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Theorem List for Metamath Proof Explorer - 23201-23300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremfimacnvinrn 23201 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 21-Jan-2017.)
 |-  ( Fun  F  ->  ( `' F " A )  =  ( `' F "
 ( A  i^i  ran  F ) ) )
 
Theoremfimacnvinrn2 23202 Taking the converse image of a set can be limited to the range of the function used. (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  (
 ( Fun  F  /\  ran 
 F  C_  B )  ->  ( `' F " A )  =  ( `' F " ( A  i^i  B ) ) )
 
Theoremsuppss2f 23203* Show that the support of a function is contained in a set. (Contributed by Thierry Arnoux, 22-Jun-2017.)
 |-  F/ k ph   &    |-  F/_ k A   &    |-  F/_ k W   &    |-  ( ( ph  /\  k  e.  ( A  \  W ) )  ->  B  =  Z )   =>    |-  ( ph  ->  ( `' ( k  e.  A  |->  B ) " ( _V  \  { Z }
 ) )  C_  W )
 
Theoremxpima 23204 The image of a constant function (or other cross product). (Contributed by Thierry Arnoux, 4-Feb-2017.)
 |-  (
 ( A  X.  B ) " C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
 
Theoremfovcld 23205 Closure law for an operation. (Contributed by NM, 19-Apr-2007.) (Revised by Thierry Arnoux, 17-Feb-2017.)
 |-  ( ph  ->  F : ( R  X.  S ) --> C )   =>    |-  ( ( ph  /\  A  e.  R  /\  B  e.  S )  ->  ( A F B )  e.  C )
 
Theoremdfrel4 23206* A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5570 for relations. (Contributed by Mario Carneiro, 16-Aug-2015.) (Revised by Thierry Arnoux, 11-May-2017.)
 |-  F/_ x R   &    |-  F/_ y R   =>    |-  ( Rel  R  <->  R  =  { <. x ,  y >.  |  x R y }
 )
 
Theoremelovimad 23207 Elementhood of the image set of an operation value (Contributed by Thierry Arnoux, 13-Mar-2017.)
 |-  ( ph  ->  A  e.  C )   &    |-  ( ph  ->  B  e.  D )   &    |-  Fun  F   &    |-  ( ph  ->  ( C  X.  D ) 
 C_  dom  F )   =>    |-  ( ph  ->  ( A F B )  e.  ( F " ( C  X.  D ) ) )
 
Theoremofrn 23208 The range of the function operation. (Contributed by Thierry Arnoux, 8-Jan-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  .+  : ( B  X.  B ) --> B )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  B )
 
Theoremofrn2 23209 The range of the function operation. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> B )   &    |-  ( ph  ->  .+  : ( B  X.  B ) --> B )   &    |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  ran  ( F  o F  .+  G )  C_  (  .+  " ( ran  F  X.  ran  G ) ) )
 
Theoremoff2 23210* The function operation produces a function - alternative form with all antecedents as deduction. (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  (
 ( ph  /\  ( x  e.  S  /\  y  e.  T ) )  ->  ( x R y )  e.  U )   &    |-  ( ph  ->  F : A --> S )   &    |-  ( ph  ->  G : B --> T )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  ( A  i^i  B )  =  C )   =>    |-  ( ph  ->  ( F  o F R G ) : C --> U )
 
Theoremunipreima 23211* Preimage of a class union. (Contributed by Thierry Arnoux, 7-Feb-2017.)
 |-  ( Fun  F  ->  ( `' F " U. A )  =  U_ x  e.  A  ( `' F " x ) )
 
Theoremsspreima 23212 The preimage of a subset is a subset of the preimage. (Contributed by Brendan Leahy, 23-Sep-2017.)
 |-  (
 ( Fun  F  /\  A  C_  B )  ->  ( `' F " A ) 
 C_  ( `' F " B ) )
 
Theoremxppreima 23213 The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
 |-  (
 ( Fun  F  /\  ran 
 F  C_  ( _V  X. 
 _V ) )  ->  ( `' F " ( Y  X.  Z ) )  =  ( ( `' ( 1st  o.  F ) " Y )  i^i  ( `' ( 2nd 
 o.  F ) " Z ) ) )
 
