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Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremnsmallnq 8601* 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 8602* 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 8603 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 8604* 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 8605* 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 8743, 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 8606 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8607* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8608* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8609* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8610 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  e.  _V
 
Theoremelnp 8611* 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 8612* 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 8613 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 8614 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 8615 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 8616 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnq 8617 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 8618 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 8619* 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 8620 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 8617 and nsmallnq 8601). (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 8621* 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 8622 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 8623* 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 8624* 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 8625* 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 8626* 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 8627* 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 8628* 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 8629* 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 8630* 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 8631* 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 8632* 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 8633* 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 8634* 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 8635* 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 8636 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  +P.  =  ( P. 
 X.  P. )
 
Theoremdmmp 8637 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  .P.  =  ( P. 
 X.  P. )
 
Theoremnqpr 8638* 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 8639 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 8640 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 8641* 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 8642 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 8643* 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 8644 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 8645 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 8646 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 8647 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 8648 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 8649 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 8650* 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 8651 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 8652 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 8653 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 8654 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 8655 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 8656 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 8657* 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 8658 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 8659 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 8660* 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 8661* 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 8662* 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 8663* 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 8664* 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 8665* 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 8666* 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 8667* 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 8668 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 8669 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 8670 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 8671* 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 8672* 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 8673* 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 8674* 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 8675* 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 8676* The union of a non-empty, 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 8677* The union of a set of positive reals (if a positive real) is its supremum (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 8678* The union of a non-empty, 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-plpr 8679* Define pre-addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |- 
 +pR  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. ) ) 
 /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( w  +P.  u ) ,  ( v  +P.  f ) >. ) ) }
 
Definitiondf-mpr 8680* Define pre-multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, and is intended to be used only by the construction. From Proposition 9-4.1 of [Gleason] p. 126. (Contributed by NM, 3-Sep-1995.) (New usage is discouraged.)
 |- 
 .pR  =  { <. <. x ,  y >. ,  z >.  |  ( ( x  e.  ( P.  X.  P. )  /\  y  e.  ( P.  X.  P. ) ) 
 /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( ( w  .P.  u )  +P.  ( v 
 .P.  f ) ) ,  ( ( w 
 .P.  f )  +P.  ( v  .P.  u ) ) >. ) ) }
 
Definitiondf-enr 8681* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8682 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8683* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 ,  v >.  +pR  <. u ,  f >. ) ]  ~R  ) ) }
 
Definitiondf-mr 8684* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 ,  v >.  .pR  <. u ,  f >. ) ]  ~R  ) ) }
 
Definitiondf-ltr 8685* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8686 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8687 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8688 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 8743, 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 8689 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 8690 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 8691 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 8692 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 8693 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 8694 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 8695 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 8696 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 ) ) >. ) )
 
Theoremaddsrpr 8697 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 8698 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 8699 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 ) )
 
Theoremgt0srpr 8700 Greater then zero in terms of positive reals. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( 0R  <R  [ <. A ,  B >. ]  ~R  <->  B  <P  A )
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