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Theorem List for Metamath Proof Explorer - 9401-9500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremltprord 9401 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 9402 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 9403 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 9404* 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 9405 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 9406 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 9407* 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 9408* 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 9409* 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 9410* 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 9411* 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 9412* 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 9413* 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 9414* 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 9415 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 9416 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 9417 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 9418* 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 9419* 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 9420* 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 9421* 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 9422* 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 9423* The union of a nonempty, bounded set of positive reals is a positive real. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  P.  A. y  e.  A  y 
 <P  x )  ->  U. A  e.  P. )
 
Theoremsuplem2pr 9424* The union of a set of positive reals (if a positive real) is its supremum (the least upper bound). Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (Revised by Mario Carneiro, 12-Jun-2013.) (New usage is discouraged.)
 |-  ( A  C_  P.  ->  ( ( y  e.  A  ->  -.  U. A  <P  y )  /\  ( y 
 <P  U. A  ->  E. z  e.  A  y  <P  z ) ) )
 
Theoremsupexpr 9425* The union of a nonempty, bounded set of positive reals has a supremum. Part of Proposition 9-3.3 of [Gleason] p. 122. (Contributed by NM, 19-May-1996.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  P.  A. y  e.  A  y 
 <P  x )  ->  E. x  e.  P.  ( A. y  e.  A  -.  x  <P  y 
 /\  A. y  e.  P.  ( y  <P  x  ->  E. z  e.  A  y  <P  z ) ) )
 
Definitiondf-enr 9426* Define equivalence relation for signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9491, 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 9427 Define class of signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9491, 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 9428* Define addition on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9491, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 +R  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( w 
 +P.  u ) ,  ( v  +P.  f
 ) >. ]  ~R  )
 ) }
 
Definitiondf-mr 9429* Define multiplication on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9491, and is intended to be used only by the construction. From Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 .R  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  R.  /\  y  e.  R. )  /\  E. w E. v E. u E. f ( ( x  =  [ <. w ,  v >. ] 
 ~R  /\  y  =  [ <. u ,  f >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u ) 
 +P.  ( v  .P.  f ) ) ,  ( ( w  .P.  f )  +P.  ( v 
 .P.  u ) )
 >. ]  ~R  ) ) }
 
Definitiondf-ltr 9430* Define ordering relation on signed reals. This is a "temporary" set used in the construction of complex numbers df-c 9491, 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 9431 Define signed real constant 0. This is a "temporary" set used in the construction of complex numbers df-c 9491, 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 9432 Define signed real constant 1. This is a "temporary" set used in the construction of complex numbers df-c 9491, 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 9433 Define signed real constant -1. This is a "temporary" set used in the construction of complex numbers df-c 9491, 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 9434 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 9435 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 9436 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 9437 The equivalence relation for signed reals exists. (Contributed by NM, 25-Jul-1995.) (New usage is discouraged.)
 |- 
 ~R  e.  _V
 
Theoremltrelsr 9438 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 9439 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 9440 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 9441 Lemma showing compatibility of multiplication. (Contributed by NM, 5-Sep-1995.) (New usage is discouraged.)
 |-  ( ( ( ( A  e.  P.  /\  B  e.  P. )  /\  ( C  e.  P.  /\  D  e.  P. )
 )  /\  ( ( F  e.  P.  /\  G  e.  P. )  /\  ( R  e.  P.  /\  S  e.  P. ) ) ) 
 ->  ( ( ( A 
 +P.  D )  =  ( B  +P.  C ) 
 /\  ( F  +P.  S )  =  ( G 
 +P.  R ) )  ->  <. ( ( A  .P.  F )  +P.  ( B 
 .P.  G ) ) ,  ( ( A  .P.  G )  +P.  ( B 
 .P.  F ) ) >.  ~R 
 <. ( ( C  .P.  R )  +P.  ( D 
 .P.  S ) ) ,  ( ( C  .P.  S )  +P.  ( D 
 .P.  R ) ) >. ) )
 
Theoremprsrlem1 9442* Decomposing signed reals into positive reals. Lemma for addsrpr 9445 and mulsrpr 9446. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( ( A  e.  ( ( P. 
 X.  P. ) /.  ~R  )  /\  B  e.  (
 ( P.  X.  P. ) /.  ~R  ) ) 
 /\  ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  ( A  =  [ <. s ,  f >. ] 
 ~R  /\  B  =  [ <. g ,  h >. ]  ~R  ) ) )  ->  ( (
 ( ( w  e. 
 P.  /\  v  e.  P. )  /\  ( s  e.  P.  /\  f  e.  P. ) )  /\  ( ( u  e. 
 P.  /\  t  e.  P. )  /\  ( g  e.  P.  /\  h  e.  P. ) ) ) 
 /\  ( ( w 
 +P.  f )  =  ( v  +P.  s
 )  /\  ( u  +P.  h )  =  ( t  +P.  g ) ) ) )
 
