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Theorem List for Intuitionistic Logic Explorer - 6601-6700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnnnq0lem1 6601* Decomposing non-negative fractions into natural numbers. Lemma for addnnnq0 6604 and mulnnnq0 6605. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( ( A  e.  ( ( om  X. 
 N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  /\  (
 ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ C ] ~Q0  )  /\  ( ( A  =  [ <. s ,  f >. ] ~Q0  /\  B  =  [ <. g ,  h >. ] ~Q0  )  /\  q  =  [ D ] ~Q0  ) ) )  ->  ( ( ( ( w  e.  om  /\  v  e.  N. )  /\  ( s  e.  om  /\  f  e.  N. )
 )  /\  ( ( u  e.  om  /\  t  e.  N. )  /\  (
 g  e.  om  /\  h  e.  N. )
 ) )  /\  (
 ( w  .o  f
 )  =  ( v  .o  s )  /\  ( u  .o  h )  =  ( t  .o  g ) ) ) )
 
Theoremaddnq0mo 6602* There is at most one result from adding non-negative fractions. (Contributed by Jim Kingdon, 23-Nov-2019.)
 |-  ( ( A  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( ( w  .o  t )  +o  (
 v  .o  u )
 ) ,  ( v  .o  t ) >. ] ~Q0  )
 )
 
Theoremmulnq0mo 6603* There is at most one result from multiplying non-negative fractions. (Contributed by Jim Kingdon, 20-Nov-2019.)
 |-  ( ( A  e.  ( ( om  X.  N. ) /. ~Q0  )  /\  B  e.  ( ( om  X.  N. ) /. ~Q0  ) )  ->  E* z E. w E. v E. u E. t ( ( A  =  [ <. w ,  v >. ] ~Q0  /\  B  =  [ <. u ,  t >. ] ~Q0  )  /\  z  =  [ <. ( w  .o  u ) ,  ( v  .o  t ) >. ] ~Q0  ) )
 
Theoremaddnnnq0 6604 Addition of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 22-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0 +Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. ( ( A  .o  D )  +o  ( B  .o  C ) ) ,  ( B  .o  D ) >. ] ~Q0  )
 
Theoremmulnnnq0 6605 Multiplication of non-negative fractions in terms of natural numbers. (Contributed by Jim Kingdon, 19-Nov-2019.)
 |-  ( ( ( A  e.  om  /\  B  e.  N. )  /\  ( C  e.  om  /\  D  e.  N. ) )  ->  ( [ <. A ,  B >. ] ~Q0 ·Q0  [ <. C ,  D >. ] ~Q0  )  =  [ <. ( A  .o  C ) ,  ( B  .o  D ) >. ] ~Q0  )
 
Theoremaddclnq0 6606 Closure of addition on non-negative fractions. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A +Q0  B )  e. Q0 )
 
Theoremmulclnq0 6607 Closure of multiplication on non-negative fractions. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A ·Q0  B )  e. Q0 )
 
Theoremnqpnq0nq 6608 A positive fraction plus a non-negative fraction is a positive fraction. (Contributed by Jim Kingdon, 30-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e. Q0 )  ->  ( A +Q0  B )  e.  Q. )
 
Theoremnqnq0a 6609 Addition of positive fractions is equal with  +Q or +Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  +Q  B )  =  ( A +Q0  B ) )
 
Theoremnqnq0m 6610 Multiplication of positive fractions is equal with  .Q or ·Q0. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( A  .Q  B )  =  ( A ·Q0  B ) )
 
Theoremnq0m0r 6611 Multiplication with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( A  e. Q0  ->  (0Q0 ·Q0  A )  = 0Q0 )
 
Theoremnq0a0 6612 Addition with zero for non-negative fractions. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( A  e. Q0  ->  ( A +Q0 0Q0 )  =  A )
 
Theoremnnanq0 6613 Addition of non-negative fractions with a common denominator. You can add two fractions with the same denominator by adding their numerators and keeping the same denominator. (Contributed by Jim Kingdon, 1-Dec-2019.)
 |-  ( ( N  e.  om 
 /\  M  e.  om  /\  A  e.  N. )  ->  [ <. ( N  +o  M ) ,  A >. ] ~Q0  =  ( [ <. N ,  A >. ] ~Q0 +Q0  [ <. M ,  A >. ] ~Q0  ) )
 
Theoremdistrnq0 6614 Multiplication of non-negative fractions is distributive. (Contributed by Jim Kingdon, 27-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( A ·Q0  ( B +Q0  C ) )  =  ( ( A ·Q0  B ) +Q0  ( A ·Q0  C ) ) )
 
Theoremmulcomnq0 6615 Multiplication of non-negative fractions is commutative. (Contributed by Jim Kingdon, 27-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0 ) 
 ->  ( A ·Q0  B )  =  ( B ·Q0  A ) )
 