Theoremxppreima2 23214* The preimage of a cross product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 7-Jun-2017.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> C )   &    |-  H  =  ( x  e.  A  |->  <. ( F `  x ) ,  ( G `  x ) >. )   =>    |-  ( ph  ->  ( `' H " ( Y  X.  Z ) )  =  ( ( `' F " Y )  i^i  ( `' G " Z ) ) )
 
Theoremfmptapd 23215* Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  _V )   &    |-  ( ph  ->  B  e.  _V )   &    |-  ( ph  ->  ( R  u.  { A } )  =  S )   &    |-  ( ( ph  /\  x  =  A )  ->  C  =  B )   =>    |-  ( ph  ->  (
 ( x  e.  R  |->  C )  u.  { <. A ,  B >. } )  =  ( x  e.  S  |->  C ) )
 
Theoremfmptpr 23216* Express a pair function in maps-to notation. (Contributed by Thierry Arnoux, 3-Jan-2017.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  C  e.  X )   &    |-  ( ph  ->  D  e.  Y )   &    |-  ( ( ph  /\  x  =  A )  ->  E  =  C )   &    |-  ( ( ph  /\  x  =  B ) 
 ->  E  =  D )   =>    |-  ( ph  ->  { <. A ,  C >. ,  <. B ,  D >. }  =  ( x  e.  { A ,  B }  |->  E ) )
 
Theoremelunirn2 23217 Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
 |-  (
 ( Fun  F  /\  B  e.  ( F `  A ) )  ->  B  e.  U. ran  F )
 
Theoremabfmpunirn 23218* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
 |-  F  =  ( x  e.  V  |->  { y  |  ph } )   &    |-  { y  |  ph }  e.  _V   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( B  e.  U.
 ran  F  <->  ( B  e.  _V 
 /\  E. x  e.  V  ps ) )
 
Theoremrabfmpunirn 23219* Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 30-Sep-2016.)
 |-  F  =  ( x  e.  V  |->  { y  e.  W  |  ph
 } )   &    |-  W  e.  _V   &    |-  (
 y  =  B  ->  (
 ph 
 <->  ps ) )   =>    |-  ( B  e.  U.
 ran  F  <->  E. x  e.  V  ( B  e.  W  /\  ps ) )
 
Theoremabfmpeld 23220* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
 |-  F  =  ( x  e.  V  |->  { y  |  ps }
 )   &    |-  ( ph  ->  { y  |  ps }  e.  _V )   &    |-  ( ph  ->  (
 ( x  =  A  /\  y  =  B )  ->  ( ps  <->  ch ) ) )   =>    |-  ( ph  ->  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ch ) ) )
 
Theoremabfmpel 23221* Membership in an element of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 19-Oct-2016.)
 |-  F  =  ( x  e.  V  |->  { y  |  ph } )   &    |-  { y  |  ph }  e.  _V   &    |-  (
 ( x  =  A  /\  y  =  B )  ->  ( ph  <->  ps ) )   =>    |-  ( ( A  e.  V  /\  B  e.  W )  ->  ( B  e.  ( F `  A )  <->  ps ) )
 
Theoremcbvmptf 23222* Rule to change the bound variable in a maps-to function, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ y A   &    |-  F/_ y B   &    |-  F/_ x C   &    |-  ( x  =  y 
 ->  B  =  C )   =>    |-  ( x  e.  A  |->  B )  =  (
 y  e.  A  |->  C )
 
TheoremfmptdF 23223 Domain and co-domain of the mapping operation; deduction form. This version of fmptd 5686 usex bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x C   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  C )   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( ph  ->  F : A --> C )
 
Theoremfmpt3d 23224* Domain and co-domain of the mapping operation; deduction form. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   =>    |-  ( ph  ->  F : A
 --> C )
 
Theoremresmptf 23225 Restriction of the mapping operation. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  F/_ x A   &    |-  F/_ x B   =>    |-  ( B  C_  A  ->  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C ) )
 
