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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrecexprlemrnd 6601* is rounded. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  q 
 Q.  q  1st `  r  Q.  q  <Q  r  r  1st `  r  Q.  r  2nd `  q  Q.  q  <Q  r  q  2nd `
 
Theoremrecexprlemdisj 6602* is disjoint. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  q  Q.  q  1st `  q  2nd `
 
Theoremrecexprlemloc 6603* is located. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  q  Q.  r  Q.  q  <Q  r  q  1st `  r  2nd `
 
Theoremrecexprlempr 6604* is a positive real. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  P.
 
Theoremrecexprlem1ssl 6605* The lower cut of one is a subset of the lower cut of  .P. . Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  1st `  1P  C_  1st `  .P.
 
Theoremrecexprlem1ssu 6606* The upper cut of one is a subset of the upper cut of  .P. . Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  2nd `  1P  C_  2nd `  .P.
 
Theoremrecexprlemss1l 6607* The lower cut of  .P. is a subset of the lower cut of one. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  1st `  .P.  C_  1st `  1P
 
Theoremrecexprlemss1u 6608* The upper cut of  .P. is a subset of the upper cut of one. Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  2nd `  .P.  C_  2nd `  1P
 
Theoremrecexprlemex 6609* is the reciprocal of . Lemma for recexpr 6610. (Contributed by Jim Kingdon, 27-Dec-2019.)
 <. {  | 
 <Q  *Q `  2nd `  } ,  {  | 
 <Q  *Q `  1st `  } >.   =>     P.  .P.  1P
 
Theoremrecexpr 6610* The reciprocal of a positive real exists. Part of Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
 P.  P.  .P.  1P
 
Theoremaptiprleml 6611 Lemma for aptipr 6613. (Contributed by Jim Kingdon, 28-Jan-2020.)
 P.  P.  <P  1st `  C_  1st `
 
Theoremaptiprlemu 6612 Lemma for aptipr 6613. (Contributed by Jim Kingdon, 28-Jan-2020.)
 P.  P.  <P  2nd `  C_  2nd `
 
Theoremaptipr 6613 Apartness of positive reals is tight. (Contributed by Jim Kingdon, 28-Jan-2020.)
 P.  P.  <P  <P
 
Theoremltmprr 6614 Ordering property of multiplication. (Contributed by Jim Kingdon, 18-Feb-2020.)
 P.  P.  C  P.  C  .P.  <P  C  .P.  <P
 
Theoremarchpr 6615* For any positive real, there is an integer that is greater than it. This is also known as the "archimedean property". The integer is embedded into the reals as described at nnprlu 6534. (Contributed by Jim Kingdon, 22-Apr-2020.)
 P.  N.  <P  <. { l  |  l  <Q  <. ,  1o >.  ~Q  } ,  {  |  <. ,  1o >.  ~Q  <Q  } >.
 
Theoremcauappcvgprlemcan 6616* Lemma for cauappcvgprlemladdrl 6629. Cancelling a term from both sides. (Contributed by Jim Kingdon, 15-Aug-2020.)
 L  P.   &     S  Q.   &     R  Q.   &     Q  Q.   =>     R  +Q  Q  1st `  L  +P.  <. { l  |  l  <Q  S  +Q  Q } ,  {  |  S  +Q  Q  <Q  } >.  R  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
 
Theoremcauappcvgprlemm 6617* Lemma for cauappcvgpr 6634. The putative limit is inhabited. (Contributed by Jim Kingdon, 18-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  Q.  s  1st `  L  r  Q.  r  2nd `  L
 
Theoremcauappcvgprlemopl 6618* Lemma for cauappcvgpr 6634. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  1st `  L  r  Q.  s  <Q  r  r  1st `  L
 
Theoremcauappcvgprlemlol 6619* Lemma for cauappcvgpr 6634. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 4-Aug-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  <Q  r  r  1st `  L  s  1st `  L
 
Theoremcauappcvgprlemopu 6620* Lemma for cauappcvgpr 6634. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 4-Aug-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     r  2nd `  L  s  Q.  s  <Q  r  s  2nd `  L
 
Theoremcauappcvgprlemupu 6621* Lemma for cauappcvgpr 6634. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 4-Aug-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  <Q  r  s  2nd `  L  r  2nd `  L
 