Theoremaddassnq0lemcl 6616 A natural number closure law. Lemma for addassnq0 6617. (Contributed by Jim Kingdon, 3-Dec-2019.)
 |-  ( ( ( I  e.  om  /\  J  e.  N. )  /\  ( K  e.  om  /\  L  e.  N. ) )  ->  ( ( ( I  .o  L )  +o  ( J  .o  K ) )  e.  om  /\  ( J  .o  L )  e.  N. ) )
 
Theoremaddassnq0 6617 Addition of non-negaative fractions is associative. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( ( A +Q0  B ) +Q0  C )  =  ( A +Q0  ( B +Q0  C ) ) )
 
Theoremdistnq0r 6618 Multiplication of non-negative fractions is distributive. Version of distrnq0 6614 with the multiplications commuted. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( ( A  e. Q0  /\  B  e. Q0  /\  C  e. Q0 )  ->  ( ( B +Q0  C ) ·Q0  A )  =  ( ( B ·Q0  A ) +Q0  ( C ·Q0  A ) ) )
 
Theoremaddpinq1 6619 Addition of one to the numerator of a fraction whose denominator is one. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  N.  ->  [ <. ( A  +N  1o ) ,  1o >. ] 
 ~Q  =  ( [ <. A ,  1o >. ] 
 ~Q  +Q  1Q )
 )
 
Theoremnq02m 6620 Multiply a non-negative fraction by two. (Contributed by Jim Kingdon, 29-Nov-2019.)
 |-  ( A  e. Q0  ->  ( [ <. 2o ,  1o >. ] ~Q0 ·Q0  A )  =  ( A +Q0  A ) )
 
Definitiondf-inp 6621* Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other.

Here we follow the definition of a Dedekind cut from Definition 11.2.1 of [HoTT], p. (varies) with the one exception that we define it over positive rational numbers rather than all rational numbers.

A Dedekind cut is an ordered pair of a lower set  l and an upper set  u which is inhabited ( E. q  e. 
Q. q  e.  l  /\  E. r  e. 
Q. r  e.  u), rounded ( A. q  e.  Q. ( q  e.  l  <->  E. r  e.  Q. ( q  <Q  r  /\  r  e.  l
) ) and likewise for  u), disjoint ( A. q  e. 
Q. -.  ( q  e.  l  /\  q  e.  u )) and located ( A. q  e. 
Q. A. r  e.  Q. ( q  <Q  r  ->  ( q  e.  l  \/  r  e.  u
) )). See HoTT for more discussion of those terms and different ways of defining Dedekind cuts.

(Note: This is a "temporary" definition used in the construction of complex numbers, and is intended to be used only by the construction.) (Contributed by Jim Kingdon, 25-Sep-2019.)

 |- 
 P.  =  { <. l ,  u >.  |  ( ( ( l  C_  Q. 
 /\  u  C_  Q. )  /\  ( E. q  e. 
 Q.  q  e.  l  /\  E. r  e.  Q.  r  e.  u )
 )  /\  ( ( A. q  e.  Q.  ( q  e.  l  <->  E. r  e.  Q.  (
 q  <Q  r  /\  r  e.  l ) )  /\  A. r  e.  Q.  (
 r  e.  u  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  u )
 ) )  /\  A. q  e.  Q.  -.  (
 q  e.  l  /\  q  e.  u )  /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  l  \/  r  e.  u ) ) ) ) }
 
Definitiondf-i1p 6622* Define the positive 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 Jim Kingdon, 25-Sep-2019.)
 |- 
 1P  =  <. { l  |  l  <Q  1Q } ,  { u  |  1Q  <Q  u } >.
 
Definitiondf-iplp 6623* Define addition on positive reals. From Section 11.2.1 of [HoTT], p. (varies). We write this definition to closely resemble the definition in HoTT although some of the conditions are redundant (for example,  r  e.  ( 1st `  x ) implies 
r  e.  Q.) and can be simplified as shown at genpdf 6663.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 26-Sep-2019.)

 |- 
 +P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y )  /\  q  =  ( r  +Q  s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
 r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
 )  /\  q  =  ( r  +Q  s
 ) ) } >. )
 
Definitiondf-imp 6624* Define multiplication on positive reals. Here we use a simple definition which is similar to df-iplp 6623 or the definition of multiplication on positive reals in Metamath Proof Explorer. This is as opposed to the more complicated definition of multiplication given in Section 11.2.1 of [HoTT], p. (varies), which appears to be motivated by handling negative numbers or handling modified Dedekind cuts in which locatedness is omitted.

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

 |- 
 .P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  x )  /\  s  e.  ( 1st `  y )  /\  q  =  ( r  .Q  s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e.  Q.  (
 r  e.  ( 2nd `  x )  /\  s  e.  ( 2nd `  y
 )  /\  q  =  ( r  .Q  s
 ) ) } >. )
 
Definitiondf-iltp 6625* Define ordering on positive reals. We define  x 
<P  y if there is a positive fraction  q which is an element of the upper cut of  x and the lower cut of  y. From the definition of < in Section 11.2.1 of [HoTT], p. (varies).

