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Theorem List for Intuitionistic Logic Explorer - 6901-7000   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremmulasssrg 6901 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) ) )
 
Theoremdistrsrg 6902 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  .R  ( B  +R  C ) )  =  ( ( A 
 .R  B )  +R  ( A  .R  C ) ) )
 
Theoremm1p1sr 6903 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 6904 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
 |-  ( -1R  .R  -1R )  =  1R
 
Theoremlttrsr 6905* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |-  ( ( f  e. 
 R.  /\  g  e.  R. 
 /\  h  e.  R. )  ->  ( ( f 
 <R  g  /\  g  <R  h )  ->  f  <R  h ) )
 
Theoremltposr 6906 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)
 |- 
 <R  Po  R.
 
Theoremltsosr 6907 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 6908 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 6909 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 6910 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
 |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
 
Theorem1idsr 6911 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
 
Theorem00sr 6912 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
 |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
 
Theoremltasrg 6913 Ordering property of addition. (Contributed by NM, 10-May-1996.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
 
Theorempn0sr 6914 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
 |-  ( A  e.  R.  ->  ( A  +R  ( A  .R  -1R ) )  =  0R )
 
Theoremnegexsr 6915* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
 |-  ( A  e.  R.  ->  E. x  e.  R.  ( A  +R  x )  =  0R )
 
Theoremrecexgt0sr 6916* The reciprocal of a positive signed real exists and is positive. (Contributed by Jim Kingdon, 6-Feb-2020.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( 0R  <R  x  /\  ( A  .R  x )  =  1R ) )
 
Theoremrecexsrlem 6917* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremaddgt0sr 6918 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  +R  B ) )
 
Theoremltadd1sr 6919 Adding one to a signed real yields a larger signed real. (Contributed by Jim Kingdon, 7-Jul-2021.)
 |-  ( A  e.  R.  ->  A  <R  ( A  +R  1R ) )
 
Theoremmulgt0sr 6920 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  .R  B ) )
 
Theoremaptisr 6921 Apartness of signed reals is tight. (Contributed by Jim Kingdon, 29-Jan-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\ 
 -.  ( A  <R  B  \/  B  <R  A ) )  ->  A  =  B )
 
Theoremmulextsr1lem 6922 Lemma for mulextsr1 6923. (Contributed by Jim Kingdon, 17-Feb-2020.)
 |-  ( ( ( X  e.  P.  /\  Y  e.  P. )  /\  ( Z  e.  P.  /\  W  e.  P. )  /\  ( U  e.  P.  /\  V  e.  P. ) )  ->  ( ( ( ( X  .P.  U ) 
 +P.  ( Y  .P.  V ) )  +P.  (
 ( Z  .P.  V )  +P.  ( W  .P.  U ) ) )  <P  ( ( ( X  .P.  V )  +P.  ( Y 
 .P.  U ) )  +P.  ( ( Z  .P.  U )  +P.  ( W 
 .P.  V ) ) ) 
 ->  ( ( X  +P.  W )  <P  ( Y  +P.  Z )  \/  ( Z  +P.  Y )  <P  ( W  +P.  X ) ) ) )
 
Theoremmulextsr1 6923 Strong extensionality of multiplication of signed reals. (Contributed by Jim Kingdon, 18-Feb-2020.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R.  /\  C  e.  R. )  ->  ( ( A  .R  C )  <R  ( B 
 .R  C )  ->  ( A  <R  B  \/  B  <R  A ) ) )
 
Theoremarchsr 6924* For any signed real, there is an integer that is greater than it. This is also known as the "archimedean property". The expression  [ <. ( <. { l  |  l 
<Q  [ <. x ,  1o >. ]  ~Q  },  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R is the embedding of the positive integer  x into the signed reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
 |-  ( A  e.  R.  ->  E. x  e.  N.  A  <R  [ <. ( <. { l  |  l  <Q  [
 <. x ,  1o >. ] 
 ~Q  } ,  { u  |  [ <. x ,  1o >. ]  ~Q  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )
 
Theoremsrpospr 6925* Mapping from a signed real greater than zero to a positive real. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( ( A  e.  R. 
 /\  0R  <R  A ) 
 ->  E! x  e.  P.  [
 <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )
 
Theoremprsrcl 6926 Mapping from a positive real to a signed real. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( A  e.  P.  ->  [ <. ( A  +P.  1P ) ,  1P >. ] 
 ~R  e.  R. )
 
