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Theorem List for Intuitionistic Logic Explorer - 7501-7600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnpex 7501 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.)
 |- 
 P.  e.  _V
 
Theoremelinp 7502* 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 7503 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 7504* 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 7505* 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 7506* 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 7507 The lower cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  L  C_ 
 Q. )
 
Theoremprssnqu 7508 The upper cut of a positive real is a subset of the positive fractions. (Contributed by Jim Kingdon, 28-Sep-2019.)
 |-  ( <. L ,  U >.  e.  P.  ->  U  C_ 
 Q. )
 
Theoremelprnql 7509 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 7510 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 7511 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnql 7512 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 7513 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 7514 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 7515 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 7516* 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 7517* 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 7518* 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 7519 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 7520 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 7521 Two possible ways of contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (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 7522* Contracting the lower side of an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (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 7523 Contracting the upper side of an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (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 7524* Induction step for prarloclem3 7525. (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 7525* Contracting an interval which straddles a Dedekind cut. Lemma for prarloc 7531. (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 7526* A slight rearrangement of prarloclem3 7525. Lemma for prarloc 7531. (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 7527* Subtracting two from a positive integer. Lemma for prarloc 7531. (Contributed by Jim Kingdon, 5-Nov-2019.)
 |-  ( ( N  e.  N. 
 /\  1o  <N  N ) 
 ->  E. x  e.  om  ( 2o  +o  x )  =  N )
 
Theoremprarloclem5 7528* A substitution of zero for  y and  N minus two for  x. Lemma for prarloc 7531. (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 7529* 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 7530 Some calculations for prarloc 7531. (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 7531* 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 7532 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 7532* A Dedekind cut is arithmetically located. This is a variation of prarloc 7531 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 7533 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
 |- 
 <P  C_  ( P.  X.  P. )
 
Theoremltdfpr 7534* More convenient form of df-iltp 7498. (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 7535* Simplification of upper or lower cut expression. Lemma for genpdf 7536. (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 7536* 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 7537* 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 7538* 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 7539* 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 7540* 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 7541* 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 7542* 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 7543* 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 7544* 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 7545* 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 7546* 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 7547* 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 7548* 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 7549* 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 7550* 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 7551* 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 7552* Associativity of lower cuts. Lemma for genpassg 7554. (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 7553* Associativity of upper cuts. Lemma for genpassg 7554. (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 7554* 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 7555 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 7556 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 7557 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 7558 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 7559 Lemma for addlocpr 7564. 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 7560 Lemma for addlocpr 7564. 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 7561 Lemma for addlocpr 7564. 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 7562 Lemma for addlocpr 7564. 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 7563 Lemma for addlocpr 7564. 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 7564* 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 7531 to both  A and  B, and uses nqtri3or 7424 rather than prloc 7519 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 7565 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 7566* 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 7567* 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 7568 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.)
 |- 
 dom  +P.  =  ( P. 
 X.  P. )
 
Theoremdmmp 7569 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.)
 |- 
 dom  .P.  =  ( P. 
 X.  P. )
 
Theoremnqprm 7570* A cut produced from a rational is inhabited. Lemma for nqprlu 7575. (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 7571* A cut produced from a rational is rounded. Lemma for nqprlu 7575. (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 7572* A cut produced from a rational is disjoint. Lemma for nqprlu 7575. (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 7573* A cut produced from a rational is located. Lemma for nqprlu 7575. (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 }
 ) ) )
 
Theoremnqprxx 7574* The canonical embedding of the rationals into the reals, expressed with the same variable for the lower and upper cuts. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( A  e.  Q.  -> 
 <. { x  |  x  <Q  A } ,  { x  |  A  <Q  x } >.  e.  P. )
 
Theoremnqprlu 7575* The canonical embedding of the rationals into the reals. (Contributed by Jim Kingdon, 24-Jun-2020.)
 |-  ( A  e.  Q.  -> 
 <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  e.  P. )
 
Theoremrecnnpr 7576* The reciprocal of a positive integer, as a positive real. (Contributed by Jim Kingdon, 27-Feb-2021.)
 |-  ( A  e.  N.  -> 
 <. { l  |  l 
 <Q  ( *Q `  [ <. A ,  1o >. ]  ~Q  ) } ,  { u  |  ( *Q `  [ <. A ,  1o >. ]  ~Q  )  <Q  u } >.  e. 
 P. )
 
