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Theorem List for Metamath Proof Explorer - 8501-8600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremprnmadd 8501* 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 8502 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 8503* 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 8504* 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 8505* 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 8506* 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 8507* 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 8508* 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 8509* 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 8510* 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 8511* 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 8512* 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 8513* 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 8514* 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 8515* 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 8516 Domain of addition on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  +P.  =  ( P. 
 X.  P. )
 
Theoremdmmp 8517 Domain of multiplication on positive reals. (Contributed by NM, 18-Nov-1995.) (New usage is discouraged.)
 |- 
 dom  .P.  =  ( P. 
 X.  P. )
 
Theoremnqpr 8518* 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 8519 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 8520 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 8521* 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 8522 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 8523* 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 8524 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 8525 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 8526 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 8527 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 8528 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 8529 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 8530* 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 8531 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 8532 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 8533 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 8534 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 8535 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 8536 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 8537* 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 8538 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 8539 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 8540* 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 8541* 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 8542* 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 8543* 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 8544* 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 8545* 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 8546* 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 8547* 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 8548 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 8549 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 8550 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 8551* 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 8552* 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 8553* 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 8554* 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 8555* 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 8556* 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 8557* 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 8558* 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 8559* Define pre-addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8560* Define pre-multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8561* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8562 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8563* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8564* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8565* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8566 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8567 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8568 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 8623, 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 8569 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 8570 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 8571 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 8572 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 8573 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 8574 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 8575 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 8576 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 8577 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 8578 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 8579 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 8580 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 8581 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 8582 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 0R  e.  R.
 
Theorem1sr 8583 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 1R  e.  R.
 
Theoremm1r 8584 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 -1R  e.  R.
 
Theoremaddclsr 8585 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. )
 
Theoremmulclsr 8586 Closure of multiplication on signed reals. (Contributed by NM, 10-Aug-1995.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  B  e.  R. )  ->  ( A  .R  B )  e.  R. )
 
Theoremdmaddsr 8587 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  +R  =  ( R.  X.  R. )
 
Theoremdmmulsr 8588 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  .R  =  ( R. 
 X.  R. )
 
Theoremaddcomsr 8589 Addition of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  +R  B )  =  ( B  +R  A )
 
Theoremaddasssr 8590 Addition of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  +R  B )  +R  C )  =  ( A  +R  ( B  +R  C ) )
 
Theoremmulcomsr 8591 Multiplication of signed reals is commutative. (Contributed by NM, 31-Aug-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  .R  B )  =  ( B  .R  A )
 
Theoremmulasssr 8592 Multiplication of signed reals is associative. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( ( A  .R  B )  .R  C )  =  ( A  .R  ( B  .R  C ) )
 
Theoremdistrsr 8593 Multiplication of signed reals is distributive. (Contributed by NM, 2-Sep-1995.) (Revised by Mario Carneiro, 28-Apr-2015.) (New usage is discouraged.)
 |-  ( A  .R  ( B  +R  C ) )  =  ( ( A 
 .R  B )  +R  ( A  .R  C ) )
 
Theoremm1p1sr 8594 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 8595 Minus one times minus one is plus one for signed reals. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( -1R  .R  -1R )  =  1R
 
Theoremltsosr 8596 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 8597 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 8598 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 8599 The signed real number 0 is an identity element for addition of signed reals. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  +R  0R )  =  A )
 
Theorem1idsr 8600 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
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