This is a "temporary" set used in the construction of complex numbers, and is intended to be used only by the construction. (Contributed by Jim Kingdon, 29-Sep-2019.)

 |- 
 <P  =  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  E. q  e.  Q.  ( q  e.  ( 2nd `  x )  /\  q  e.  ( 1st `  y ) ) ) }
 
Theoremnpsspw 6626 Lemma for proving existence of reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |- 
 P.  C_  ( ~P Q.  X. 
 ~P Q. )
 
Theorempreqlu 6627 Two reals are equal if and only if their lower and upper cuts are. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  =  B 
 <->  ( ( 1st `  A )  =  ( 1st `  B )  /\  ( 2nd `  A )  =  ( 2nd `  B ) ) ) )
 
Theoremnpex 6628 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
 |- 
 P.  e.  _V
 
Theoremelinp 6629* Membership in positive reals. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  <->  ( ( ( L  C_  Q.  /\  U  C_ 
 Q. )  /\  ( E. q  e.  Q.  q  e.  L  /\  E. r  e.  Q.  r  e.  U ) )  /\  ( ( A. q  e.  Q.  ( q  e.  L  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  L )
 )  /\  A. r  e. 
 Q.  ( r  e.  U  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  U )
 ) )  /\  A. q  e.  Q.  -.  (
 q  e.  L  /\  q  e.  U )  /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  L  \/  r  e.  U )
 ) ) ) )
 
Theoremprop 6630 A positive real is an ordered pair of a lower cut and an upper cut. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( A  e.  P.  -> 
 <. ( 1st `  A ) ,  ( 2nd `  A ) >.  e.  P. )
 
Theoremelnp1st2nd 6631* Membership in positive reals, using  1st and  2nd to refer to the lower and upper cut. (Contributed by Jim Kingdon, 3-Oct-2019.)
 |-  ( A  e.  P.  <->  (
 ( A  e.  ( ~P Q.  X.  ~P Q. )  /\  ( E. q  e.  Q.  q  e.  ( 1st `  A )  /\  E. r  e.  Q.  r  e.  ( 2nd `  A ) ) )  /\  ( ( A. q  e.  Q.  ( q  e.  ( 1st `  A ) 
 <-> 
 E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  A ) ) ) 
 /\  A. r  e.  Q.  ( r  e.  ( 2nd `  A )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  A ) ) ) )  /\  A. q  e.  Q.  -.  ( q  e.  ( 1st `  A )  /\  q  e.  ( 2nd `  A ) ) 
 /\  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  A )  \/  r  e.  ( 2nd `  A ) ) ) ) ) )
 
Theoremprml 6632* A positive real's lower cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  E. x  e.  Q.  x  e.  L )
 
Theoremprmu 6633* A positive real's upper cut is inhabited. (Contributed by Jim Kingdon, 27-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  E. x  e.  Q.  x  e.  U )
 
Theoremprssnql 6634 A positive real's lower cut is a subset of the positive fractions. It would presumably be possible to also prove  L 
C.  Q., but we only need  L  C_  Q. so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  L  C_ 
 Q. )
 
Theoremprssnqu 6635 A positive real's upper cut is a subset of the positive fractions. It would presumably be possible to also prove  U 
C.  Q., but we only need  U  C_  Q. so far. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  U  C_ 
 Q. )
 
Theoremelprnql 6636 An element of a positive real's lower cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  B  e.  Q. )
 
Theoremelprnqu 6637 An element of a positive real's upper cut is a positive fraction. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  U )  ->  B  e.  Q. )
 
Theorem0npr 6638 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnql 6639 A lower cut is closed downwards under the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  ( C  <Q  B  ->  C  e.  L ) )
 
Theoremprcunqu 6640 An upper cut is closed upwards under the positive fractions. (Contributed by Jim Kingdon, 25-Nov-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  C  e.  U )  ->  ( C  <Q  B  ->  B  e.  U ) )
 
Theoremprubl 6641 A positive fraction not in a lower cut is an upper bound. (Contributed by Jim Kingdon, 29-Sep-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  B  e.  L ) 
 /\  C  e.  Q. )  ->  ( -.  C  e.  L  ->  B  <Q  C ) )
 
Theoremprltlu 6642 An element of a lower cut is less than an element of the corresponding upper cut. (Contributed by Jim Kingdon, 15-Oct-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L  /\  C  e.  U )  ->  B  <Q  C )
 
Theoremprnmaxl 6643* A lower cut has no largest member. (Contributed by Jim Kingdon, 29-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  E. x  e.  L  B  <Q  x )
 
Theoremprnminu 6644* An upper cut has no smallest member. (Contributed by Jim Kingdon, 7-Nov-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  U )  ->  E. x  e.  U  x  <Q  B )
 
