlin 7845 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 7887. (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 7846 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 7888. (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 7847 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 7889.

(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 7848 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 7890. (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 7849 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 7891. (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 7850 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 7892.

(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 7851* The reciprocal from axprecex 7842 is unique. (Contributed by Jim Kingdon, 15-Jul-2021.)
 |-  ( ( A  e.  RR  /\  0  <RR  A ) 
 ->  E! x  e.  RR  ( A  x.  x )  =  1 )
 
Theoremrecriota 7852* 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 7853* 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 8880 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 7893. (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 7854* Peano's inductive postulate. This is a counterpart to peano5nni 8881 designed for real number axioms which involve natural numbers (notably, axcaucvg 7862). (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 7855* Principle of Mathematical Induction (inference schema). This is a counterpart to nnind 8894 designed for real number axioms which involve natural numbers (notably, axcaucvg 7862). (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 7856* 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 7857* Lemma for axcaucvg 7862. 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 7858* Lemma for axcaucvg 7862. 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 7859* Lemma for axcaucvg 7862. 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 7860* Lemma for axcaucvg 7862. 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 7861* Lemma for axcaucvg 7862. 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 7862* 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 7894. (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 7863* Lemma for axpre-suploc 7864. 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 7771. (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 7864* 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 7895. (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 7865 The complex numbers form a set. Proofs should normally use cnex 7898 instead. (New usage is discouraged.) (Contributed by NM, 1-Mar-1995.)
 |- 
 CC  e.  _V
 
Axiomax-resscn 7866 The real numbers are a subset of the complex numbers. Axiom for real and complex numbers, justified by Theorem axresscn 7822. (Contributed by NM, 1-Mar-1995.)
 |- 
 RR  C_  CC
 
Axiomax-1cn 7867 1 is a complex number. Axiom for real and complex numbers, justified by Theorem ax1cn 7823. (Contributed by NM, 1-Mar-1995.)
 |-  1  e.  CC
 
Axiomax-1re 7868 1 is a real number. Axiom for real and complex numbers, justified by Theorem ax1re 7824. Proofs should use 1re 7919 instead. (Contributed by Jim Kingdon, 13-Jan-2020.) (New usage is discouraged.)
 |-  1  e.  RR
 
Axiomax-icn 7869  _i is a complex number. Axiom for real and complex numbers, justified by Theorem axicn 7825. (Contributed by NM, 1-Mar-1995.)
 |-  _i  e.  CC
 
Axiomax-addcl 7870 Closure law for addition of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddcl 7826. Proofs should normally use addcl 7899 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 7871 Closure law for addition in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axaddrcl 7827. Proofs should normally use readdcl 7900 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Axiomax-mulcl 7872 Closure law for multiplication of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulcl 7828. Proofs should normally use mulcl 7901 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
 
Axiomax-mulrcl 7873 Closure law for multiplication in the real subfield of complex numbers. Axiom for real and complex numbers, justified by Theorem axmulrcl 7829. Proofs should normally use remulcl 7902 instead. (New usage is discouraged.) (Contributed by NM, 22-Nov-1994.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  x.  B )  e.  RR )
 
Axiomax-addcom 7874 Addition commutes. Axiom for real and complex numbers, justified by Theorem axaddcom 7832. Proofs should normally use addcom 8056 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 17-Jan-2020.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  =  ( B  +  A )
 )
 
Axiomax-mulcom 7875 Multiplication of complex numbers is commutative. Axiom for real and complex numbers, justified by Theorem axmulcom 7833. Proofs should normally use mulcom 7903 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 7876 Addition of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axaddass 7834. Proofs should normally use addass 7904 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 7877 Multiplication of complex numbers is associative. Axiom for real and complex numbers, justified by Theorem axmulass 7835. Proofs should normally use mulass 7905 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 7878 Distributive law for complex numbers (left-distributivity). Axiom for real and complex numbers, justified by Theorem axdistr 7836. Proofs should normally use adddi 7906 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 7879 i-squared equals -1 (expressed as i-squared plus 1 is 0). Axiom for real and complex numbers, justified by Theorem axi2m1 7837. (Contributed by NM, 29-Jan-1995.)
 |-  ( ( _i  x.  _i )  +  1
 )  =  0
 
Axiomax-0lt1 7880 0 is less than 1. Axiom for real and complex numbers, justified by Theorem ax0lt1 7838. Proofs should normally use 0lt1 8046 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 12-Jan-2020.)
 |-  0  <RR  1
 
Axiomax-1rid 7881  1 is an identity element for real multiplication. Axiom for real and complex numbers, justified by Theorem ax1rid 7839. (Contributed by NM, 29-Jan-1995.)
 |-  ( A  e.  RR  ->  ( A  x.  1
 )  =  A )
 
Axiomax-0id 7882  0 is an identity element for real addition. Axiom for real and complex numbers, justified by Theorem ax0id 7840.