Theoremprsrpos 6927 Mapping from a positive real to a signed real yields a result greater than zero. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( A  e.  P.  ->  0R  <R  [ <. ( A 
 +P.  1P ) ,  1P >. ]  ~R  )
 
Theoremprsradd 6928 Mapping from positive real addition to signed real addition. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  [ <. ( ( A  +P.  B ) 
 +P.  1P ) ,  1P >. ]  ~R  =  ( [ <. ( A  +P.  1P ) ,  1P >. ]  ~R  +R 
 [ <. ( B  +P.  1P ) ,  1P >. ] 
 ~R  ) )
 
Theoremprsrlt 6929 Mapping from positive real ordering to signed real ordering. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  <P  B  <->  [ <. ( A  +P.  1P ) ,  1P >. ] 
 ~R  <R  [ <. ( B 
 +P.  1P ) ,  1P >. ]  ~R  ) )
 
Theoremprsrriota 6930* Mapping a restricted iota from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ( A  e.  R. 
 /\  0R  <R  A ) 
 ->  [ <. ( ( iota_ x  e.  P.  [ <. ( x  +P.  1P ) ,  1P >. ]  ~R  =  A )  +P.  1P ) ,  1P >. ]  ~R  =  A )
 
Theoremcaucvgsrlemcl 6931* Lemma for caucvgsr 6944. Terms of the sequence from caucvgsrlemgt1 6937 can be mapped to positive reals. (Contributed by Jim Kingdon, 2-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   =>    |-  ( ( ph  /\  A  e.  N. )  ->  ( iota_
 y  e.  P.  ( F `  A )  =  [ <. ( y  +P.  1P ) ,  1P >. ] 
 ~R  )  e.  P. )
 
Theoremcaucvgsrlemasr 6932* Lemma for caucvgsr 6944. The lower bound is a signed real. (Contributed by Jim Kingdon, 4-Jul-2021.)
 |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `
  m ) )   =>    |-  ( ph  ->  A  e.  R. )
 
Theoremcaucvgsrlemfv 6933* Lemma for caucvgsr 6944. Coercing sequence value from a positive real to a signed real. (Contributed by Jim Kingdon, 29-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ( ph  /\  A  e.  N. )  ->  [ <. ( ( G `
  A )  +P.  1P ) ,  1P >. ] 
 ~R  =  ( F `
  A ) )
 
Theoremcaucvgsrlemf 6934* Lemma for caucvgsr 6944. Defining the sequence in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ph  ->  G : N. --> P. )
 
Theoremcaucvgsrlemcau 6935* Lemma for caucvgsr 6944. Defining the Cauchy condition in terms of positive reals. (Contributed by Jim Kingdon, 23-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( G `  n )  <P  ( ( G `
  k )  +P.  <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) 
 /\  ( G `  k )  <P  ( ( G `  n ) 
 +P.  <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >. ) ) ) )
 
Theoremcaucvgsrlembound 6936* Lemma for caucvgsr 6944. Defining the boundedness condition in terms of positive reals. (Contributed by Jim Kingdon, 25-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   &    |-  G  =  ( x  e.  N.  |->  (
 iota_ y  e.  P.  ( F `  x )  =  [ <. ( y 
 +P.  1P ) ,  1P >. ]  ~R  ) )   =>    |-  ( ph  ->  A. m  e.  N.  1P  <P  ( G `  m ) )
 
Theoremcaucvgsrlemgt1 6937* Lemma for caucvgsr 6944. A Cauchy sequence whose terms are greater than one converges. (Contributed by Jim Kingdon, 22-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( F `  m ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e. 
 R.  ( 0R  <R  x 
 ->  E. j  e.  N.  A. i  e.  N.  (
 j  <N  i  ->  (
 ( F `  i
 )  <R  ( y  +R  x )  /\  y  <R  ( ( F `  i
 )  +R  x )
 ) ) ) )
 
Theoremcaucvgsrlemoffval 6938* Lemma for caucvgsr 6944. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ( ph  /\  J  e.  N. )  ->  ( ( G `  J )  +R  A )  =  ( ( F `
  J )  +R  1R ) )
 
Theoremcaucvgsrlemofff 6939* Lemma for caucvgsr 6944. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  G : N. --> R. )
 
Theoremcaucvgsrlemoffcau 6940* Lemma for caucvgsr 6944. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( 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  ) ) ) )
 