Theoremcauappcvgprlemrnd 6622* Lemma for cauappcvgpr 6634. The putative limit is rounded. (Contributed by Jim Kingdon, 18-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  Q.  s  1st `  L  r  Q.  s  <Q  r  r  1st `  L  r  Q.  r  2nd `  L  s  Q.  s  <Q  r  s  2nd `  L
 
Theoremcauappcvgprlemdisj 6623* Lemma for cauappcvgpr 6634. The putative limit is disjoint. (Contributed by Jim Kingdon, 18-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  Q.  s  1st `  L  s  2nd `  L
 
Theoremcauappcvgprlemloc 6624* Lemma for cauappcvgpr 6634. The putative limit is located. (Contributed by Jim Kingdon, 18-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     s  Q.  r  Q.  s  <Q  r  s  1st `  L  r  2nd `  L
 
Theoremcauappcvgprlemcl 6625* Lemma for cauappcvgpr 6634. The putative limit is a positive real. (Contributed by Jim Kingdon, 20-Jun-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     L  P.
 
Theoremcauappcvgprlemladdfu 6626* Lemma for cauappcvgprlemladd 6630. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     S  Q.   =>     2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  2nd `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q 
 +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
 
Theoremcauappcvgprlemladdfl 6627* Lemma for cauappcvgprlemladd 6630. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     S  Q.   =>     1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  1st `  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `  q 
 +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
 
Theoremcauappcvgprlemladdru 6628* Lemma for cauappcvgprlemladd 6630. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     S  Q.   =>     2nd `  <. { l 
 Q.  |  q  Q.  l  +Q  q  <Q  F `  q 
 +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.  C_  2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
 
Theoremcauappcvgprlemladdrl 6629* Lemma for cauappcvgprlemladd 6630. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 11-Jul-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     S  Q.   =>     1st `  <. { l 
 Q.  |  q  Q.  l  +Q  q  <Q  F `  q 
 +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.  C_  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
 
Theoremcauappcvgprlemladd 6630* Lemma for cauappcvgpr 6634. This takes  L and offsets it by the positive fraction  S. (Contributed by Jim Kingdon, 23-Jun-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     S  Q.   =>     L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >. 
 <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q  +Q  S } ,  {  Q.  |  q  Q.  F `  q  +Q  q  +Q  S  <Q  } >.
 
Theoremcauappcvgprlem1 6631* Lemma for cauappcvgpr 6634. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     Q  Q.   &     R  Q.   =>     <. { l  |  l  <Q  F `  Q } ,  {  |  F `  Q  <Q  } >.  <P  L  +P.  <. { l  |  l  <Q  Q  +Q  R } ,  {  |  Q  +Q  R 
 <Q  } >.
 
Theoremcauappcvgprlem2 6632* Lemma for cauappcvgpr 6634. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 23-Jun-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   &     Q  Q.   &     R  Q.   =>     L  <P 
 <. { l  |  l 
 <Q  F `  Q  +Q  Q  +Q  R } ,  {  |  F `  Q  +Q  Q  +Q  R 
 <Q  } >.
 
Theoremcauappcvgprlemlim 6633* Lemma for cauappcvgpr 6634. The putative limit is a limit. (Contributed by Jim Kingdon, 20-Jun-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   &     L  <. { l  Q.  |  q  Q.  l  +Q  q  <Q  F `
  q } ,  {  Q.  |  q  Q.  F `  q  +Q  q  <Q  } >.   =>     q  Q.  r  Q.  <. { l  |  l  <Q  F `
  q } ,  {  |  F `  q 
 <Q  } >.  <P  L 
 +P.  <. { l  |  l  <Q  q  +Q  r } ,  {  |  q  +Q  r  <Q  } >.  L  <P 
 <. { l  |  l 
 <Q  F `  q  +Q  q  +Q  r } ,  {  |  F `  q  +Q  q  +Q  r 
 <Q  } >.
 