Theoremprnmaddl 6645* A lower cut has no largest member. Addition version. (Contributed by Jim Kingdon, 29-Sep-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  B  e.  L )  ->  E. x  e.  Q.  ( B  +Q  x )  e.  L )
 
Theoremprloc 6646 A Dedekind cut is located. (Contributed by Jim Kingdon, 23-Oct-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  A  <Q  B )  ->  ( A  e.  L  \/  B  e.  U ) )
 
Theoremprdisj 6647 A Dedekind cut is disjoint. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  A  e.  Q. )  ->  -.  ( A  e.  L  /\  A  e.  U ) )
 
Theoremprarloclemlt 6648 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  y  e. 
 om )  ->  ( A  +Q  ( [ <. ( y  +o  1o ) ,  1o >. ]  ~Q  .Q  P ) )  <Q  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) ) )
 
Theoremprarloclemlo 6649* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  y  e. 
 om )  ->  (
 ( A  +Q  ( [ <. ( y  +o  1o ) ,  1o >. ] 
 ~Q  .Q  P )
 )  e.  L  ->  ( ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  suc 
 X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) 
 ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) ) ) )
 
Theoremprarloclemup 6650 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 10-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  y  e. 
 om )  ->  (
 ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U  ->  ( ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  suc 
 X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) 
 ->  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) ) ) )
 
Theoremprarloclem3step 6651* Induction step for prarloclem3 6652. (Contributed by Jim Kingdon, 9-Nov-2019.)
 |-  ( ( ( X  e.  om  /\  ( <. L ,  U >.  e. 
 P.  /\  A  e.  L  /\  P  e.  Q. ) )  /\  E. y  e.  om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  suc  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )  ->  E. y  e.  om  (
 ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
 
Theoremprarloclem3 6652* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 27-Oct-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L ) 
 /\  ( X  e.  om 
 /\  P  e.  Q. )  /\  E. y  e. 
 om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  X ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )  ->  E. j  e.  om  (
 ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ] 
 ~Q  .Q  P )
 )  e.  U ) )
 
Theoremprarloclem4 6653* A slight rearrangement of prarloclem3 6652. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 4-Nov-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L ) 
 /\  P  e.  Q. )  ->  ( E. x  e.  om  E. y  e. 
 om  ( ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U )  ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) ) )
 
Theoremprarloclemn 6654* Subtracting two from a positive integer. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( ( N  e.  N. 
 /\  1o  <N  N ) 
 ->  E. x  e.  om  ( 2o  +o  x )  =  N )
 
Theoremprarloclem5 6655* A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 6658. (Contributed by Jim Kingdon, 4-Nov-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L ) 
 /\  ( N  e.  N. 
 /\  P  e.  Q.  /\ 
 1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) 
 ->  E. x  e.  om  E. y  e.  om  (
 ( A +Q0  ( [ <. y ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( ( y  +o  2o )  +o  x ) ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) )
 
Theoremprarloclem 6656* A special case of Lemma 6.16 from [BauerTaylor], p. 32. Given evenly spaced rational numbers from 
A to  A  +Q  ( N  .Q  P ) (which are in the lower and upper cuts, respectively, of a real number), there are a pair of numbers, two positions apart in the even spacing, which straddle the cut. (Contributed by Jim Kingdon, 22-Oct-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  A  e.  L ) 
 /\  ( N  e.  N. 
 /\  P  e.  Q.  /\ 
 1o  <N  N )  /\  ( A  +Q  ( [ <. N ,  1o >. ]  ~Q  .Q  P ) )  e.  U ) 
 ->  E. j  e.  om  ( ( A +Q0  ( [ <. j ,  1o >. ] ~Q0 ·Q0  P ) )  e.  L  /\  ( A  +Q  ( [ <. ( j  +o  2o ) ,  1o >. ] 
 ~Q  .Q  P )
 )  e.  U ) )
 
Theoremprarloclemcalc 6657 Some calculations for prarloc 6658. (Contributed by Jim Kingdon, 26-Oct-2019.)
 |-  ( ( ( A  =  ( X +Q0  ( [ <. M ,  1o >. ] ~Q0 ·Q0  Q ) )  /\  B  =  ( X  +Q  ( [ <. ( M  +o  2o ) ,  1o >. ] 
 ~Q  .Q  Q )
 ) )  /\  (
 ( Q  e.  Q.  /\  ( Q  +Q  Q )  <Q  P )  /\  ( X  e.  Q.  /\  M  e.  om )
 ) )  ->  B  <Q  ( A  +Q  P ) )
 
Theoremprarloc 6658* A Dedekind cut is arithmetically located. Part of Proposition 11.15 of [BauerTaylor], p. 52, slightly modified. It states that given a tolerance  P, there are elements of the lower and upper cut which are within that tolerance of each other.