Theoremltnqex 7577 The class of rationals less than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |- 
 { x  |  x  <Q  A }  e.  _V
 
Theoremgtnqex 7578 The class of rationals greater than a given rational is a set. (Contributed by Jim Kingdon, 13-Dec-2019.)
 |- 
 { x  |  A  <Q  x }  e.  _V
 
Theoremnqprl 7579* Comparing a fraction to a real can be done by whether it is an element of the lower cut, or by 
<P. (Contributed by Jim Kingdon, 8-Jul-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  P. )  ->  ( A  e.  ( 1st `  B )  <->  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  <P  B ) )
 
Theoremnqpru 7580* Comparing a fraction to a real can be done by whether it is an element of the upper cut, or by 
<P. (Contributed by Jim Kingdon, 29-Nov-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  P. )  ->  ( A  e.  ( 2nd `  B )  <->  B 
 <P  <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >. ) )
 
Theoremnnprlu 7581* The canonical embedding of positive integers into the positive reals. (Contributed by Jim Kingdon, 23-Apr-2020.)
 |-  ( A  e.  N.  -> 
 <. { l  |  l 
 <Q  [ <. A ,  1o >. ]  ~Q  } ,  { u  |  [ <. A ,  1o >. ]  ~Q  <Q  u } >.  e.  P. )
 
Theorem1pr 7582 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.)
 |- 
 1P  e.  P.
 
Theorem1prl 7583 The lower cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  ( 1st `  1P )  =  { x  |  x  <Q  1Q }
 
Theorem1pru 7584 The upper cut of the positive real number 'one'. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  ( 2nd `  1P )  =  { x  |  1Q  <Q  x }
 
Theoremaddnqprlemrl 7585* Lemma for addnqpr 7589. The reverse subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 1st `  ( <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) 
 C_  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B ) 
 <Q  u } >. ) )
 
Theoremaddnqprlemru 7586* Lemma for addnqpr 7589. The reverse subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 2nd `  ( <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) 
 C_  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B ) 
 <Q  u } >. ) )
 
Theoremaddnqprlemfl 7587* Lemma for addnqpr 7589. The forward subset relationship for the lower cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 1st `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B ) 
 <Q  u } >. )  C_  ( 1st `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
 
Theoremaddnqprlemfu 7588* Lemma for addnqpr 7589. The forward subset relationship for the upper cut. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  ( 2nd `  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B ) 
 <Q  u } >. )  C_  ( 2nd `  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) ) )
 
Theoremaddnqpr 7589* Addition of fractions embedded into positive reals. One can either add the fractions as fractions, or embed them into positive reals and add them as positive reals, and get the same result. (Contributed by Jim Kingdon, 19-Aug-2020.)
 |-  ( ( A  e.  Q. 
 /\  B  e.  Q. )  ->  <. { l  |  l  <Q  ( A  +Q  B ) } ,  { u  |  ( A  +Q  B )  <Q  u } >.  =  ( <. { l  |  l 
 <Q  A } ,  { u  |  A  <Q  u } >.  +P.  <. { l  |  l  <Q  B } ,  { u  |  B  <Q  u } >. ) )
 
Theoremaddnqpr1 7590* Addition of one to a fraction embedded into a positive real. One can either add the fraction one to the fraction, or the positive real one to the positive real, and get the same result. Special case of addnqpr 7589. (Contributed by Jim Kingdon, 26-Apr-2020.)
 |-  ( A  e.  Q.  -> 
 <. { l  |  l 
 <Q  ( A  +Q  1Q ) } ,  { u  |  ( A  +Q  1Q )  <Q  u } >.  =  ( <. { l  |  l  <Q  A } ,  { u  |  A  <Q  u } >.  +P.  1P ) )
 
Theoremappdivnq 7591* Approximate division for positive rationals. Proposition 12.7 of [BauerTaylor], p. 55 (a special case where  A and  B are positive, as well as  C). Our proof is simpler than the one in BauerTaylor because we have reciprocals. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( A  <Q  B 
 /\  C  e.  Q. )  ->  E. m  e.  Q.  ( A  <Q  ( m  .Q  C )  /\  ( m  .Q  C ) 
 <Q  B ) )
 