Theoremaddsrmo 9443* There is at most one result from adding signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  E* z E. w E. v E. u E. t
 ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( w 
 +P.  u ) ,  ( v  +P.  t
 ) >. ]  ~R  )
 )
 
Theoremmulsrmo 9444* There is at most one result from multiplying signed reals. (Contributed by Jim Kingdon, 30-Dec-2019.)
 |-  ( ( A  e.  ( ( P.  X.  P. ) /.  ~R  )  /\  B  e.  ( ( P.  X.  P. ) /.  ~R  ) )  ->  E* z E. w E. v E. u E. t
 ( ( A  =  [ <. w ,  v >. ]  ~R  /\  B  =  [ <. u ,  t >. ]  ~R  )  /\  z  =  [ <. ( ( w  .P.  u ) 
 +P.  ( v  .P.  t ) ) ,  ( ( w  .P.  t )  +P.  ( v 
 .P.  u ) )
 >. ]  ~R  ) )
 
Theoremaddsrpr 9445 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 9446 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 9447 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 9448 Greater than 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 9449 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 9450 The constant  0R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 0R  e.  R.
 
Theorem1sr 9451 The constant  1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 1R  e.  R.
 
Theoremm1r 9452 The constant  -1R is a signed real. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 -1R  e.  R.
 
Theoremaddclsr 9453 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 9454 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 9455 Domain of addition on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  +R  =  ( R.  X.  R. )
 
Theoremdmmulsr 9456 Domain of multiplication on signed reals. (Contributed by NM, 25-Aug-1995.) (New usage is discouraged.)
 |- 
 dom  .R  =  ( R. 
 X.  R. )
 
Theoremaddcomsr 9457 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 9458 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 9459 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 9460 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 9461 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 9462 Minus one plus one is zero for signed reals. (Contributed by NM, 5-May-1996.) (New usage is discouraged.)
 |-  ( -1R  +R  1R )  =  0R
 
Theoremm1m1sr 9463 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 9464 Signed real 'less than' is a strict ordering. (Contributed by NM, 19-Feb-1996.) (New usage is discouraged.)
 |- 
 <R  Or  R.
 
Theorem0lt1sr 9465 0 is less than 1 for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 0R  <R  1R
 
Theorem1ne0sr 9466 1 and 0 are distinct for signed reals. (Contributed by NM, 26-Mar-1996.) (New usage is discouraged.)
 |- 
 -.  1R  =  0R
 
Theorem0idsr 9467 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 9468 1 is an identity element for multiplication. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  1R )  =  A )
 
Theorem00sr 9469 A signed real times 0 is 0. (Contributed by NM, 10-Apr-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  .R  0R )  =  0R )
 
Theoremltasr 9470 Ordering property of addition. (Contributed by NM, 10-May-1996.) (New usage is discouraged.)
 |-  ( C  e.  R.  ->  ( A  <R  B  <->  ( C  +R  A )  <R  ( C  +R  B ) ) )
 
Theorempn0sr 9471 A signed real plus its negative is zero. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  ( A  +R  ( A  .R  -1R ) )  =  0R )
 
Theoremnegexsr 9472* Existence of negative signed real. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 2-May-1996.) (New usage is discouraged.)
 |-  ( A  e.  R.  ->  E. x  e.  R.  ( A  +R  x )  =  0R )
 
Theoremrecexsrlem 9473* The reciprocal of a positive signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( 0R  <R  A  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremaddgt0sr 9474 The sum of two positive signed reals is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  +R  B ) )
 
Theoremmulgt0sr 9475 The product of two positive signed reals is positive. (Contributed by NM, 13-May-1996.) (New usage is discouraged.)
 |-  ( ( 0R  <R  A 
 /\  0R  <R  B ) 
 ->  0R  <R  ( A  .R  B ) )
 
Theoremsqgt0sr 9476 The square of a nonzero signed real is positive. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  A  =/=  0R )  ->  0R  <R  ( A 
 .R  A ) )
 
Theoremrecexsr 9477* The reciprocal of a nonzero signed real exists. Part of Proposition 9-4.3 of [Gleason] p. 126. (Contributed by NM, 15-May-1996.) (New usage is discouraged.)
 |-  ( ( A  e.  R. 
 /\  A  =/=  0R )  ->  E. x  e.  R.  ( A  .R  x )  =  1R )
 