Theoremcaucvgsrlemoffgt1 6941* Lemma for caucvgsr 6944. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  A. m  e.  N.  1R  <R  ( G `  m ) )
 
Theoremcaucvgsrlemoffres 6942* Lemma for caucvgsr 6944. Offsetting the values of the sequence so they are greater than one. (Contributed by Jim Kingdon, 3-Jul-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   &    |-  G  =  ( a  e.  N.  |->  ( ( ( F `  a )  +R  1R )  +R  ( A  .R  -1R ) ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e.  R.  ( 0R  <R  x  ->  E. j  e.  N.  A. k  e. 
 N.  ( j  <N  k 
 ->  ( ( F `  k )  <R  ( y  +R  x )  /\  y  <R  ( ( F `
  k )  +R  x ) ) ) ) )
 
Theoremcaucvgsrlembnd 6943* Lemma for caucvgsr 6944. A Cauchy sequence with a lower bound converges. (Contributed by Jim Kingdon, 19-Jun-2021.)
 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   &    |-  ( ph  ->  A. m  e.  N.  A  <R  ( F `  m ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e. 
 R.  ( 0R  <R  x 
 ->  E. j  e.  N.  A. k  e.  N.  (
 j  <N  k  ->  (
 ( F `  k
 )  <R  ( y  +R  x )  /\  y  <R  ( ( F `  k
 )  +R  x )
 ) ) ) )
 
Theoremcaucvgsr 6944* A Cauchy sequence of signed reals with a modulus of convergence converges to a signed real. 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).

This is similar to caucvgprpr 6868 but is for signed reals rather than positive reals.

Here is an outline of how we prove it:

1. Choose a lower bound for the sequence (see caucvgsrlembnd 6943).

2. Offset each element of the sequence so that each element of the resulting sequence is greater than one (greater than zero would not suffice, because the limit as well as the elements of the sequence need to be positive) (see caucvgsrlemofff 6939).

3. Since a signed real (element of  R.) which is greater than zero can be mapped to a positive real (element of  P.), perform that mapping on each element of the sequence and invoke caucvgprpr 6868 to get a limit (see caucvgsrlemgt1 6937).

4. Map the resulting limit from positive reals back to signed reals (see caucvgsrlemgt1 6937).

5. Offset that limit so that we get the limit of the original sequence rather than the limit of the offsetted sequence (see caucvgsrlemoffres 6942). (Contributed by Jim Kingdon, 20-Jun-2021.)

 |-  ( ph  ->  F : N. --> R. )   &    |-  ( ph  ->  A. n  e.  N.  A. k  e.  N.  ( n  <N  k  ->  (
 ( F `  n )  <R  ( ( F `
  k )  +R  [
 <. ( <. { l  |  l  <Q  ( *Q ` 
 [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P.  1P ) ,  1P >. ]  ~R  )  /\  ( F `  k )  <R  ( ( F `  n )  +R  [ <. ( <. { l  |  l  <Q  ( *Q `  [ <. n ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. n ,  1o >. ]  ~Q  )  <Q  u } >.  +P. 
 1P ) ,  1P >. ]  ~R  ) ) ) )   =>    |-  ( ph  ->  E. y  e.  R.  A. x  e. 
 R.  ( 0R  <R  x 
 ->  E. j  e.  N.  A. k  e.  N.  (
 j  <N  k  ->  (
 ( F `  k
 )  <R  ( y  +R  x )  /\  y  <R  ( ( F `  k
 )  +R  x )
 ) ) ) )
 
Syntaxcc 6945 Class of complex numbers.
 class  CC
 
Syntaxcr 6946 Class of real numbers.
 class  RR
 
Syntaxcc0 6947 Extend class notation to include the complex number 0.
 class 
 0
 
Syntaxc1 6948 Extend class notation to include the complex number 1.
 class 
 1
 
Syntaxci 6949 Extend class notation to include the complex number i.
 class  _i
 
Syntaxcaddc 6950 Addition on complex numbers.
 class  +
 
Syntaxcltrr 6951 'Less than' predicate (defined over real subset of complex numbers).
 class  <RR
 
Syntaxcmul 6952 Multiplication on complex numbers. The token  x. is a center dot.
 class  x.
 
