HomeHome Metamath Proof Explorer
Theorem List (p. 374 of 402)
< Previous  Next >
Browser slow? Try the
Unicode version.

Mirrors  >  Metamath Home Page  >  MPE Home Page  >  Theorem List Contents  >  Recent Proofs       This page: Page List

Color key:    Metamath Proof Explorer  Metamath Proof Explorer
(1-26506)
  Hilbert Space Explorer  Hilbert Space Explorer
(26507-28029)
  Users' Mathboxes  Users' Mathboxes
(28030-40127)
 

Theorem List for Metamath Proof Explorer - 37301-37400   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremrexsngf 37301* Restricted existential quantification over a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ps   &    |-  ( x  =  A  ->  ( ph  <->  ps ) )   =>    |-  ( A  e.  V  ->  ( E. x  e. 
 { A } ph  <->  ps ) )
 
Theoremuzwo4 37302* Well-ordering principle: any nonempty subset of an upper set of integers has the least element. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ j ps   &    |-  ( j  =  k  ->  ( ph  <->  ps ) )   =>    |-  ( ( S  C_  ( ZZ>= `  M )  /\  E. j  e.  S  ph )  ->  E. j  e.  S  ( ph  /\  A. k  e.  S  (
 k  <  j  ->  -. 
 ps ) ) )
 
Theorem0in 37303 The intersection of the empty set with a class is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( (/) 
 i^i  A )  =  (/)
 
Theoremunisn0 37304 The union of the singleton of the empty set is the empty set. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  U. { (/)
 }  =  (/)
 
Theoremssin0 37305 If two classes are disjoint, two respective subclasses are disjoint. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ( A  i^i  B )  =  (/)  /\  C  C_  A  /\  D  C_  B )  ->  ( C  i^i  D )  =  (/) )
 
Theoreminabs3 37306 Absorption law for intersection. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( C  C_  B  ->  (
 ( A  i^i  B )  i^i  C )  =  ( A  i^i  C ) )
 
Theorempwpwuni 37307 Relationship between power class and union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  e.  V  ->  ( A  e.  ~P ~P B 
 <-> 
 U. A  e.  ~P B ) )
 
Theoremdisjiun2 37308* In a disjoint collection, an indexed union is disjoint from an additional term. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  -> Disj  x  e.  A  B )   &    |-  ( ph  ->  C  C_  A )   &    |-  ( ph  ->  D  e.  ( A  \  C ) )   &    |-  ( x  =  D  ->  B  =  E )   =>    |-  ( ph  ->  (
 U_ x  e.  C  B  i^i  E )  =  (/) )
 
Theorem0pwfi 37309 The empty set is in any power set, and it's finite. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (/)  e.  ( ~P A  i^i  Fin )
 
Theoremssinss2d 37310 Intersection preserves subclass relationship. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  B  C_  C )   =>    |-  ( ph  ->  ( A  i^i  B )  C_  C )
 
Theoremzct 37311 The set of integer numbers is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ZZ  ~<_  om
 
Theoremiunxsngf2 37312* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  ( A  e.  V  -> 
 U_ x  e.  { A } B  =  C )
 
Theorempwfin0 37313 A finite set always belongs to a power class. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ~P A  i^i  Fin )  =/= 
 (/)
 
Theoremuzct 37314 An upper integer set is countable. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  Z  =  ( ZZ>= `  N )   =>    |-  Z  ~<_  om
 
Theoremiunxsnf 37315* A singleton index picks out an instance of an indexed union's argument. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  A  e.  _V   &    |-  ( x  =  A  ->  B  =  C )   =>    |-  U_ x  e.  { A } B  =  C
 
Theoremfiiuncl 37316* If a set is closed under the union of two sets, then it is closed under finite indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  D )   &    |-  ( ( ph  /\  y  e.  D  /\  z  e.  D )  ->  (
 y  u.  z )  e.  D )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/= 
 (/) )   =>    |-  ( ph  ->  U_ x  e.  A  B  e.  D )
 
Theoremiunp1 37317* The addition of the next set to a union indexed by a finite set of sequential integers. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ k B   &    |-  ( ph  ->  N  e.  ( ZZ>= `  M )
 )   &    |-  ( k  =  ( N  +  1 ) 
 ->  A  =  B )   =>    |-  ( ph  ->  U_ k  e.  ( M ... ( N  +  1 )
 ) A  =  (
 U_ k  e.  ( M ... N ) A  u.  B ) )
 