Theoremappdiv0nq 7592* Approximate division for positive rationals. This can be thought of as a variation of appdivnq 7591 in which  A is zero, although it can be stated and proved in terms of positive rationals alone, without zero as such. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ( B  e.  Q. 
 /\  C  e.  Q. )  ->  E. m  e.  Q.  ( m  .Q  C ) 
 <Q  B )
 
Theoremprmuloclemcalc 7593 Calculations for prmuloc 7594. (Contributed by Jim Kingdon, 9-Dec-2019.)
 |-  ( ph  ->  R  <Q  U )   &    |-  ( ph  ->  U 
 <Q  ( D  +Q  P ) )   &    |-  ( ph  ->  ( A  +Q  X )  =  B )   &    |-  ( ph  ->  ( P  .Q  B )  <Q  ( R  .Q  X ) )   &    |-  ( ph  ->  A  e.  Q. )   &    |-  ( ph  ->  B  e.  Q. )   &    |-  ( ph  ->  D  e.  Q. )   &    |-  ( ph  ->  P  e.  Q. )   &    |-  ( ph  ->  X  e.  Q. )   =>    |-  ( ph  ->  ( U  .Q  A ) 
 <Q  ( D  .Q  B ) )
 
Theoremprmuloc 7594* Positive reals are multiplicatively located. Lemma 12.8 of [BauerTaylor], p. 56. (Contributed by Jim Kingdon, 8-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  A  <Q  B )  ->  E. d  e.  Q.  E. u  e.  Q.  (
 d  e.  L  /\  u  e.  U  /\  ( u  .Q  A ) 
 <Q  ( d  .Q  B ) ) )
 
Theoremprmuloc2 7595* Positive reals are multiplicatively located. This is a variation of prmuloc 7594 which only constructs one (named) point and is therefore often easier to work with. It states that given a ratio  B, there are elements of the lower and upper cut which have exactly that ratio between them. (Contributed by Jim Kingdon, 28-Dec-2019.)
 |-  ( ( <. L ,  U >.  e.  P.  /\  1Q  <Q  B )  ->  E. x  e.  L  ( x  .Q  B )  e.  U )
 
Theoremmulnqprl 7596 Lemma to prove downward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-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 ) ) ) )
 
Theoremmulnqpru 7597 Lemma to prove upward closure in positive real multiplication. (Contributed by Jim Kingdon, 10-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 ) ) ) )
 
Theoremmullocprlem 7598 Calculations for mullocpr 7599. (Contributed by Jim Kingdon, 10-Dec-2019.)
 |-  ( ph  ->  ( A  e.  P.  /\  B  e.  P. ) )   &    |-  ( ph  ->  ( U  .Q  Q )  <Q  ( E  .Q  ( D  .Q  U ) ) )   &    |-  ( ph  ->  ( E  .Q  ( D  .Q  U ) )  <Q  ( T  .Q  ( D  .Q  U ) ) )   &    |-  ( ph  ->  ( T  .Q  ( D  .Q  U ) )  <Q  ( D  .Q  R ) )   &    |-  ( ph  ->  ( Q  e.  Q.  /\  R  e.  Q. ) )   &    |-  ( ph  ->  ( D  e.  Q.  /\  U  e.  Q. )
 )   &    |-  ( ph  ->  ( D  e.  ( 1st `  A )  /\  U  e.  ( 2nd `  A ) ) )   &    |-  ( ph  ->  ( E  e.  Q. 
 /\  T  e.  Q. ) )   =>    |-  ( ph  ->  ( Q  e.  ( 1st `  ( A  .P.  B ) )  \/  R  e.  ( 2nd `  ( A  .P.  B ) ) ) )
 
Theoremmullocpr 7599* Locatedness of multiplication on positive reals. Lemma 12.9 in [BauerTaylor], p. 56 (but where both  A and  B are positive, not just  A). (Contributed by Jim Kingdon, 8-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 ) ) ) ) )
 
Theoremmulclpr 7600 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  .P.  B )  e.  P. )
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