Theoremfvmpt2f 23226 Value of a function given by the "maps to" notation. (Contributed by Thierry Arnoux, 9-Mar-2017.)
 |-  F/_ x A   =>    |-  ( ( x  e.  A  /\  B  e.  C )  ->  ( ( x  e.  A  |->  B ) `  x )  =  B )
 
Theoremfvmpt2d 23227* Deduction version of fvmpt2 5610. (Contributed by Thierry Arnoux, 8-Dec-2016.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   =>    |-  ( ( ph  /\  x  e.  A )  ->  ( F `  x )  =  B )
 
Theoremmptfnf 23228 The maps-to notation defines a function with domain. (Contributed by Scott Fenton, 21-Mar-2011.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   =>    |-  ( A. x  e.  A  B  e.  _V  <->  ( x  e.  A  |->  B )  Fn  A )
 
Theoremfnmptf 23229 The maps-to notation defines a function with domain. (Contributed by NM, 9-Apr-2013.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   =>    |-  ( A. x  e.  A  B  e.  V  ->  ( x  e.  A  |->  B )  Fn  A )
 
Theoremfeqmptdf 23230 Deduction form of dffn5f 5579. (Contributed by Mario Carneiro, 8-Jan-2015.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   &    |-  F/_ x F   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  =  ( x  e.  A  |->  ( F `  x ) ) )
 
Theoremfmptcof2 23231* Composition of two functions expressed as ordered-pair class abstractions. (Contributed by FL, 21-Jun-2012.) (Revised by Mario Carneiro, 24-Jul-2014.) (Revised by Thierry Arnoux, 10-May-2017.)
 |-  F/_ x A   &    |-  F/_ x B   &    |-  F/ x ph   &    |-  ( ph  ->  A. x  e.  A  R  e.  B )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  R ) )   &    |-  ( ph  ->  G  =  ( y  e.  B  |->  S ) )   &    |-  ( y  =  R  ->  S  =  T )   =>    |-  ( ph  ->  ( G  o.  F )  =  ( x  e.  A  |->  T ) )
 
Theoremfcomptf 23232* Express composition of two functions as a maps-to applying both in sequence. This version has one less distinct variable restriction compared to fcompt 5696. (Contributed by Thierry Arnoux, 30-Jun-2017.)
 |-  F/_ x B   =>    |-  ( ( A : D
 --> E  /\  B : C
 --> D )  ->  ( A  o.  B )  =  ( x  e.  C  |->  ( A `  ( B `
  x ) ) ) )
 
Theoremcofmpt 23233* Express composition of a maps-to function with another function in a maps-to notation. (Contributed by Thierry Arnoux, 29-Jun-2017.)
 |-  ( ph  ->  F : C --> D )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   =>    |-  ( ph  ->  ( F  o.  ( x  e.  A  |->  B ) )  =  ( x  e.  A  |->  ( F `  B ) ) )
 
Theoremofoprabco 23234* Function operation as a composition with an operation. (Contributed by Thierry Arnoux, 4-Jun-2017.)
 |-  F/_ a M   &    |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  G : A --> C )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  M  =  ( a  e.  A  |->  <.
 ( F `  a
 ) ,  ( G `
  a ) >. ) )   &    |-  ( ph  ->  N  =  ( x  e.  B ,  y  e.  C  |->  ( x R y ) ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( N  o.  M ) )
 
Theoremoffval2f 23235* The function operation expressed as a mapping. (Contributed by Thierry Arnoux, 23-Jun-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  W )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  C  e.  X )   &    |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   &    |-  ( ph  ->  G  =  ( x  e.  A  |->  C ) )   =>    |-  ( ph  ->  ( F  o F R G )  =  ( x  e.  A  |->  ( B R C ) ) )
 
TheoremfuncnvmptOLD 23236* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x ( x  e.  A  /\  y  =  B ) ) )
 
Theoremfuncnvmpt 23237* Condition for a function in maps-to notation to be single-rooted. (Contributed by Thierry Arnoux, 28-Feb-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. y E* x  e.  A y  =  B ) )
 
Theoremfuncnv5mpt 23238* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 1-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( x  =  z  ->  B  =  C )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. x  e.  A  A. z  e.  A  ( x  =  z  \/  B  =/=  C ) ) )
 