Theoremcauappcvgpr 6634* A Cauchy approximation has a limit. A Cauchy approximation, here  F, is similar to a Cauchy sequence but is indexed by the desired tolerance (that is, how close together terms needs to be) rather than by natural numbers. This is basically Theorem 11.2.12 of [HoTT], p. (varies) with a few differences such as that we are proving the existence of a limit without anything about how fast it converges (that is, mere existence instead of existence, in HoTT terms), and that the codomain of  F is  Q. rather than  P.. We also specify that every term needs to be larger than a fraction , to avoid the case where we have positive fractions which converge to zero (which is not a positive real). (Contributed by Jim Kingdon, 19-Jun-2020.)
 F : Q. --> Q.   &     p  Q.  q  Q.  F `
  p  <Q  F `  q  +Q  p  +Q  q  F `  q  <Q  F `  p  +Q  p  +Q  q   &     p  Q.  <Q  F `
  p   =>     P.  q 
 Q.  r  Q.  <. { l  |  l  <Q  F `  q } ,  {  |  F `  q  <Q  } >.  <P 
 +P.  <. { l  |  l  <Q  q  +Q  r } ,  {  |  q  +Q  r  <Q  } >.  <P 
 <. { l  |  l 
 <Q  F `  q  +Q  q  +Q  r } ,  {  |  F `  q  +Q  q  +Q  r 
 <Q  } >.
 
Theoremarchrecnq 6635* Archimedean principle for fractions (reciprocal version). (Contributed by Jim Kingdon, 27-Sep-2020.)
 Q.  j  N.  *Q `  <. j ,  1o >.  ~Q  <Q
 
Theoremcaucvgprlemk 6636 Lemma for caucvgpr 6653. Reciprocals of positive integers decrease as the positive integers increase. (Contributed by Jim Kingdon, 9-Oct-2020.)
 J  <N  K   &     *Q `  <. J ,  1o >.  ~Q  <Q  Q   =>     *Q `  <. K ,  1o >.  ~Q  <Q  Q
 
Theoremcaucvgprlemnkj 6637* Lemma for caucvgpr 6653. Part of disjointness. (Contributed by Jim Kingdon, 23-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     K  N.   &     J  N.   &     S  Q.   =>     S  +Q  *Q `  <. K ,  1o >.  ~Q 
 <Q  F `  K  F `
  J  +Q  *Q `  <. J ,  1o >.  ~Q  <Q  S
 
Theoremcaucvgprlemnbj 6638* Lemma for caucvgpr 6653. Non-existence of two elements of the sequence which are too far from each other. (Contributed by Jim Kingdon, 18-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     N.   &     J  N.   =>     F `  +Q  *Q `  <. ,  1o >.  ~Q 
 +Q  *Q `  <. J ,  1o >.  ~Q  <Q  F `
  J
 
Theoremcaucvgprlemm 6639* Lemma for caucvgpr 6653. The putative limit is inhabited. (Contributed by Jim Kingdon, 27-Sep-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  Q.  s  1st `  L  r 
 Q.  r  2nd `  L
 
Theoremcaucvgprlemopl 6640* Lemma for caucvgpr 6653. The lower cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  1st `  L  r  Q.  s  <Q  r  r  1st `  L
 
Theoremcaucvgprlemlol 6641* Lemma for caucvgpr 6653. The lower cut of the putative limit is lower. (Contributed by Jim Kingdon, 20-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  <Q  r  r  1st `  L  s  1st `  L
 
Theoremcaucvgprlemopu 6642* Lemma for caucvgpr 6653. The upper cut of the putative limit is open. (Contributed by Jim Kingdon, 20-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     r  2nd `  L  s  Q.  s  <Q  r  s  2nd `  L
 
Theoremcaucvgprlemupu 6643* Lemma for caucvgpr 6653. The upper cut of the putative limit is upper. (Contributed by Jim Kingdon, 20-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  <Q  r  s  2nd `  L  r  2nd `  L
 
Theoremcaucvgprlemrnd 6644* Lemma for caucvgpr 6653. The putative limit is rounded. (Contributed by Jim Kingdon, 27-Sep-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  Q.  s  1st `  L  r  Q.  s  <Q  r  r  1st `  L  r  Q.  r  2nd `  L  s  Q.  s  <Q  r  s  2nd `  L
 