Proofs should normally use addid1 8057 instead. (New usage is discouraged.) (Contributed by Jim Kingdon, 16-Jan-2020.)

 |-  ( A  e.  CC  ->  ( A  +  0 )  =  A )
 
Axiomax-rnegex 7883* Existence of negative of real number. Axiom for real and complex numbers, justified by Theorem axrnegex 7841. (Contributed by Eric Schmidt, 21-May-2007.)
 |-  ( A  e.  RR  ->  E. x  e.  RR  ( A  +  x )  =  0 )
 
Axiomax-precex 7884* Existence of reciprocal of positive real number. Axiom for real and complex numbers, justified by Theorem axprecex 7842. (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 7885* 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 7843. For naming consistency, use cnre 7916 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 7886 Real number less-than is irreflexive. Axiom for real and complex numbers, justified by Theorem ax-pre-ltirr 7886. (Contributed by Jim Kingdon, 12-Jan-2020.)
 |-  ( A  e.  RR  ->  -.  A  <RR  A )
 
Axiomax-pre-ltwlin 7887 Real number less-than is weakly linear. Axiom for real and complex numbers, justified by Theorem axpre-ltwlin 7845. (Contributed by Jim Kingdon, 12-Jan-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( A  <RR  B  ->  ( A  <RR  C  \/  C  <RR  B ) ) )
 
Axiomax-pre-lttrn 7888 Ordering on reals is transitive. Axiom for real and complex numbers, justified by Theorem axpre-lttrn 7846. (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-apti 7889 Apartness of reals is tight. Axiom for real and complex numbers, justified by Theorem axpre-apti 7847. (Contributed by Jim Kingdon, 29-Jan-2020.)
 |-  ( ( A  e.  RR  /\  B  e.  RR  /\ 
 -.  ( A  <RR  B  \/  B  <RR  A ) )  ->  A  =  B )
 
Axiomax-pre-ltadd 7890 Ordering property of addition on reals. Axiom for real and complex numbers, justified by Theorem axpre-ltadd 7848. (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 7891 The product of two positive reals is positive. Axiom for real and complex numbers, justified by Theorem axpre-mulgt0 7849. (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-mulext 7892 Strong extensionality of multiplication (expressed in terms of  <RR). Axiom for real and complex numbers, justified by Theorem axpre-mulext 7850

(Contributed by Jim Kingdon, 18-Feb-2020.)

 |-  ( ( A  e.  RR  /\  B  e.  RR  /\  C  e.  RR )  ->  ( ( A  x.  C )  <RR  ( B  x.  C )  ->  ( A  <RR  B  \/  B  <RR  A ) ) )
 
Axiomax-arch 7893* Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by Theorem axarch 7853.

This axiom should not be used directly; instead use arch 9132 (which is the same, but stated in terms of 
NN and  <). (Contributed by Jim Kingdon, 2-May-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 )
 
Axiomax-caucvg 7894* Completeness. Axiom for real and complex numbers, justified by Theorem axcaucvg 7862.

A Cauchy sequence (as defined here, which has a rate convergence built in) of real numbers converges to a real number. Specifically on rate of convergence, all terms after the nth term must be within  1  /  n of the nth term.

This axiom should not be used directly; instead use caucvgre 10945 (which is the same, but stated in terms of the  NN and  1  /  n notations). (Contributed by Jim Kingdon, 19-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 ) ) ) ) )
 
Axiomax-pre-suploc 7895* 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.

Although this and ax-caucvg 7894 are both completeness properties, countable choice would probably be needed to derive this from ax-caucvg 7894.

(Contributed by Jim Kingdon, 23-Jan-2024.)

 |-  ( ( ( 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 ) ) )
 
Axiomax-addf 7896 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 https://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 7899 should be used. Note that uses of ax-addf 7896 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 7830. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 +  : ( CC 
 X.  CC ) --> CC
 
Axiomax-mulf 7897 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 https://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 7872 should be used. Note that uses of ax-mulf 7897 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 7831. (New usage is discouraged.) (Contributed by NM, 19-Oct-2004.)

 |- 
 x.  : ( CC 
 X.  CC ) --> CC
 
4.2  Derive the basic properties from the field axioms
 
4.2.1  Some deductions from the field axioms for complex numbers
 
Theoremcnex 7898 Alias for ax-cnex 7865. (Contributed by Mario Carneiro, 17-Nov-2014.)
 |- 
 CC  e.  _V
 
Theoremaddcl 7899 Alias for ax-addcl 7870, for naming consistency with addcli 7924. Use this theorem instead of ax-addcl 7870 or axaddcl 7826. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremreaddcl 7900 Alias for ax-addrcl 7871, for naming consistency with readdcli 7933. (Contributed by NM, 10-Mar-2008.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
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