Theoremfuncnv4mpt 23239* Two ways to say that a function in maps-to notation is single-rooted. (Contributed by Thierry Arnoux, 2-Mar-2017.)
 |-  F/ x ph   &    |-  F/_ x A   &    |-  F/_ x F   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  e.  V )   =>    |-  ( ph  ->  ( Fun  `' F  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) ) )
 
Theoremrnmptss 23240* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( A. x  e.  A  B  e.  C  ->  ran  F  C_  C )
 
Theoremrnmpt2ss 23241* The range of an operation given by the "maps to" notation as a subset. (Contributed by Thierry Arnoux, 23-May-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   =>    |-  ( A. x  e.  A  A. y  e.  B  C  e.  D  ->  ran  F  C_  D )
 
Theorempartfun 23242 Rewrite a function defined by parts, using a mapping and an if construct, into a union of functions on disjoint domains. (Contributed by Thierry Arnoux, 30-Mar-2017.)
 |-  ( x  e.  A  |->  if ( x  e.  B ,  C ,  D )
 )  =  ( ( x  e.  ( A  i^i  B )  |->  C )  u.  ( x  e.  ( A  \  B )  |->  D ) )
 
Theoremgtiso 23243 Two ways to write a strictly decreasing function on the reals. (Contributed by Thierry Arnoux, 6-Apr-2017.)
 |-  (
 ( A  C_  RR*  /\  B  C_  RR* )  ->  ( F 
 Isom  <  ,  `'  <  ( A ,  B )  <->  F  Isom  <_  ,  `'  <_  ( A ,  B ) ) )
 
Theoremisoun 23244* Infer an isomorphism from for a union of two isomorphisms. (Contributed by Thierry Arnoux, 30-Mar-2017.)
 |-  ( ph  ->  H  Isom  R ,  S  ( A ,  B ) )   &    |-  ( ph  ->  G 
 Isom  R ,  S  ( C ,  D ) )   &    |-  ( ( ph  /\  x  e.  A  /\  y  e.  C )  ->  x R y )   &    |-  ( ( ph  /\  z  e.  B  /\  w  e.  D )  ->  z S w )   &    |-  ( ( ph  /\  x  e.  C  /\  y  e.  A )  ->  -.  x R y )   &    |-  ( ( ph  /\  z  e.  D  /\  w  e.  B )  ->  -.  z S w )   &    |-  ( ph  ->  ( A  i^i  C )  =  (/) )   &    |-  ( ph  ->  ( B  i^i  D )  =  (/) )   =>    |-  ( ph  ->  ( H  u.  G )  Isom  R ,  S  ( ( A  u.  C ) ,  ( B  u.  D ) ) )
 
18.3.12  First and second members of an ordered pair - misc additions
 
Theoremdf1stres 23245* Definition for a restriction of the  1st (first member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( 1st  |`  ( A  X.  B ) )  =  ( x  e.  A ,  y  e.  B  |->  x )
 
Theoremdf2ndres 23246* Definition for a restriction of the  2nd (second member of an ordered pair) function. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( 2nd  |`  ( A  X.  B ) )  =  ( x  e.  A ,  y  e.  B  |->  y )
 
Theorem1stnpr 23247 Value of the first-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( -.  A  e.  ( _V 
 X.  _V )  ->  ( 1st `  A )  =  (/) )
 
Theorem2ndnpr 23248 Value of the second-member function at non-pairs. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( -.  A  e.  ( _V 
 X.  _V )  ->  ( 2nd `  A )  =  (/) )
 
Theoremcurry2ima 23249* The image of a curried function with a constant second argument. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  G  =  ( F  o.  `' ( 1st  |`  ( _V  X.  { C } ) ) )   =>    |-  ( ( F  Fn  ( A  X.  B ) 
 /\  C  e.  B  /\  D  C_  A )  ->  ( G " D )  =  { y  |  E. x  e.  D  y  =  ( x F C ) } )
 
