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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnsmallnq 9401* The is no smallest positive fraction. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x  x  <Q  A )
 
Theoremltbtwnnq 9402* There exists a number between any two positive fractions. Proposition 9-2.6(i) of [Gleason] p. 120. (Contributed by NM, 17-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  <Q  B  <->  E. x ( A 
 <Q  x  /\  x  <Q  B ) )
 
Theoremltrnq 9403 Ordering property of reciprocal for positive fractions. Proposition 9-2.6(iv) of [Gleason] p. 120. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  <Q  B  <->  ( *Q `  B )  <Q  ( *Q `  A ) )
 
Theoremarchnq 9404* For any fraction, there is an integer that is greater than it. This is also known as the "archimedean property". (Contributed by Mario Carneiro, 10-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  E. x  e.  N.  A  <Q  <. x ,  1o >.
 )
 
Definitiondf-np 9405* 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. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of [Gleason] p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c 9544, and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  =  { x  |  ( ( (/)  C.  x  /\  x  C.  Q. )  /\  A. y  e.  x  ( A. z ( z 
 <Q  y  ->  z  e.  x )  /\  E. z  e.  x  y  <Q  z ) ) }
 
Definitiondf-1p 9406 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9544, and is intended to be used only by the construction. Definition of [Gleason] p. 122. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |- 
 1P  =  { x  |  x  <Q  1Q }
 
Definitiondf-plp 9407* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, and is intended to be used only by the construction. From Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 +P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  +Q  u ) } )
 
Definitiondf-mp 9408* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, and is intended to be used only by the construction. From Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 .P.  =  ( x  e.  P. ,  y  e. 
 P.  |->  { w  |  E. v  e.  x  E. u  e.  y  w  =  ( v  .Q  u ) } )
 
Definitiondf-ltp 9409* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, and is intended to be used only by the construction. From Proposition 9-3.2 of [Gleason] p. 122. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
 |- 
 <P  =  { <. x ,  y >.  |  ( ( x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
 
Theoremnpex 9410 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  e.  _V
 
Theoremelnp 9411* Membership in positive reals. (Contributed by NM, 16-Feb-1996.) (New usage is discouraged.)
 |-  ( A  e.  P.  <->  (
 ( (/)  C.  A  /\  A  C.  Q. )  /\  A. x  e.  A  (
 A. y ( y 
 <Q  x  ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
 
Theoremelnpi 9412* Membership in positive reals. (Contributed by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  <->  (
 ( A  e.  _V  /\  (/)  C.  A  /\  A  C. 
 Q. )  /\  A. x  e.  A  ( A. y ( y  <Q  x 
 ->  y  e.  A )  /\  E. y  e.  A  x  <Q  y ) ) )
 
Theoremprn0 9413 A positive real is not empty. (Contributed by NM, 15-May-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  A  =/=  (/) )
 
Theoremprpssnq 9414 A positive real is a subset of the positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  A  C.  Q. )
 
Theoremelprnq 9415 A positive real is a set of positive fractions. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  B  e.  Q. )
 
Theorem0npr 9416 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnq 9417 A positive real is closed downwards under the positive fractions. Definition 9-3.1 (ii) of [Gleason] p. 121. (Contributed by NM, 25-Feb-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  ( C  <Q  B 
 ->  C  e.  A ) )
 
Theoremprub 9418 A positive fraction not in a positive real is an upper bound. Remark (1) of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  A )  /\  C  e.  Q. )  ->  ( -.  C  e.  A  ->  B 
 <Q  C ) )
 
Theoremprnmax 9419* A positive real has no largest member. Definition 9-3.1(iii) of [Gleason] p. 121. (Contributed by NM, 9-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  E. x  e.  A  B  <Q  x )
 
Theoremnpomex 9420 A simplifying observation, and an indication of why any attempt to develop a theory of the real numbers without the Axiom of Infinity is doomed to failure: since every member of  P. is an infinite set, the negation of Infinity implies that  P., and hence 
RR, is empty. (Note that this proof, which used the fact that Dedekind cuts have no maximum, could just as well have used that they have no minimum, since they are downward-closed by prcdnq 9417 and nsmallnq 9401). (Contributed by Mario Carneiro, 11-May-2013.) (Revised by Mario Carneiro, 16-Nov-2014.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  om  e.  _V )
 