Theoremmappsrpr 9478 Mapping from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  -1R )  <R  ( C  +R  [ <. A ,  1P >. ]  ~R  ) 
 <->  A  e.  P. )
 
Theoremltpsrpr 9479 Mapping of order from positive signed reals to positive reals. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  [ <. A ,  1P >. ]  ~R  )  <R  ( C  +R  [ <. B ,  1P >. ] 
 ~R  )  <->  A  <P  B )
 
Theoremmap2psrpr 9480* Equivalence for positive signed real. (Contributed by NM, 17-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  C  e.  R.   =>    |-  ( ( C  +R  -1R )  <R  A  <->  E. x  e.  P.  ( C  +R  [ <. x ,  1P >. ]  ~R  )  =  A )
 
Theoremsupsrlem 9481* Lemma for supremum theorem. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  B  =  { w  |  ( C  +R  [ <. w ,  1P >. ] 
 ~R  )  e.  A }   &    |-  C  e.  R.   =>    |-  ( ( C  e.  A  /\  E. x  e.  R.  A. y  e.  A  y  <R  x ) 
 ->  E. x  e.  R.  ( A. y  e.  A  -.  x  <R  y  /\  A. y  e.  R.  (
 y  <R  x  ->  E. z  e.  A  y  <R  z ) ) )
 
Theoremsupsr 9482* A nonempty, bounded set of signed reals has a supremum. (Contributed by NM, 21-May-1996.) (Revised by Mario Carneiro, 15-Jun-2013.) (New usage is discouraged.)
 |-  ( ( A  =/=  (/)  /\  E. x  e.  R.  A. y  e.  A  y 
 <R  x )  ->  E. x  e.  R.  ( A. y  e.  A  -.  x  <R  y 
 /\  A. y  e.  R.  ( y  <R  x  ->  E. z  e.  A  y  <R  z ) ) )
 
Syntaxcc 9483 Class of complex numbers.
 class  CC
 
Syntaxcr 9484 Class of real numbers.
 class  RR
 
Syntaxcc0 9485 Extend class notation to include the complex number 0.
 class 
 0
 
Syntaxc1 9486 Extend class notation to include the complex number 1.
 class 
 1
 
Syntaxci 9487 Extend class notation to include the complex number i.
 class  _i
 
Syntaxcaddc 9488 Addition on complex numbers.
 class  +
 
Syntaxcltrr 9489 'Less than' predicate (defined over real subset of complex numbers).
 class  <RR
 
Syntaxcmul 9490 Multiplication on complex numbers. The token  x. is a center dot.
 class  x.
 
Definitiondf-c 9491 Define the set of complex numbers. The 23 axioms for complex numbers start at axresscn 9518. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 CC  =  ( R. 
 X.  R. )
 
Definitiondf-0 9492 Define the complex number 0. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  0  =  <. 0R ,  0R >.
 
Definitiondf-1 9493 Define the complex number 1. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  1  =  <. 1R ,  0R >.
 
Definitiondf-i 9494 Define the complex number  _i (the imaginary unit). (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  _i  =  <. 0R ,  1R >.
 
Definitiondf-r 9495 Define the set of real numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 RR  =  ( R. 
 X.  { 0R } )
 
Definitiondf-add 9496* Define addition over complex numbers. (Contributed by NM, 28-May-1995.) (New usage is discouraged.)
 |- 
 +  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( w  +R  u ) ,  ( v  +R  f ) >. ) ) }
 
Definitiondf-mul 9497* Define multiplication over complex numbers. (Contributed by NM, 9-Aug-1995.) (New usage is discouraged.)
 |- 
 x.  =  { <. <. x ,  y >. ,  z >.  |  (
 ( x  e.  CC  /\  y  e.  CC )  /\  E. w E. v E. u E. f ( ( x  =  <. w ,  v >.  /\  y  =  <. u ,  f >. )  /\  z  = 
 <. ( ( w  .R  u )  +R  ( -1R  .R  ( v  .R  f ) ) ) ,  ( ( v 
 .R  u )  +R  ( w  .R  f ) ) >. ) ) }
 
Definitiondf-lt 9498* Define 'less than' on the real subset of complex numbers. Proofs should typically use  < instead; see df-ltxr 9626. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |- 
 <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  = 
 <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
 
Theoremopelcn 9499 Ordered pair membership in the class of complex numbers. (Contributed by NM, 14-May-1996.) (New usage is discouraged.)
 |-  ( <. A ,  B >.  e.  CC  <->  ( A  e.  R. 
 /\  B  e.  R. ) )
 
Theoremopelreal 9500 Ordered pair membership in class of real subset of complex numbers. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
 |-  ( <. A ,  0R >.  e.  RR  <->  A  e.  R. )
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