Theoremcaucvgprlemdisj 6645* Lemma for caucvgpr 6653. The putative limit is disjoint. (Contributed by Jim Kingdon, 27-Sep-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  Q.  s  1st `  L  s  2nd `  L
 
Theoremcaucvgprlemloc 6646* Lemma for caucvgpr 6653. The putative limit is located. (Contributed by Jim Kingdon, 27-Sep-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     s  Q.  r 
 Q.  s  <Q  r  s  1st `  L  r  2nd `  L
 
Theoremcaucvgprlemcl 6647* Lemma for caucvgpr 6653. The putative limit is a positive real. (Contributed by Jim Kingdon, 26-Sep-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     L  P.
 
Theoremcaucvgprlemladdfu 6648* Lemma for caucvgpr 6653. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 9-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   &     S  Q.   =>     2nd `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.  C_  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q 
 +Q  S  <Q  }
 
Theoremcaucvgprlemladdrl 6649* Lemma for caucvgpr 6653. Adding  S after embedding in positive reals, or adding it as a rational. (Contributed by Jim Kingdon, 8-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   &     S  Q.   =>     {
 l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q 
 <Q  F `  j  +Q  S }  C_  1st `  L  +P.  <. { l  |  l  <Q  S } ,  {  |  S  <Q  } >.
 
Theoremcaucvgprlem1 6650* Lemma for caucvgpr 6653. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   &     Q  Q.   &     J  <N  K   &     *Q `  <. J ,  1o >.  ~Q  <Q  Q   =>     <. { l  |  l  <Q  F `  K } ,  {  |  F `  K  <Q  } >.  <P  L  +P.  <. { l  |  l  <Q  Q } ,  {  |  Q  <Q  } >.
 
Theoremcaucvgprlem2 6651* Lemma for caucvgpr 6653. Part of showing the putative limit to be a limit. (Contributed by Jim Kingdon, 3-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   &     Q  Q.   &     J  <N  K   &     *Q `  <. J ,  1o >.  ~Q  <Q  Q   =>     L  <P 
 <. { l  |  l 
 <Q  F `  K  +Q  Q } ,  {  |  F `  K  +Q  Q 
 <Q  } >.
 
Theoremcaucvgprlemlim 6652* Lemma for caucvgpr 6653. The putative limit is a limit. (Contributed by Jim Kingdon, 1-Oct-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   &     L  <. { l  Q.  |  j  N.  l  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  F `
  j } ,  {  Q.  |  j  N.  F `  j  +Q  *Q `  <. j ,  1o >.  ~Q  <Q  } >.   =>     Q.  j 
 N.  k  N.  j  <N  k  <. { l  |  l  <Q  F `  k } ,  {  |  F `  k  <Q  } >.  <P  L 
 +P.  <. { l  |  l  <Q  } ,  {  |  <Q  } >.  L  <P  <. { l  |  l  <Q  F `  k  +Q  } ,  {  |  F `  k  +Q  <Q  } >.
 
Theoremcaucvgpr 6653* A Cauchy sequence of positive fractions with a modulus of convergence converges to a positive real. This is basically Corollary 11.2.13 of [HoTT], p. (varies) (one key difference being that this is for positive reals rather than signed reals). Also, 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). We also specify that every term needs to be larger than a fraction , to avoid the case where we have positive fractions which converge to zero (which is not a positive real). (Contributed by Jim Kingdon, 18-Jun-2020.)
 F : N. --> Q.   &     n  N.  k  N.  n  <N  k  F `  n  <Q  F `  k 
 +Q  *Q `  <. n ,  1o >.  ~Q  F `
  k  <Q  F `  n  +Q  *Q `  <. n ,  1o >.  ~Q    &     j  N.  <Q  F `
  j   =>     P.  Q.  j  N.  k  N.  j  <N  k  <. { l  |  l 
 <Q  F `  k } ,  {  |  F `  k  <Q  } >.  <P  +P.  <. { l  |  l  <Q  } ,  {  |  <Q  } >.  <P  <. { l  |  l  <Q  F `  k  +Q  } ,  {  |  F `  k  +Q  <Q  } >.
 