Theoremprnmadd 9421* A positive real has no largest member. Addition version. (Contributed by NM, 7-Apr-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  A )  ->  E. x ( B  +Q  x )  e.  A )
 
Theoremltrelpr 9422 Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
 |- 
 <P  C_  ( P.  X.  P. )
 
Theoremgenpv 9423* Value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  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 )  =  { f  |  E. g  e.  A  E. h  e.  B  f  =  ( g G h ) } )
 
Theoremgenpelv 9424* Membership in value of general operation (addition or multiplication) on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  ->  ( C  e.  ( A F B )  <->  E. g  e.  A  E. h  e.  B  C  =  ( g G h ) ) )
 
Theoremgenpprecl 9425* Pre-closure law for general operation on positive reals. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  ->  ( ( C  e.  A  /\  D  e.  B )  ->  ( C G D )  e.  ( A F B ) ) )
 
Theoremgenpdm 9426* Domain of general operation on positive reals. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  dom  F  =  ( P.  X.  P. )
 
Theoremgenpn0 9427* The result of an operation on positive reals is not empty. (Contributed by NM, 28-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   =>    |-  (
 ( A  e.  P.  /\  B  e.  P. )  -> 
 (/)  C.  ( A F B ) )
 
Theoremgenpss 9428* The result of an operation on positive reals is a subset of the positive fractions. (Contributed by NM, 18-Nov-1995.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  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 )  C_  Q. )
 
Theoremgenpnnp 9429* The result of an operation on positive reals is different from the set of positive fractions. (Contributed by NM, 29-Feb-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  (
 z  e.  Q.  ->  ( x  <Q  y  <->  ( z G x )  <Q  ( z G y ) ) )   &    |-  ( x G y )  =  ( y G x )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  -.  ( A F B )  =  Q. )
 
Theoremgenpcd 9430* Downward closure of an operation on positive reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  (
 ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e.  Q. )  ->  ( x  <Q  ( g G h )  ->  x  e.  ( A F B ) ) )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  ( x  <Q  f 
 ->  x  e.  ( A F B ) ) ) )
 
Theoremgenpnmax 9431* An operation on positive reals has no largest member. (Contributed by NM, 10-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  (
 v  e.  Q.  ->  ( z  <Q  w  <->  ( v G z )  <Q  ( v G w ) ) )   &    |-  ( z G w )  =  ( w G z )   =>    |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( f  e.  ( A F B )  ->  E. x  e.  ( A F B ) f 
 <Q  x ) )
 
Theoremgenpcl 9432* Closure of an operation on reals. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 17-Nov-2014.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  v  x  =  ( y G z ) } )   &    |-  (
 ( y  e.  Q.  /\  z  e.  Q. )  ->  ( y G z )  e.  Q. )   &    |-  ( h  e.  Q.  ->  ( f  <Q  g  <->  ( h G f )  <Q  ( h G g ) ) )   &    |-  ( x G y )  =  ( y G x )   &    |-  ( ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e. 
 Q. )  ->  ( x  <Q  ( g G h )  ->  x  e.  ( A F B ) ) )   =>    |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A F B )  e. 
 P. )
 
Theoremgenpass 9433* Associativity of an operation on reals. (Contributed by NM, 18-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  F  =  ( w  e.  P. ,  v  e.  P.  |->  { x  |  E. y  e.  w  E. z  e.  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 G g ) G h )  =  ( f G ( g G h ) )   =>    |-  ( ( A F B ) F C )  =  ( A F ( B F C ) )
 
Theoremplpv 9434* Value of addition on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  +P.  B )  =  { x  |  E. y  e.  A  E. z  e.  B  x  =  ( y  +Q  z ) } )
 
Theoremmpv 9435* Value of multiplication on positive reals. (Contributed by NM, 28-Feb-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  .P.  B )  =  { x  |  E. y  e.  A  E. z  e.  B  x  =  ( y  .Q  z ) } )
 
Theoremdmplp 9436 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  +P.  =  ( P. 
 X.  P. )
 
Theoremdmmp 9437 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  .P.  =  ( P. 
 X.  P. )
 
Theoremnqpr 9438* The canonical embedding of the rationals into the reals. (Contributed by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( A  e.  Q.  ->  { x  |  x  <Q  A }  e.  P. )
 
Theorem1pr 9439 The positive real number 'one'. (Contributed by NM, 13-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |- 
 1P  e.  P.
 