18.3.13  Supremum - misc additions
 
Theoremsupssd 23250* Inequality deduction for supremum of a subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  R  Or  A )   &    |-  ( ph  ->  B  C_  C )   &    |-  ( ph  ->  C 
 C_  A )   &    |-  ( ph  ->  E. x  e.  A  ( A. y  e.  B  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  B  y R z ) ) )   &    |-  ( ph  ->  E. x  e.  A  (
 A. y  e.  C  -.  x R y  /\  A. y  e.  A  ( y R x  ->  E. z  e.  C  y R z ) ) )   =>    |-  ( ph  ->  -.  sup ( C ,  A ,  R ) R sup ( B ,  A ,  R ) )
 
18.3.14  Ordering on reals - misc additions
 
Theoremlt2addrd 23251* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  ( B  +  C )
 )   =>    |-  ( ph  ->  E. b  e.  RR  E. c  e. 
 RR  ( A  =  ( b  +  c
 )  /\  b  <  B 
 /\  c  <  C ) )
 
18.3.15  Extended reals - misc additions
 
Theoremxrlelttric 23252 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremxrre3FL 23253 A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.) (TODO remove and use xrre3 10502, which was imported )
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR )  /\  ( B  <_  A  /\  A  <  +oo )
 )  ->  A  e.  RR )
 
Theoremxraddge02 23254 A number is less than or equal to itself plus a nonnegative number. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( 0  <_  B  ->  A  <_  ( A + e B ) ) )
 
Theoremxlt2addrd 23255* If the right-side of a 'less-than' relationship is an addition, then we can express the left-side as an addition, too, where each term is respectively less than each term of the original right side. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR* )   &    |-  ( ph  ->  B  =/=  -oo )   &    |-  ( ph  ->  C  =/=  -oo )   &    |-  ( ph  ->  A  <  ( B + e C ) )   =>    |-  ( ph  ->  E. b  e.  RR*  E. c  e.  RR*  ( A  =  ( b + e
 c )  /\  b  <  B  /\  c  <  C ) )
 
Theoremxrsupssd 23256 Inequality deduction for supremum of an extended real subset. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  ( ph  ->  B  C_  C )   &    |-  ( ph  ->  C  C_  RR* )   =>    |-  ( ph  ->  sup ( B ,  RR* ,  <  ) 
 <_  sup ( C ,  RR*
 ,  <  ) )
 
Theoremxrofsup 23257 The supremum is preserved by extended addition set operation. (provided minus infinity is not involved as it does not behave well with addition) (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  ( ph  ->  X  C_  RR* )   &    |-  ( ph  ->  Y  C_  RR* )   &    |-  ( ph  ->  sup ( X ,  RR*
 ,  <  )  =/=  -oo )   &    |-  ( ph  ->  sup ( Y ,  RR* ,  <  )  =/=  -oo )   &    |-  ( ph  ->  Z  =  ( + e "
 ( X  X.  Y ) ) )   =>    |-  ( ph  ->  sup ( Z ,  RR* ,  <  )  =  ( sup ( X ,  RR*
 ,  <  ) + e sup ( Y ,  RR*
 ,  <  ) )
 )
 
Theoremsupxrnemnf 23258 The supremum of a non-empty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
 |-  (
 ( A  C_  RR*  /\  A  =/= 
 (/)  /\  -.  -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  =/=  -oo )
 
Theoremxrhaus 23259 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  (ordTop ` 
 <_  )  e.  Haus
 
18.3.16  Real number intervals - misc additions
 
Theoremicossicc 23260 A closed-below, open-above interval is a subset of its closure. (Contributed by Thierry Arnoux, 25-Oct-2016.)
 |-  ( A [,) B )  C_  ( A [,] B )
 
Theoremiocssicc 23261 A closed-above, open-below interval is a subset of its closure. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  ( A (,] B )  C_  ( A [,] B )
 
Theoremioossico 23262 An open interval is a subset of its closure-below. (Contributed by Thierry Arnoux, 3-Mar-2017.)
 |-  ( A (,) B )  C_  ( A [,) B )
 
Theoremiocssioo 23263 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <  B ) )  ->  ( C (,] D )  C_  ( A (,) B ) )
 