Usually, proofs will be shorter if they use prarloc2 6659 instead. (Contributed by Jim Kingdon, 22-Oct-2019.)

 |-  ( ( <. L ,  U >.  e.  P.  /\  P  e.  Q. )  ->  E. a  e.  L  E. b  e.  U  b  <Q  ( a  +Q  P ) )
 
Theoremprarloc2 6659* A Dedekind cut is arithmetically located. This is a variation of prarloc 6658 which only constructs one (named) point and is therefore often easier to work with. It states that given a tolerance  P, there are elements of the lower and upper cut which are exactly that tolerance from each other. (Contributed by Jim Kingdon, 26-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  P  e.  Q. )  ->  E. a  e.  L  ( a  +Q  P )  e.  U )
 
Theoremltrelpr 6660 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
 |- 
 <P  C_  ( P.  X.  P. )
 
Theoremltdfpr 6661* More convenient form of df-iltp 6625. (Contributed by Jim Kingdon, 15-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  <P  B  <->  E. q  e.  Q.  ( q  e.  ( 2nd `  A )  /\  q  e.  ( 1st `  B ) ) ) )
 
Theoremgenpdflem 6662* Simplification of upper or lower cut expression. Lemma for genpdf 6663. (Contributed by Jim Kingdon, 30-Sep-2019.)
 |-  ( ( ph  /\  r  e.  A )  ->  r  e.  Q. )   &    |-  ( ( ph  /\  s  e.  B ) 
 ->  s  e.  Q. )   =>    |-  ( ph  ->  { q  e.  Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  A  /\  s  e.  B  /\  q  =  ( r G s ) ) }  =  { q  e.  Q.  |  E. r  e.  A  E. s  e.  B  q  =  ( r G s ) }
 )
 
Theoremgenpdf 6663* Simplified definition of addition or multiplication on positive reals. (Contributed by Jim Kingdon, 30-Sep-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e. 
 Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 1st `  w )  /\  s  e.  ( 1st `  v )  /\  q  =  ( r G s ) ) } ,  { q  e.  Q.  |  E. r  e.  Q.  E. s  e. 
 Q.  ( r  e.  ( 2nd `  w )  /\  s  e.  ( 2nd `  v )  /\  q  =  ( r G s ) ) } >. )   =>    |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { q  e. 
 Q.  |  E. r  e.  ( 1st `  w ) E. s  e.  ( 1st `  v ) q  =  ( r G s ) } ,  { q  e.  Q.  |  E. r  e.  ( 2nd `  w ) E. s  e.  ( 2nd `  v ) q  =  ( r G s ) } >. )
 
Theoremgenipv 6664* Value of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingon, 3-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  = 
 <. { q  e.  Q.  |  E. r  e.  ( 1st `  A ) E. s  e.  ( 1st `  B ) q  =  ( r G s ) } ,  {
 q  e.  Q.  |  E. r  e.  ( 2nd `  A ) E. s  e.  ( 2nd `  B ) q  =  ( r G s ) } >. )
 
Theoremgenplt2i 6665* Operating on both sides of two inequalities, when the operation is consistent with  <Q. (Contributed by Jim Kingdon, 6-Oct-2019.)
 |-  ( ( x  e. 
 Q.  /\  y  e.  Q. 
 /\  z  e.  Q. )  ->  ( x  <Q  y  <-> 
 ( z G x )  <Q  ( z G y ) ) )   &    |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ( A  <Q  B 
 /\  C  <Q  D ) 
 ->  ( A G C )  <Q  ( B G D ) )
 
Theoremgenpelxp 6666* Set containing the result of adding or multiplying positive reals. (Contributed by Jim Kingdon, 5-Dec-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A F B )  e.  ( ~P Q.  X.  ~P Q. ) )
 
Theoremgenpelvl 6667* Membership in lower cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 1st `  ( A F B ) )  <->  E. g  e.  ( 1st `  A ) E. h  e.  ( 1st `  B ) C  =  ( g G h ) ) )
 
Theoremgenpelvu 6668* Membership in upper cut of general operation (addition or multiplication) on positive reals. (Contributed by Jim Kingdon, 15-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( 2nd `  ( A F B ) )  <->  E. g  e.  ( 2nd `  A ) E. h  e.  ( 2nd `  B ) C  =  ( g G h ) ) )
 
Theoremgenpprecll 6669* Pre-closure law for general operation on lower cuts. (Contributed by Jim Kingdon, 2-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  (
 ( C  e.  ( 1st `  A )  /\  D  e.  ( 1st `  B ) )  ->  ( C G D )  e.  ( 1st `  ( A F B ) ) ) )
 
Theoremgenppreclu 6670* Pre-closure law for general operation on upper cuts. (Contributed by Jim Kingdon, 7-Nov-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  (
 ( C  e.  ( 2nd `  A )  /\  D  e.  ( 2nd `  B ) )  ->  ( C G D )  e.  ( 2nd `  ( A F B ) ) ) )
 