Theoremaddclprlem1 9440 Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  g  e.  A )  /\  x  e.  Q. )  ->  ( x  <Q  ( g  +Q  h )  ->  ( ( x  .Q  ( *Q `  ( g  +Q  h ) ) )  .Q  g )  e.  A ) )
 
Theoremaddclprlem2 9441* Lemma to prove downward closure in positive real addition. Part of proof of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |-  ( ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e. 
 Q. )  ->  ( x  <Q  ( g  +Q  h )  ->  x  e.  ( A  +P.  B ) ) )
 
Theoremaddclpr 9442 Closure of addition on positive reals. First statement of Proposition 9-3.5 of [Gleason] p. 123. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  +P.  B )  e.  P. )
 
Theoremmulclprlem 9443* Lemma to prove downward closure in positive real multiplication. Part of proof of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 14-Mar-1996.) (New usage is discouraged.)
 |-  ( ( ( ( A  e.  P.  /\  g  e.  A )  /\  ( B  e.  P.  /\  h  e.  B ) )  /\  x  e. 
 Q. )  ->  ( x  <Q  ( g  .Q  h )  ->  x  e.  ( A  .P.  B ) ) )
 
Theoremmulclpr 9444 Closure of multiplication on positive reals. First statement of Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 13-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  .P.  B )  e.  P. )
 
Theoremaddcompr 9445 Addition of positive reals is commutative. Proposition 9-3.5(ii) of [Gleason] p. 123. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
 |-  ( A  +P.  B )  =  ( B  +P.  A )
 
Theoremaddasspr 9446 Addition of positive reals is associative. Proposition 9-3.5(i) of [Gleason] p. 123. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  +P.  B )  +P.  C )  =  ( A  +P.  ( B  +P.  C ) )
 
Theoremmulcompr 9447 Multiplication of positive reals is commutative. Proposition 9-3.7(ii) of [Gleason] p. 124. (Contributed by NM, 19-Nov-1995.) (New usage is discouraged.)
 |-  ( A  .P.  B )  =  ( B  .P.  A )
 
Theoremmulasspr 9448 Multiplication of positive reals is associative. Proposition 9-3.7(i) of [Gleason] p. 124. (Contributed by NM, 18-Mar-1996.) (New usage is discouraged.)
 |-  ( ( A  .P.  B )  .P.  C )  =  ( A  .P.  ( B  .P.  C ) )
 
Theoremdistrlem1pr 9449 Lemma for distributive law for positive reals. (Contributed by NM, 1-May-1996.) (Revised by Mario Carneiro, 13-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( A  .P.  ( B  +P.  C ) ) 
 C_  ( ( A 
 .P.  B )  +P.  ( A  .P.  C ) ) )
 
Theoremdistrlem4pr 9450* Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P.  /\  C  e.  P. )  /\  ( ( x  e.  A  /\  y  e.  B )  /\  ( f  e.  A  /\  z  e.  C ) ) )  ->  ( ( x  .Q  y )  +Q  (
 f  .Q  z )
 )  e.  ( A 
 .P.  ( B  +P.  C ) ) )
 
Theoremdistrlem5pr 9451 Lemma for distributive law for positive reals. (Contributed by NM, 2-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P.  /\  C  e.  P. )  ->  ( ( A  .P.  B )  +P.  ( A 
 .P.  C ) )  C_  ( A  .P.  ( B 
 +P.  C ) ) )
 