Theoremicossioo 23264 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 29-Mar-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <  C  /\  D  <_  B )
 )  ->  ( C [,) D )  C_  ( A (,) B ) )
 
Theoremicossico 23265 Condition for a closed interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B )
 )  ->  ( C [,) D )  C_  ( A [,) B ) )
 
Theoremioossioo 23266 Condition for an open interval to be a subset of an open interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR* )  /\  ( A  <_  C  /\  D  <_  B )
 )  ->  ( C (,) D )  C_  ( A (,) B ) )
 
Theoremjoiniooico 23267 Disjoint joining an open interval with a closed below, open above interval to form a closed below, open above interval. (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <  B  /\  B  <_  C ) )  ->  ( ( ( A (,) B )  i^i  ( B [,) C ) )  =  (/)  /\  (
 ( A (,) B )  u.  ( B [,) C ) )  =  ( A (,) C ) ) )
 
Theoremiccgelb 23268 An element of a closed interval is more than or equal to its lower bound (Contributed by Thierry Arnoux, 23-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  ( A [,] B ) )  ->  A  <_  C )
 
Theoremsnunioc 23269 The closure of the open end of a left-open real interval. (Contributed by Thierry Arnoux, 28-Mar-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( { A }  u.  ( A (,] B ) )  =  ( A [,] B ) )
 
Theoremubico 23270 A right-open interval does not contain its right endpoint. (Contributed by Thierry Arnoux, 5-Apr-2017.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR* )  ->  -.  B  e.  ( A [,) B ) )
 
Theoremxeqlelt 23271 Equality in terms of 'less than or equal to', 'less than'. (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  =  B  <->  ( A  <_  B  /\  -.  A  <  B ) ) )
 
Theoremeliccelico 23272 Relate elementhood to a closed interval with elementhood to the same closed-below, opened-above interval or to its upper bound. (Contributed by Thierry Arnoux, 3-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <_  B )  ->  ( C  e.  ( A [,] B )  <->  ( C  e.  ( A [,) B )  \/  C  =  B ) ) )
 
Theoremelicoelioo 23273 Relate elementhood to a closed-below, opened-above interval with elementhood to the same opened interval or to its lower bound. (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  A  <  B )  ->  ( C  e.  ( A [,) B )  <->  ( C  =  A  \/  C  e.  ( A (,) B ) ) ) )
 
Theoremiocinioc2 23274 Intersection between two opened below, closed above intervals sharing the same upper bound. (Contributed by Thierry Arnoux, 7-Aug-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <_  B )  ->  (
 ( A (,] C )  i^i  ( B (,] C ) )  =  ( B (,] C ) )
 
Theoremxrdifh 23275 Set difference of a half-opened interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
 |-  A  e.  RR*   =>    |-  ( RR*  \  ( A [,]  +oo ) )  =  (  -oo [,) A )
 
Theoremiocinif 23276 Relate intersection of two opened below, closed above intervals with the same upper bound with a conditional construct. (Contributed by Thierry Arnoux, 7-Aug-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  ->  (
 ( A (,] C )  i^i  ( B (,] C ) )  =  if ( A  <  B ,  ( B (,] C ) ,  ( A (,] C ) ) )
 
Theoremdifioo 23277 The difference between two opened intervals sharing the same lower bound (Contributed by Thierry Arnoux, 26-Sep-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  A  <  B )  ->  (
 ( A (,) C )  \  ( A (,) B ) )  =  ( B [,) C ) )
 
18.3.17  Finite intervals of integers - misc additions
 
Theoremfzssnn 23278 Finite sets of sequential integers starting from a natural are a subset of the natural numbers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( M  e.  NN  ->  ( M ... N ) 
 C_  NN )
 
Theoremssnnssfz 23279* For any finite subset of  NN, find a superset in the form of a set of sequential integers. (Contributed by Thierry Arnoux, 13-Sep-2017.)
 |-  ( A  e.  ( ~P NN  i^i  Fin )  ->  E. n  e.  NN  A  C_  (
 1 ... n ) )
 
18.3.18  Half-open integer ranges - misc additions
 
Theoremfzossnn 23280 Half-opened integer ranges starting with 1 are subsets of NN. (Contributed by Thierry Arnoux, 28-Dec-2016.)
 |-  (
 1..^ N )  C_  NN
 