Definitiondf-c 6953 Define the set of complex numbers. (Contributed by NM, 22-Feb-1996.)
 |- 
 CC  =  ( R. 
 X.  R. )
 
Definitiondf-0 6954 Define the complex number 0. (Contributed by NM, 22-Feb-1996.)
 |-  0  =  <. 0R ,  0R >.
 
Definitiondf-1 6955 Define the complex number 1. (Contributed by NM, 22-Feb-1996.)
 |-  1  =  <. 1R ,  0R >.
 
Definitiondf-i 6956 Define the complex number  _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.)
 |-  _i  =  <. 0R ,  1R >.
 
Definitiondf-r 6957 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.)
 |- 
 RR  =  ( R. 
 X.  { 0R } )
 
Definitiondf-add 6958* Define addition over complex numbers. (Contributed by NM, 28-May-1995.)
 |- 
 +  =  { <. <. 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 6959* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.)
 |- 
 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 6960* 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 6961 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 6962 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 6963* Membership in class of real numbers. (Contributed by NM, 31-Mar-1996.)
 |-  ( A  e.  RR  <->  E. x  e.  R.  <. x ,  0R >.  =  A )
 
Theoremelrealeu 6964* The real number mapping in elreal 6963 is unique. (Contributed by Jim Kingdon, 11-Jul-2021.)
 |-  ( A  e.  RR  <->  E! x  e.  R.  <. x ,  0R >.  =  A )
 
Theoremelreal2 6965 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 6966 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.)
 |- 
 -.  (/)  e.  CC
 
Theoremltrelre 6967 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
 |- 
 <RR  C_  ( RR  X.  RR )
 
Theoremaddcnsr 6968 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 6969 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 6970 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 6971 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 6972 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 6973 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 6974 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 6975 Technical trick to permit reuse of previous lemmas to prove arithmetic operation laws in  CC from those in  R.. The trick involves qsid 6202, 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 6953), 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 6976 Technical trick to permit re-use of some equivalence class lemmas for operation laws. See dfcnqs 6975 and mulcnsrec 6977. (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 6977 Technical trick to permit re-use of some equivalence class lemmas for operation laws. The trick involves ecidg 6201, which shows that the coset of the converse epsilon relation (which is not an equivalence relation) leaves a set unchanged. See also dfcnqs 6975. (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 6978 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 7064. (Contributed by Jim Kingdon, 14-Jul-2021.)
 |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( A  +  B )  e.  _V )
 
Theorempitonnlem1 6979* Lemma for pitonn 6982. 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 6980 Lemma for pitonn 6982. 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 6981* Lemma for pitonn 6982. 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 6982* 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 6983* 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 6984* Embedding from  N. to  RR. Similar to pitonn 6982 but separate in the sense that we have not proved nnssre 7994 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 6985* 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 6986* One is an element of  NN. This is a counterpart to 1nn 8001 designed for real number axioms which involve natural numbers (notably, axcaucvg 7032). (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 6987* A successor of a positive integer is a positive integer. This is a counterpart to peano2nn 8002 designed for real number axioms which involve to natural numbers (notably, axcaucvg 7032). (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 6988* 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 6989* 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 6990* 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 6991 Lemma for recidpirq 6992. 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 6992* 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 )
 
3.1.2  Final derivation of real and complex number postulates
 
Theoremaxcnex 6993 The complex numbers form a set. Use cnex 7063 instead. (Contributed by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |- 
 CC  e.  _V
 
Theoremaxresscn 6994 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 7034. (Contributed by NM, 1-Mar-1995.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (New usage is discouraged.)
 |- 
 RR  C_  CC
 
Theoremax1cn 6995 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 7035. (Contributed by NM, 12-Apr-2007.) (New usage is discouraged.)
 |-  1  e.  CC
 
Theoremax1re 6996 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 7036.

In the Metamath Proof Explorer, this is not a complex number axiom but is proved from ax-1cn 7035 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 6997  _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 7037. (Contributed by NM, 23-Feb-1996.) (New usage is discouraged.)
 |-  _i  e.  CC
 
Theoremaxaddcl 6998 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 7038 be used later. Instead, in most cases use addcl 7064. (Contributed by NM, 14-Jun-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  +  B )  e.  CC )
 
Theoremaxaddrcl 6999 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 7039 be used later. Instead, in most cases use readdcl 7065. (Contributed by NM, 31-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A  +  B )  e.  RR )
 
Theoremaxmulcl 7000 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 7040 be used later. Instead, in most cases use mulcl 7066. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B )  e.  CC )
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