Theoremfiunicl 37318* If a set is closed under the union of two sets, then it is closed under finite union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A  /\  y  e.  A )  ->  ( x  u.  y )  e.  A )   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  =/=  (/) )   =>    |-  ( ph  ->  U. A  e.  A )
 
Theoremnnnfi 37319 The set of positive integers is infinite. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  -.  NN  e.  Fin
 
Theoremixpeq2d 37320 Equality theorem for infinite Cartesian product. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  =  C )   =>    |-  ( ph  ->  X_ x  e.  A  B  =  X_ x  e.  A  C )
 
Theoremdisjxp1 37321* The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  -> Disj  x  e.  A  ( B  X.  C ) )
 
Theoremelexd 37322 If a class is a member of another class, it is a set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   =>    |-  ( ph  ->  A  e.  _V )
 
Theoremelpwd 37323 Membership in a power class. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  A  C_  B )   =>    |-  ( ph  ->  A  e.  ~P B )
 
Theoremdisjsnxp 37324* The sets in the cartesian product of singletons with other sets, are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |- Disj  j  e.  A  ( { j }  X.  B )
 
Theoremssfid 37325 A subset of a finite set is finite. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  B  C_  A )   =>    |-  ( ph  ->  B  e.  Fin )
 
21.30.2  Functions
 
Theoremunima 37326 Image of a union. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F  Fn  A  /\  B  C_  A  /\  C  C_  A )  ->  ( F " ( B  u.  C ) )  =  ( ( F
 " B )  u.  ( F " C ) ) )
 
Theoremfeq1dd 37327 Equality deduction for functions. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  =  G )   &    |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  G : A --> B )
 
Theoremfnresdmss 37328 A function does not change when restricted to a set that contains its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F  Fn  A  /\  A  C_  B )  ->  ( F  |`  B )  =  F )
 
Theoremfmptsnxp 37329* Maps-to notation and cross product for a singleton function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  ( x  e.  { A }  |->  B )  =  ( { A }  X.  { B }
 ) )
 
Theoremmptex2 37330* If a class given as a map-to notation is a set, it's image values are set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  ( t  e.  A  |->  B ) : A --> C )   =>    |-  ( ( ph  /\  t  e.  A ) 
 ->  B  e.  C )
 
Theoremfvmpt2bd 37331* Value of a function given by the "maps to" notation. Deduction version. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  F  =  ( x  e.  A  |->  B ) )   =>    |-  ( ( ph  /\  x  e.  A  /\  B  e.  C )  ->  ( F `
  x )  =  B )
 
Theoremrnmptfi 37332* The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  A  =  ( x  e.  B  |->  C )   =>    |-  ( B  e.  Fin  ->  ran  A  e.  Fin )
 
Theoremfresin2 37333 Restriction of a function with respect to the intersection with its domain. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( F : A --> B  ->  ( F  |`  ( C  i^i  A ) )  =  ( F  |`  C ) )
 
Theoremrnmptc 37334* Range of a constant function in map to notation. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   &    |-  ( ph  ->  A  =/=  (/) )   =>    |-  ( ph  ->  ran  F  =  { B } )
 
Theoremffi 37335 A function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F : A --> B  /\  A  e.  Fin )  ->  F  e.  Fin )
 
Theoremsuprnmpt 37336* An explicit bound for the range of a bounded function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  =/=  (/) )   &    |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  RR )   &    |-  ( ph  ->  E. y  e.  RR  A. x  e.  A  B  <_  y )   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  C  =  sup ( ran  F ,  RR ,  <  )   =>    |-  ( ph  ->  ( C  e.  RR  /\  A. x  e.  A  B  <_  C ) )
 
Theoremrnffi 37337 The range of a function with finite domain is finite. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( F : A --> B  /\  A  e.  Fin )  ->  ran  F  e.  Fin )
 
Theoremmptelpm 37338* A function in maps-to notation is a partial map . (Contributed by Glauco Siliprandi, 5-Apr-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  C )   &    |-  ( ph  ->  A 
 C_  D )   &    |-  ( ph  ->  C  e.  V )   &    |-  ( ph  ->  D  e.  W )   =>    |-  ( ph  ->  ( x  e.  A  |->  B )  e.  ( C  ^pm  D ) )
 