Theoremdistrpr 9452 Multiplication of positive reals is distributive. Proposition 9-3.7(iii) of [Gleason] p. 124. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  .P.  ( B  +P.  C ) )  =  ( ( A 
 .P.  B )  +P.  ( A  .P.  C ) )
 
Theorem1idpr 9453 1 is an identity element for positive real multiplication. Theorem 9-3.7(iv) of [Gleason] p. 124. (Contributed by NM, 2-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  ( A  .P.  1P )  =  A )
 
Theoremltprord 9454 Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B ) )
 
Theorempsslinpr 9455 Proper subset is a linear ordering on positive reals. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( A  C.  B  \/  A  =  B  \/  B  C.  A )
 )
 
Theoremltsopr 9456 Positive real 'less than' is a strict ordering. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 25-Feb-1996.) (New usage is discouraged.)
 |- 
 <P  Or  P.
 
Theoremprlem934 9457* Lemma 9-3.4 of [Gleason] p. 122. (Contributed by NM, 25-Mar-1996.) (Revised by Mario Carneiro, 11-May-2013.) (New usage is discouraged.)
 |-  B  e.  _V   =>    |-  ( A  e.  P. 
 ->  E. x  e.  A  -.  ( x  +Q  B )  e.  A )
 
Theoremltaddpr 9458 The sum of two positive reals is greater than one of them. Proposition 9-3.5(iii) of [Gleason] p. 123. (Contributed by NM, 26-Mar-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  A  <P  ( A 
 +P.  B ) )
 
Theoremltaddpr2 9459 The sum of two positive reals is greater than one of them. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( C  e.  P.  ->  ( ( A  +P.  B )  =  C  ->  A 
 <P  C ) )
 
Theoremltexprlem1 9460* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( B  e.  P.  ->  ( A  C.  B  ->  C  =/=  (/) ) )
 
Theoremltexprlem2 9461* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 3-Apr-1996.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( B  e.  P.  ->  C  C.  Q. )
 
Theoremltexprlem3 9462* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( B  e.  P.  ->  ( x  e.  C  ->  A. z ( z 
 <Q  x  ->  z  e.  C ) ) )
 
Theoremltexprlem4 9463* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( B  e.  P.  ->  ( x  e.  C  ->  E. z ( z  e.  C  /\  x  <Q  z ) ) )
 
Theoremltexprlem5 9464* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 6-Apr-1996.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( ( B  e.  P. 
 /\  A  C.  B ) 
 ->  C  e.  P. )
 
Theoremltexprlem6 9465* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  ( A  +P.  C )  C_  B )
 
Theoremltexprlem7 9466* Lemma for Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  C  =  { x  |  E. y ( -.  y  e.  A  /\  ( y  +Q  x )  e.  B ) }   =>    |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B )  ->  B  C_  ( A  +P.  C ) )
 
Theoremltexpri 9467* Proposition 9-3.5(iv) of [Gleason] p. 123. (Contributed by NM, 13-May-1996.) (Revised by Mario Carneiro, 14-Jun-2013.) (New usage is discouraged.)
 |-  ( A  <P  B  ->  E. x  e.  P.  ( A  +P.  x )  =  B )
 
Theoremltaprlem 9468 Lemma for Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
 |-  ( C  e.  P.  ->  ( A  <P  B  ->  ( C  +P.  A ) 
 <P  ( C  +P.  B ) ) )
 
Theoremltapr 9469 Ordering property of addition. Proposition 9-3.5(v) of [Gleason] p. 123. (Contributed by NM, 8-Apr-1996.) (New usage is discouraged.)
 |-  ( C  e.  P.  ->  ( A  <P  B  <->  ( C  +P.  A )  <P  ( C  +P.  B ) ) )
 
Theoremaddcanpr 9470 Addition cancellation law for positive reals. Proposition 9-3.5(vi) of [Gleason] p. 123. (Contributed by NM, 9-Apr-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  B  e.  P. )  ->  ( ( A 
 +P.  B )  =  ( A  +P.  C ) 
 ->  B  =  C ) )
 