Theoremgenipdm 6671* Domain of general operation on positive reals. (Contributed by Jim Kingdon, 2-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  dom  F  =  ( P.  X.  P. )
 
Theoremgenpml 6672* The lower cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 1st `  ( A F B ) ) )
 
Theoremgenpmu 6673* The upper cut produced by addition or multiplication on positive reals is inhabited. (Contributed by Jim Kingdon, 5-Dec-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  E. q  e.  Q.  q  e.  ( 2nd `  ( A F B ) ) )
 
Theoremgenpcdl 6674* Downward closure of an operation on positive reals. (Contributed by Jim Kingdon, 14-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  (
 ( ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B ) ) )  /\  x  e.  Q. )  ->  ( x  <Q  ( g G h )  ->  x  e.  ( 1st `  ( A F B ) ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( f  e.  ( 1st `  ( A F B ) ) 
 ->  ( x  <Q  f  ->  x  e.  ( 1st `  ( A F B ) ) ) ) )
 
Theoremgenpcuu 6675* Upward closure of an operation on positive reals. (Contributed by Jim Kingdon, 8-Nov-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  (
 ( ( ( A  e.  P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B ) ) )  /\  x  e.  Q. )  ->  ( ( g G h )  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( f  e.  ( 2nd `  ( A F B ) ) 
 ->  ( f  <Q  x  ->  x  e.  ( 2nd `  ( A F B ) ) ) ) )
 
Theoremgenprndl 6676* The lower cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  (
 ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  y  <->  ( z G x )  <Q  ( z G y ) ) )   &    |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )   &    |-  ( ( ( ( A  e.  P.  /\  g  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 1st `  B ) ) ) 
 /\  x  e.  Q. )  ->  ( x  <Q  ( g G h ) 
 ->  x  e.  ( 1st `  ( A F B ) ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A. q  e.  Q.  ( q  e.  ( 1st `  ( A F B ) )  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  ( 1st `  ( A F B ) ) ) ) )
 
Theoremgenprndu 6677* The upper cut produced by addition or multiplication on positive reals is rounded. (Contributed by Jim Kingdon, 7-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  (
 ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  y  <->  ( z G x )  <Q  ( z G y ) ) )   &    |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )   &    |-  ( ( ( ( A  e.  P.  /\  g  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  h  e.  ( 2nd `  B ) ) ) 
 /\  x  e.  Q. )  ->  ( ( g G h )  <Q  x 
 ->  x  e.  ( 2nd `  ( A F B ) ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A. r  e.  Q.  ( r  e.  ( 2nd `  ( A F B ) )  <->  E. q  e.  Q.  ( q  <Q  r  /\  q  e.  ( 2nd `  ( A F B ) ) ) ) )
 
Theoremgenpdisj 6678* The lower and upper cuts produced by addition or multiplication on positive reals are disjoint. (Contributed by Jim Kingdon, 15-Oct-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  (
 ( x  e.  Q.  /\  y  e.  Q.  /\  z  e.  Q. )  ->  ( x  <Q  y  <->  ( z G x )  <Q  ( z G y ) ) )   &    |-  ( ( x  e.  Q.  /\  y  e.  Q. )  ->  ( x G y )  =  ( y G x ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A. q  e.  Q.  -.  ( q  e.  ( 1st `  ( A F B ) )  /\  q  e.  ( 2nd `  ( A F B ) ) ) )
 
Theoremgenpassl 6679* Associativity of lower cuts. Lemma for genpassg 6681. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  dom  F  =  ( P.  X.  P. )   &    |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  (
 f F g )  e.  P. )   &    |-  (
 ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  ( ( f G g ) G h )  =  ( f G ( g G h ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( 1st `  (
 ( A F B ) F C ) )  =  ( 1st `  ( A F ( B F C ) ) ) )
 
Theoremgenpassu 6680* Associativity of upper cuts. Lemma for genpassg 6681. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  dom  F  =  ( P.  X.  P. )   &    |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  (
 f F g )  e.  P. )   &    |-  (
 ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  ( ( f G g ) G h )  =  ( f G ( g G h ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( 2nd `  (
 ( A F B ) F C ) )  =  ( 2nd `  ( A F ( B F C ) ) ) )
 
Theoremgenpassg 6681* Associativity of an operation on reals. (Contributed by Jim Kingdon, 11-Dec-2019.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  <. { x  e. 
 Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 1st `  w )  /\  z  e.  ( 1st `  v )  /\  x  =  ( y G z ) ) } ,  { x  e.  Q.  |  E. y  e.  Q.  E. z  e. 
 Q.  ( y  e.  ( 2nd `  w )  /\  z  e.  ( 2nd `  v )  /\  x  =  ( y G z ) ) } >. )   &    |-  ( ( y  e.  Q.  /\  z  e.  Q. )  ->  (
 y G z )  e.  Q. )   &    |-  dom  F  =  ( P.  X.  P. )   &    |-  ( ( f  e.  P.  /\  g  e.  P. )  ->  (
 f F g )  e.  P. )   &    |-  (
 ( f  e.  Q.  /\  g  e.  Q.  /\  h  e.  Q. )  ->  ( ( f G g ) G h )  =  ( f G ( g G h ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( ( A F B ) F C )  =  ( A F ( B F C ) ) )
 
