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Theorem List for Metamath Proof Explorer - 8601-8700   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Definitiondf-1p 8601 Define the positive real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8602* Define addition on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8603* Define multiplication on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8604* Define ordering on positive reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8605 The class of positive reals is a set. (Contributed by NM, 31-Oct-1995.) (New usage is discouraged.)
 |- 
 P.  e.  _V
 
Theoremelnp 8606* 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 8607* 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 8608 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 8609 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 8610 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 8611 The empty set is not a positive real. (Contributed by NM, 15-Nov-1995.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  P.
 
Theoremprcdnq 8612 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 8613 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 8614* 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 8615 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 8612 and nsmallnq 8596). (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 8616* 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 8617 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 8618* 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 8619* 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 8620* 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 8621* 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 8622* 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 8623* 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 8624* 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 8625* 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 8626* 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 8627* 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 8628* 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 8629* 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 8630* 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 8631 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  +P.  =  ( P. 
 X.  P. )
 
Theoremdmmp 8632 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  .P.  =  ( P. 
 X.  P. )
 
Theoremnqpr 8633* 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 8634 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 8635 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 8636* 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 8637 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 8638* 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 8639 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 8640 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 8641 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 8642 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 8643 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 8644 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 8645* 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 8646 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 8647 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 8648 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 8649 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 8650 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 8651 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 8652* 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 8653 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 8654 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 8655* 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 8656* 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 8657* 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 8658* 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 8659* 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 8660* 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 8661* 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 8662* 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 8663 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 8664 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 8665 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 8666* 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 8667* 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 8668* 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 8669* 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 8670* 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 8671* 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 8672* 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 8673* 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 8674* Define pre-addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8675* Define pre-multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8676* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8677 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8678* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8679* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8680* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8681 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8682 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8683 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 8738, 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 8684 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 8685 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 8686 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 8687 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 8688 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 8689 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 8690 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 8691 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 8692 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 8693 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 8694 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 8695 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 )
 
Theorem0nsr 8696 The empty set is not a signed real. (Contributed by NM, 25-Aug-1995.) (Revised by Mario Carneiro, 10-Jul-2014.) (New usage is discouraged.)
 |- 
 -.  (/)  e.  R.
 
Theorem0r 8697 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 0R  e.  R.
 
Theorem1sr 8698 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 1R  e.  R.
 
Theoremm1r 8699 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 -1R  e.  R.
 
Theoremaddclsr 8700 Closure of addition on signed reals. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  +R  B )  e.  R. )
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