Theoremelfzo1 23281 Membership in a half-open integer range based at 1. (Contributed by Thierry Arnoux, 14-Feb-2017.)
 |-  ( N  e.  ( 1..^ M )  <->  ( N  e.  NN  /\  M  e.  NN  /\  N  <  M ) )
 
18.3.19  Closed unit
 
Theoremunitsscn 23282 The closed unit is a subset of the set of the complex numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 0 [,] 1 )  C_  CC
 
Theoremelunitrn 23283 The closed unit is a subset of the set of the real numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  RR )
 
Theoremelunitcn 23284 The closed unit is a subset of the set of the complext numbers Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 21-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  A  e.  CC )
 
Theoremelunitge0 23285 An element of the closed unit is positive Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 20-Dec-2016.)
 |-  ( A  e.  ( 0 [,] 1 )  ->  0  <_  A )
 
Theoremunitssxrge0 23286 The closed unit is a subset of the set of the extended non-negative reals. Useful lemma for manipulating probabilities within the closed unit. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 0 [,] 1 )  C_  ( 0 [,]  +oo )
 
Theoremunitdivcld 23287 Necessary conditions for a quotient to be in the closed unit. (somewhat too strong, it would be sufficient that A and B are in RR+) (Contributed by Thierry Arnoux, 20-Dec-2016.)
 |-  (
 ( A  e.  (
 0 [,] 1 )  /\  B  e.  ( 0 [,] 1 )  /\  B  =/=  0 )  ->  ( A  <_  B  <->  ( A  /  B )  e.  (
 0 [,] 1 ) ) )
 
Theoremiistmd 23288 The closed unit monoid is a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
 |-  I  =  ( (mulGrp ` fld )s  ( 0 [,] 1
 ) )   =>    |-  I  e. TopMnd
 
18.3.20  Topology of ` ( RR X. RR ) `
 
Theoremtpr2tp 23289 The usual topology on  ( RR  X.  RR ) is the product topology of the usual topology on  RR. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  ( J  tX  J )  e.  (TopOn `  ( RR  X. 
 RR ) )
 
Theoremtpr2uni 23290 The usual topology on  ( RR  X.  RR ) is the product topology of the usual topology on  RR. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   =>    |-  U. ( J  tX  J )  =  ( RR  X.  RR )
 
Theoremclduni 23291 For any topology, the union of the closed sets is the base set. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  ( J  e.  Top  ->  U. ( Clsd `  J )  = 
 U. J )
 
Theoremxpinpreima 23292 Rewrite the cartesian product of two sets as the intersection of their preimage by  1st and  2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( _V  X.  _V ) ) " A )  i^i  ( `' ( 2nd  |`  ( _V  X.  _V ) ) " B ) )
 
Theoremxpinpreima2 23293 Rewrite the cartesian product of two sets as the intersection of their preimage by  1st and  2nd, the projections on the first and second elements. (Contributed by Thierry Arnoux, 22-Sep-2017.)
 |-  (
 ( A  C_  E  /\  B  C_  F )  ->  ( A  X.  B )  =  ( ( `' ( 1st  |`  ( E  X.  F ) )
 " A )  i^i  ( `' ( 2nd  |`  ( E  X.  F ) ) " B ) ) )
 
Theoremsqsscirc1 23294 The complex square of side  D is a subset of the complex circle of radius  D. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  (
 ( ( ( X  e.  RR  /\  0  <_  X )  /\  ( Y  e.  RR  /\  0  <_  Y ) )  /\  D  e.  RR+ )  ->  ( ( X  <_  ( D  /  2 ) 
 /\  Y  <_  ( D  /  2 ) ) 
 ->  ( sqr `  (
 ( X ^ 2
 )  +  ( Y ^ 2 ) ) )  <  D ) )
 