Theoremaddnqprllem 6682 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  G  e.  L ) 
 /\  X  e.  Q. )  ->  ( X  <Q  S 
 ->  ( ( X  .Q  ( *Q `  S ) )  .Q  G )  e.  L ) )
 
Theoremaddnqprulem 6683 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 7-Dec-2019.)
 |-  ( ( ( <. L ,  U >.  e.  P.  /\  G  e.  U ) 
 /\  X  e.  Q. )  ->  ( S  <Q  X 
 ->  ( ( X  .Q  ( *Q `  S ) )  .Q  G )  e.  U ) )
 
Theoremaddnqprl 6684 Lemma to prove downward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
 |-  ( ( ( ( A  e.  P.  /\  G  e.  ( 1st `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 1st `  B ) ) ) 
 /\  X  e.  Q. )  ->  ( X  <Q  ( G  +Q  H ) 
 ->  X  e.  ( 1st `  ( A  +P.  B ) ) ) )
 
Theoremaddnqpru 6685 Lemma to prove upward closure in positive real addition. (Contributed by Jim Kingdon, 5-Dec-2019.)
 |-  ( ( ( ( A  e.  P.  /\  G  e.  ( 2nd `  A ) )  /\  ( B  e.  P.  /\  H  e.  ( 2nd `  B ) ) ) 
 /\  X  e.  Q. )  ->  ( ( G  +Q  H )  <Q  X 
 ->  X  e.  ( 2nd `  ( A  +P.  B ) ) ) )
 
Theoremaddlocprlemlt 6686 Lemma for addlocpr 6691. The  Q  <Q  ( D  +Q  E ) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
 |-  ( ph  ->  A  e.  P. )   &    |-  ( ph  ->  B  e.  P. )   &    |-  ( ph  ->  Q  <Q  R )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )   &    |-  ( ph  ->  D  e.  ( 1st `  A ) )   &    |-  ( ph  ->  U  e.  ( 2nd `  A )
 )   &    |-  ( ph  ->  U  <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  E  e.  ( 1st `  B ) )   &    |-  ( ph  ->  T  e.  ( 2nd `  B ) )   &    |-  ( ph  ->  T 
 <Q  ( E  +Q  P ) )   =>    |-  ( ph  ->  ( Q  <Q  ( D  +Q  E )  ->  Q  e.  ( 1st `  ( A  +P.  B ) ) ) )
 
Theoremaddlocprlemeqgt 6687 Lemma for addlocpr 6691. This is a step used in both the  Q  =  ( D  +Q  E ) and  ( D  +Q  E
)  <Q  Q cases. (Contributed by Jim Kingdon, 7-Dec-2019.)
 |-  ( ph  ->  A  e.  P. )   &    |-  ( ph  ->  B  e.  P. )   &    |-  ( ph  ->  Q  <Q  R )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )   &    |-  ( ph  ->  D  e.  ( 1st `  A ) )   &    |-  ( ph  ->  U  e.  ( 2nd `  A )
 )   &    |-  ( ph  ->  U  <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  E  e.  ( 1st `  B ) )   &    |-  ( ph  ->  T  e.  ( 2nd `  B ) )   &    |-  ( ph  ->  T 
 <Q  ( E  +Q  P ) )   =>    |-  ( ph  ->  ( U  +Q  T )  <Q  ( ( D  +Q  E )  +Q  ( P  +Q  P ) ) )
 
Theoremaddlocprlemeq 6688 Lemma for addlocpr 6691. The  Q  =  ( D  +Q  E ) case. (Contributed by Jim Kingdon, 6-Dec-2019.)
 |-  ( ph  ->  A  e.  P. )   &    |-  ( ph  ->  B  e.  P. )   &    |-  ( ph  ->  Q  <Q  R )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )   &    |-  ( ph  ->  D  e.  ( 1st `  A ) )   &    |-  ( ph  ->  U  e.  ( 2nd `  A )
 )   &    |-  ( ph  ->  U  <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  E  e.  ( 1st `  B ) )   &    |-  ( ph  ->  T  e.  ( 2nd `  B ) )   &    |-  ( ph  ->  T 
 <Q  ( E  +Q  P ) )   =>    |-  ( ph  ->  ( Q  =  ( D  +Q  E )  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
 