Definitiondf-enr 6654* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)

 ~R  { <. ,  >.  |  P.  X.  P.  P.  X.  P.  <. ,  >.  <. ,  >.  +P.  +P.  }
 
Definitiondf-nr 6655 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 25-Jul-1995.)

 R.  P.  X.  P. /.  ~R
 
Definitiondf-plr 6656* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)

 +R  { <. <. ,  >. ,  >.  |  R.  R.  <. ,  >. 
 ~R 
 <. ,  >. 
 ~R  <.  +P.  , 
 +P.  >. 
 ~R  }
 
Definitiondf-mr 6657* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.)

 .R  { <. <. ,  >. ,  >.  |  R.  R.  <. ,  >. 
 ~R 
 <. ,  >. 
 ~R  <. 
 .P.  +P.  .P.  , 
 .P.  +P.  .P.  >.  ~R  }
 
Definitiondf-ltr 6658* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.4 of [Gleason] p. 127. (Contributed by NM, 14-Feb-1996.)

 <R  { <. ,  >.  |  R.  R.  <. ,  >.  ~R  <. ,  >.  ~R  +P.  <P  +P.  }
 
Definitiondf-0r 6659 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)

 0R  <. 1P ,  1P >.  ~R
 
Definitiondf-1r 6660 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. From Proposition 9-4.2 of [Gleason] p. 126. (Contributed by NM, 9-Aug-1995.)

 1R  <. 1P  +P.  1P ,  1P >.  ~R
 
Definitiondf-m1r 6661 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.)

 -1R  <. 1P ,  1P  +P.  1P >.  ~R
 
Theoremenrbreq 6662 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.)
 P.  P.  C  P.  D  P.  <. ,  >.  ~R  <. C ,  D >.  +P.  D  +P.  C
 
Theoremenrer 6663 The equivalence relation for signed reals is an equivalence relation. Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 6-Jul-2015.)

 ~R  Er  P.  X. 
 P.
 
Theoremenreceq 6664 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.)
 P.  P.  C  P.  D  P.  <. ,  >.  ~R  <. C ,  D >. 
 ~R  +P.  D  +P.  C
 
Theoremenrex 6665 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.)

 ~R  _V
 
Theoremltrelsr 6666 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.)

 <R  C_  R.  X.  R.
 
Theoremaddcmpblnr 6667 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.)
 P.  P.  C  P.  D  P.  F  P.  G  P.  R  P.  S  P. 
 +P.  D  +P.  C  F  +P.  S  G  +P.  R  <.  +P.  F ,  +P.  G >.  ~R  <. C  +P.  R ,  D  +P.  S >.
 
Theoremmulcmpblnrlemg 6668 Lemma used in lemma showing compatibility of multiplication. (Contributed by Jim Kingdon, 1-Jan-2020.)
 P.  P.  C  P.  D  P.  F  P.  G  P.  R  P.  S  P. 
 +P.  D  +P.  C  F  +P.  S  G  +P.  R  D  .P.  F  +P.  .P.  F 
 +P.  .P.  G  +P.  C  .P.  S 
 +P.  D  .P.  R  D 
 .P.  F  +P.  .P.  G  +P.  .P.  F  +P.  C  .P.  R 
 +P.  D  .P.  S
 
Theoremmulcmpblnr 6669 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.)
 P.  P.  C  P.  D  P.  F  P.  G  P.  R  P.  S  P. 
 +P.  D  +P.  C  F  +P.  S  G  +P.  R  <.  .P.  F  +P. 
 .P.  G ,  .P.  G  +P.  .P.  F >.  ~R  <. C  .P.  R  +P.  D  .P.  S ,  C  .P.  S 
 +P.  D  .P.  R >.
 
Theoremprsrlem1 6670* Decomposing signed reals into positive reals. Lemma for addsrpr 6673 and mulsrpr 6674. (Contributed by Jim Kingdon, 30-Dec-2019.)
 P.  X. 
 P. /.  ~R  P.  X.  P. /.  ~R  <. ,  >. 
 ~R 
 <. ,  t >. 
 ~R  <. s ,  >.  ~R  <. ,  h >.  ~R  P.  P.  s 
 P.  P.  P.  t  P.  P.  h  P.  +P.  +P.  s 
 +P.  h  t  +P.
 