Theoremprlem936 9471* Lemma 9-3.6 of [Gleason] p. 124. (Contributed by NM, 26-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  e.  P. 
 /\  1Q  <Q  B ) 
 ->  E. x  e.  A  -.  ( x  .Q  B )  e.  A )
 
Theoremreclem2pr 9472* Lemma for Proposition 9-3.7 of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  B  =  { x  |  E. y ( x 
 <Q  y  /\  -.  ( *Q `  y )  e.  A ) }   =>    |-  ( A  e.  P. 
 ->  B  e.  P. )
 
Theoremreclem3pr 9473* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  B  =  { x  |  E. y ( x 
 <Q  y  /\  -.  ( *Q `  y )  e.  A ) }   =>    |-  ( A  e.  P. 
 ->  1P  C_  ( A  .P.  B ) )
 
Theoremreclem4pr 9474* Lemma for Proposition 9-3.7(v) of [Gleason] p. 124. (Contributed by NM, 30-Apr-1996.) (New usage is discouraged.)
 |-  B  =  { x  |  E. y ( x 
 <Q  y  /\  -.  ( *Q `  y )  e.  A ) }   =>    |-  ( A  e.  P. 
 ->  ( A  .P.  B )  =  1P )
 
Theoremrecexpr 9475* 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.) (New usage is discouraged.)
 |-  ( A  e.  P.  ->  E. x  e.  P.  ( A  .P.  x )  =  1P )
 
Theoremsuplem1pr 9476* The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  P.  A. y  e.  A  y 
 <P  x )  ->  U. A  e.  P. )
 
Theoremsuplem2pr 9477* The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( A  C_  P.  ->  ( ( y  e.  A  ->  -.  U. A  <P  y )  /\  ( y 
 <P  U. A  ->  E. z  e.  A  y  <P  z ) ) )
 
Theoremsupexpr 9478* The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  P.  A. y  e.  A  y 
 <P  x )  ->  E. x  e.  P.  ( A. y  e.  A  -.  x  <P  y 
 /\  A. y  e.  P.  ( y  <P  x  ->  E. z  e.  A  y  <P  z ) ) )
 
Definitiondf-enr 9479* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 ~R  =  { <. x ,  y >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X. 
 P. ) )  /\  E. z E. w E. v E. u ( ( x  =  <. z ,  w >.  /\  y  = 
 <. v ,  u >. ) 
 /\  ( z  +P.  u )  =  ( w 
 +P.  v ) ) ) }
 
Definitiondf-nr 9480 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 R.  =  ( ( P.  X.  P. ) /.  ~R  )
 
Definitiondf-plr 9481* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 +R  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( w 
 +P.  u ) ,  ( v  +P.  f
 ) >. ]  ~R  )
 ) }
 
Definitiondf-mr 9482* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 .R  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u ) 
 +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  ( v 
 .P.  u ) )
 >. ]  ~R  ) ) }
 
Definitiondf-ltr 9483* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 <R  =  { <. x ,  y >.  |  ( ( x  e.  R.  /\  y  e.  R. )  /\  E. z E. w E. v E. u ( ( x  =  [ <. z ,  w >. ] 
 ~R  /\  y  =  [ <. v ,  u >. ]  ~R  )  /\  ( z  +P.  u ) 
 <P  ( w  +P.  v
 ) ) ) }
 
Definitiondf-0r 9484 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 0R  =  [ <. 1P ,  1P >. ]  ~R
 
Definitiondf-1r 9485 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9544, 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.) (New usage is discouraged.)
 |- 
 1R  =  [ <. ( 1P  +P.  1P ) ,  1P >. ]  ~R
 
Definitiondf-m1r 9486 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 9544, and is intended to be used only by the construction. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 -1R  =  [ <. 1P ,  ( 1P  +P.  1P ) >. ]  ~R
 
Theoremenrbreq 9487 Equivalence relation for signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( <. A ,  B >.  ~R  <. C ,  D >.  <-> 
 ( A  +P.  D )  =  ( B  +P.  C ) ) )
 
Theoremenrer 9488 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.) (New usage is discouraged.)
 |- 
 ~R  Er  ( P.  X. 
 P. )
 