Theoremaddlocprlemgt 6689 Lemma for addlocpr 6691. The  ( D  +Q  E
)  <Q  Q case. (Contributed by Jim Kingdon, 6-Dec-2019.)
 |-  ( ph  ->  A  e.  P. )   &    |-  ( ph  ->  B  e.  P. )   &    |-  ( ph  ->  Q  <Q  R )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )   &    |-  ( ph  ->  D  e.  ( 1st `  A ) )   &    |-  ( ph  ->  U  e.  ( 2nd `  A )
 )   &    |-  ( ph  ->  U  <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  E  e.  ( 1st `  B ) )   &    |-  ( ph  ->  T  e.  ( 2nd `  B ) )   &    |-  ( ph  ->  T 
 <Q  ( E  +Q  P ) )   =>    |-  ( ph  ->  (
 ( D  +Q  E )  <Q  Q  ->  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
 
Theoremaddlocprlem 6690 Lemma for addlocpr 6691. The result, in deduction form. (Contributed by Jim Kingdon, 6-Dec-2019.)
 |-  ( ph  ->  A  e.  P. )   &    |-  ( ph  ->  B  e.  P. )   &    |-  ( ph  ->  Q  <Q  R )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  ( Q  +Q  ( P  +Q  P ) )  =  R )   &    |-  ( ph  ->  D  e.  ( 1st `  A ) )   &    |-  ( ph  ->  U  e.  ( 2nd `  A )
 )   &    |-  ( ph  ->  U  <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  E  e.  ( 1st `  B ) )   &    |-  ( ph  ->  T  e.  ( 2nd `  B ) )   &    |-  ( ph  ->  T 
 <Q  ( E  +Q  P ) )   =>    |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  +P.  B ) )  \/  R  e.  ( 2nd `  ( A  +P.  B ) ) ) )
 
Theoremaddlocpr 6691* Locatedness of addition on positive reals. Lemma 11.16 in [BauerTaylor], p. 53. The proof in BauerTaylor relies on signed rationals, so we replace it with another proof which applies prarloc 6658 to both  A and  B, and uses nqtri3or 6551 rather than prloc 6646 to decide whether  q is too big to be in the lower cut of  A  +P.  B (and deduce that if it is, then  r must be in the upper cut). What the two proofs have in common is that they take the difference between  q and  r to determine how tight a range they need around the real numbers. (Contributed by Jim Kingdon, 5-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  ( 1st `  ( A  +P.  B ) )  \/  r  e.  ( 2nd `  ( A  +P.  B ) ) ) ) )
 
Theoremaddclpr 6692 Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. Combination of Lemma 11.13 and Lemma 11.16 in [BauerTaylor], p. 53. (Contributed by NM, 13-Mar-1996.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  +P.  B )  e.  P. )
 
Theoremplpvlu 6693* Value of addition on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  +P.  B )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y  +Q  z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y  +Q  z
 ) } >. )
 
Theoremmpvlu 6694* Value of multiplication on positive reals. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  .P.  B )  =  <. { x  e.  Q.  |  E. y  e.  ( 1st `  A ) E. z  e.  ( 1st `  B ) x  =  ( y  .Q  z ) } ,  { x  e.  Q.  |  E. y  e.  ( 2nd `  A ) E. z  e.  ( 2nd `  B ) x  =  ( y  .Q  z
 ) } >. )
 
Theoremdmplp 6695 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
 |- 
 dom  +P.  =  ( P. 
 X.  P. )
 
Theoremdmmp 6696 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
 |- 
 dom  .P.  =  ( P. 
 X.  P. )
 
Theoremnqprm 6697* A cut produced from a rational is inhabited. Lemma for nqprlu 6702. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  ->  ( E. q  e. 
 Q.  q  e.  { x  |  x  <Q  A }  /\  E. r  e.  Q.  r  e.  { x  |  A  <Q  x } ) )
 
Theoremnqprrnd 6698* A cut produced from a rational is rounded. Lemma for nqprlu 6702. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  ->  ( A. q  e. 
 Q.  ( q  e. 
 { x  |  x  <Q  A }  <->  E. r  e.  Q.  ( q  <Q  r  /\  r  e.  { x  |  x  <Q  A }
 ) )  /\  A. r  e.  Q.  (
 r  e.  { x  |  A  <Q  x }  <->  E. q  e.  Q.  (
 q  <Q  r  /\  q  e.  { x  |  A  <Q  x } ) ) ) )
 
Theoremnqprdisj 6699* A cut produced from a rational is disjoint. Lemma for nqprlu 6702. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  ->  A. q  e.  Q.  -.  ( q  e.  { x  |  x  <Q  A }  /\  q  e. 
 { x  |  A  <Q  x } ) )
 
Theoremnqprloc 6700* A cut produced from a rational is located. Lemma for nqprlu 6702. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  ->  A. q  e.  Q.  A. r  e.  Q.  (
 q  <Q  r  ->  (
 q  e.  { x  |  x  <Q  A }  \/  r  e.  { x  |  A  <Q  x }
 ) ) )
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