Theoremaddsrmo 6671* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 P.  X.  P. /.  ~R  P.  X. 
 P. /.  ~R  t  <. ,  >.  ~R  <. ,  t >.  ~R  <. 
 +P.  ,  +P.  t >. 
 ~R
 
Theoremmulsrmo 6672* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 P.  X.  P. /.  ~R  P.  X. 
 P. /.  ~R  t  <. ,  >.  ~R  <. ,  t >.  ~R  <.  .P. 
 +P.  .P.  t ,  .P.  t 
 +P.  .P.  >.  ~R
 
Theoremaddsrpr 6673 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 P.  P.  C  P.  D  P.  <. ,  >.  ~R  +R  <. C ,  D >. 
 ~R  <.  +P.  C ,  +P.  D >.  ~R
 
Theoremmulsrpr 6674 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.)
 P.  P.  C  P.  D  P.  <. ,  >.  ~R  .R  <. C ,  D >. 
 ~R  <.  .P.  C  +P. 
 .P.  D ,  .P.  D  +P.  .P.  C >.  ~R
 
Theoremltsrprg 6675 Ordering of signed reals in terms of positive reals. (Contributed by Jim Kingdon, 2-Jan-2019.)
 P.  P.  C  P.  D  P.  <. ,  >.  ~R  <R  <. C ,  D >.  ~R  +P.  D 
 <P  +P.  C
 
Theoremgt0srpr 6676 Greater than zero in terms of positive reals. (Contributed by NM, 13-May-1996.)
 0R  <R  <. ,  >.  ~R 
 <P
 
Theorem0nsr 6677 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.)
 (/)  R.
 
Theorem0r 6678 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.)

 0R  R.
 
Theorem1sr 6679 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.)

 1R  R.
 
Theoremm1r 6680 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.)

 -1R  R.
 
Theoremaddclsr 6681 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.)
 R.  R.  +R  R.
 
Theoremmulclsr 6682 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.)
 R.  R.  .R  R.
 
Theoremaddcomsrg 6683 Addition of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 R.  R.  +R  +R
 
Theoremaddasssrg 6684 Addition of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 R.  R.  C  R.  +R  +R  C  +R  +R  C
 
Theoremmulcomsrg 6685 Multiplication of signed reals is commutative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 R.  R.  .R  .R
 
Theoremmulasssrg 6686 Multiplication of signed reals is associative. (Contributed by Jim Kingdon, 3-Jan-2020.)
 R.  R.  C  R.  .R  .R  C  .R  .R  C
 
Theoremdistrsrg 6687 Multiplication of signed reals is distributive. (Contributed by Jim Kingdon, 4-Jan-2020.)
 R.  R.  C  R.  .R  +R  C 
 .R  +R  .R  C
 
Theoremm1p1sr 6688 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.)
 -1R  +R  1R  0R
 
Theoremm1m1sr 6689 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.)
 -1R  .R  -1R  1R
 
Theoremlttrsr 6690* Signed real 'less than' is a transitive relation. (Contributed by Jim Kingdon, 4-Jan-2019.)

 R.  R.  h  R. 
 <R  <R  h  <R  h
 
Theoremltposr 6691 Signed real 'less than' is a partial order. (Contributed by Jim Kingdon, 4-Jan-2019.)

 <R  Po  R.
 
Theoremltsosr 6692 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.)

 <R  Or  R.
 
Theorem0lt1sr 6693 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.)

 0R  <R  1R
 
Theorem1ne0sr 6694 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.)
 1R  0R
 
Theorem0idsr 6695 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.)
 R.  +R  0R
 
Theorem1idsr 6696 1 is an identity element for multiplication. (Contributed by Jim Kingdon, 5-Jan-2020.)
 R.  .R  1R
 
Theorem00sr 6697 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.)
 R.  .R  0R  0R
 
Theoremltasrg 6698 Ordering property of addition. (Contributed by NM, 10-May-1996.)
 R.  R.  C  R.  <R  C  +R  <R  C  +R
 
Theorempn0sr 6699 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.)
 R.  +R  .R  -1R  0R
 
Theoremnegexsr 6700* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.)
 R.  R.  +R  0R
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