Theoremenreceq 9489 Equivalence class equality of positive fractions in terms of positive integers. (Contributed by NM, 29-Nov-1995.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  =  [ <. C ,  D >. ] 
 ~R 
 <->  ( A  +P.  D )  =  ( B  +P.  C ) ) )
 
Theoremenrex 9490 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 9491 Signed real 'less than' is a relation on signed reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)
 |- 
 <R  C_  ( R.  X.  R. )
 
Theoremaddcmpblnr 9492 Lemma showing compatibility of addition. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |-  ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
 )  /\  ( ( F  e.  P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) ) 
 ->  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  <. ( A  +P.  F ) ,  ( B  +P.  G ) >.  ~R  <. ( C  +P.  R ) ,  ( D  +P.  S ) >. ) )
 
Theoremmulcmpblnrlem 9493 Lemma used in lemma showing compatibility of multiplication. (Contributed by NM, 4-Sep-1995.) (New usage is discouraged.)
 |-  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  ( ( D  .P.  F )  +P.  ( ( ( A  .P.  F )  +P.  ( B  .P.  G ) )  +P.  (
 ( C  .P.  S )  +P.  ( D  .P.  R ) ) ) )  =  ( ( D 
 .P.  F )  +P.  (
 ( ( A  .P.  G )  +P.  ( B 
 .P.  F ) )  +P.  ( ( C  .P.  R )  +P.  ( D 
 .P.  S ) ) ) ) )
 
Theoremmulcmpblnr 9494 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
 |-  ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
 )  /\  ( ( F  e.  P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) ) 
 ->  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  <. ( ( A  .P.  F )  +P.  ( B 
 .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B 
 .P.  F ) ) >.  ~R 
 <. ( ( C  .P.  R )  +P.  ( D 
 .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D 
 .P.  R ) ) >. ) )
 
Theoremprsrlem1 9495* Decomposing signed reals into positive reals. Lemma for addsrpr 9498 and mulsrpr 9499. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( ( A  e.  ( ( P. 
 X.  P. ) /.  ~R  )  /\  B  e.  (
 ( P.  X.  P. ) /.  ~R  ) ) 
 /\  ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ] 
 ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  ( (
 ( ( w  e. 
 P.  /\  v  e.  P. )  /\  ( s  e.  P.  /\  f  e.  P. ) )  /\  ( ( u  e. 
 P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. ) ) ) 
 /\  ( ( w 
 +P.  f )  =  ( v  +P.  s
 )  /\  ( u  +P.  h )  =  ( t  +P.  g ) ) ) )
 
Theoremaddsrmo 9496* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  E* z E. w E. v E. u E. t
 ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w 
 +P.  u ) ,  ( v  +P.  t
 ) >. ]  ~R  )
 )
 
Theoremmulsrmo 9497* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  E* z E. w E. v E. u E. t
 ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u ) 
 +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  ( v 
 .P.  u ) )
 >. ]  ~R  ) )
 
Theoremaddsrpr 9498 Addition of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  +R  [ <. C ,  D >. ] 
 ~R  )  =  [ <. ( A  +P.  C ) ,  ( B  +P.  D ) >. ]  ~R  )
 
Theoremmulsrpr 9499 Multiplication of signed reals in terms of positive reals. (Contributed by NM, 3-Sep-1995.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
 |-  ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. ) )  ->  ( [ <. A ,  B >. ]  ~R  .R  [ <. C ,  D >. ] 
 ~R  )  =  [ <. ( ( A  .P.  C )  +P.  ( B 
 .P.  D ) ) ,  ( ( A  .P.  D )  +P.  ( B 
 .P.  C ) ) >. ] 
 ~R  )
 
Theoremltsrpr 9500 Ordering of signed reals in terms of positive reals. (Contributed by NM, 20-Feb-1996.) (Revised by Mario Carneiro, 12-Aug-2015.) (New usage is discouraged.)
 |-  ( [ <. A ,  B >. ]  ~R  <R  [
 <. C ,  D >. ] 
 ~R 
 <->  ( A  +P.  D )  <P  ( B  +P.  C ) )
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