Theoremsqsscirc2 23295 The complex square of side  D is a subset of the complex circle of radius  D. (Contributed by Thierry Arnoux, 25-Sep-2017.)
 |-  (
 ( ( A  e.  CC  /\  B  e.  CC )  /\  D  e.  RR+ )  ->  ( ( ( abs `  ( Re `  ( B  -  A ) ) )  <_  ( D  /  2
 )  /\  ( abs `  ( Im `  ( B  -  A ) ) )  <_  ( D  /  2 ) )  ->  ( abs `  ( B  -  A ) )  <  D ) )
 
Theoremcnre2csqlem 23296* Lemma for cnre2csqima 23297 (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  ( G  |`  ( RR  X.  RR ) )  =  ( H  o.  F )   &    |-  F  Fn  ( RR  X.  RR )   &    |-  G  Fn  _V   &    |-  ( x  e.  ( RR  X. 
 RR )  ->  ( G `  x )  e. 
 RR )   &    |-  ( ( x  e.  ran  F  /\  y  e.  ran  F ) 
 ->  ( H `  ( x  -  y ) )  =  ( ( H `
  x )  -  ( H `  y ) ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR 
 X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( `' ( G  |`  ( RR 
 X.  RR ) ) "
 ( ( ( G `
  X )  -  D ) [,) (
 ( G `  X )  +  D )
 ) )  ->  ( abs `  ( H `  ( ( F `  Y )  -  ( F `  X ) ) ) )  <_  D ) )
 
Theoremcnre2csqima 23297* Image of a centered square by the canonical bijection from  ( RR  X.  RR ) to  CC. (Contributed by Thierry Arnoux, 27-Sep-2017.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) )   =>    |-  ( ( X  e.  ( RR  X.  RR )  /\  Y  e.  ( RR 
 X.  RR )  /\  D  e.  RR+ )  ->  ( Y  e.  ( (
 ( ( 1st `  X )  -  D ) [,) ( ( 1st `  X )  +  D )
 )  X.  ( (
 ( 2nd `  X )  -  D ) [,) (
 ( 2nd `  X )  +  D ) ) ) 
 ->  ( ( abs `  ( Re `  ( ( F `
  Y )  -  ( F `  X ) ) ) )  <_  D  /\  ( abs `  ( Im `  ( ( F `
  Y )  -  ( F `  X ) ) ) )  <_  D ) ) )
 
Theoremtpr2rico 23298* For any point of an open set of the usual topology on  ( RR  X.  RR ) there is a closed below opened above square which contains that point and is entirely in the open set. This is square is actually similar to a ball by the  ( l ^  +oo ) norm, closed below, centered on  X. (Contributed by Thierry Arnoux, 21-Sep-2017.)
 |-  J  =  ( topGen `  ran  (,) )   &    |-  G  =  ( u  e.  RR ,  v  e.  RR  |->  ( u  +  ( _i  x.  v ) ) )   &    |-  B  =  ran  ( x  e.  ran  [,)
 ,  y  e.  ran  [,)  |->  ( x  X.  y
 ) )   =>    |-  ( ( A  e.  ( J  tX  J ) 
 /\  X  e.  A )  ->  E. r  e.  B  ( X  e.  r  /\  r  C_  A ) )
 
18.3.21  Order topology - misc. additions
 
Theoremcnvordtrestixx 23299* The restriction of the 'greater than' order to an interval gives the same topology as the subspace topology. (Contributed by Thierry Arnoux, 1-Apr-2017.)
 |-  A  C_  RR*   &    |-  ( ( x  e.  A  /\  y  e.  A )  ->  ( x [,] y )  C_  A )   =>    |-  ( (ordTop `  <_  )t  A )  =  (ordTop `  ( `'  <_  i^i  ( A  X.  A ) ) )
 
18.3.22  Continuity in topological spaces - misc. additions
 
Theoremressplusf 23300 The group operation function  + f of a structure's restriction is the operation function's restriction to the new base. (Contributed by Thierry Arnoux, 26-Mar-2017.)
 |-  B  =  ( Base `  G )   &    |-  H  =  ( Gs  A )   &    |-  .+^  =  ( +g  `  G )   &    |-  .+^  Fn  ( B  X.  B )   &    |-  A  C_  B   =>    |-  ( + f `  H )  =  (  .+^  |`  ( A  X.  A ) )
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