Theoremf1oeq3d 37339 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : C -1-1-onto-> A  <->  F : C -1-1-onto-> B ) )
 
Theoremfimass 37340 The image of a class is a subset of its codomain. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( F : A --> B  ->  ( F " X ) 
 C_  B )
 
Theoremrnmptpr 37341* Range of a function defined on an unordered pair. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  F  =  ( x  e.  { A ,  B }  |->  C )   &    |-  ( x  =  A  ->  C  =  D )   &    |-  ( x  =  B  ->  C  =  E )   =>    |-  ( ph  ->  ran  F  =  { D ,  E }
 )
 
Theoremffnd 37342 A mapping is a function. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F : A --> B )   =>    |-  ( ph  ->  F  Fn  A )
 
Theoremresmpti 37343* Restriction of the mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  B  C_  A   =>    |-  ( ( x  e.  A  |->  C )  |`  B )  =  ( x  e.  B  |->  C )
 
Theoremfouniiun 37344* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( F : A -onto-> B  ->  U. B  =  U_ x  e.  A  ( F `  x ) )
 
Theoremf1oeq2d 37345 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  =  B )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> C  <->  F : B -1-1-onto-> C ) )
 
Theoremrnresun 37346 Distribution law for range of a restriction over a union. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ran  ( F  |`  ( A  u.  B ) )  =  ( ran  ( F  |`  A )  u. 
 ran  ( F  |`  B ) )
 
Theoremf1oeq1d 37347 Equality deduction for one-to-one onto functions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  =  G )   =>    |-  ( ph  ->  ( F : A -1-1-onto-> B  <->  G : A -1-1-onto-> B ) )
 
Theoremdffo3f 37348* An onto mapping expressed in terms of function values. As dffo3 5989 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x F   =>    |-  ( F : A -onto-> B 
 <->  ( F : A --> B  /\  A. y  e.  B  E. x  e.  A  y  =  ( F `  x ) ) )
 
Theoremrnresss 37349 The range of a restriction is a subset of the whole range. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ran  ( A  |`  B ) 
 C_  ran  A
 
Theoremmpteq1i 37350* An equality theorem for the maps to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  A  =  B   =>    |-  ( x  e.  A  |->  C )  =  ( x  e.  B  |->  C )
 
Theoremelrnmptd 37351* The range of a function in maps-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ph  ->  E. x  e.  A  C  =  B )   &    |-  ( ph  ->  C  e.  V )   =>    |-  ( ph  ->  C  e.  ran  F )
 
Theoremelrnmptf 37352 The range of a function in maps-to notation. Same as elrnmpt 5036, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  F  =  ( x  e.  A  |->  B )   =>    |-  ( C  e.  V  ->  ( C  e.  ran  F  <->  E. x  e.  A  C  =  B )
 )
 
Theoremmptss 37353* Sufficient condition for inclusion in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( A  C_  B  ->  ( x  e.  A  |->  C ) 
 C_  ( x  e.  B  |->  C ) )
 
Theoremrnmptssrn 37354* Inclusion relation for two ranges expressed in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  E. y  e.  C  B  =  D )   =>    |-  ( ph  ->  ran  ( x  e.  A  |->  B )  C_  ran  ( y  e.  C  |->  D ) )
 
Theoremdisjf1 37355* A 1 to 1 mapping built from disjoint, nonempty sets . (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ( ph  /\  x  e.  A )  ->  B  =/= 
 (/) )   &    |-  ( ph  -> Disj  x  e.  A  B )   =>    |-  ( ph  ->  F : A -1-1-> V )
 
Theoremrnsnf 37356 The range of a function whose domain is a singleton. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : { A } --> B )   =>    |-  ( ph  ->  ran  F  =  { ( F `  A ) } )
 
Theoremwessf1ornlem 37357* Given a function  F on a well ordered domain  A there exists a subset of  A such that  F restricted to such subset is injective and onto the range of  F (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  R  We  A )   &    |-  G  =  ( y  e.  ran  F 
 |->  ( iota_ x  e.  ( `' F " { y } ) A. z  e.  ( `' F " { y } )  -.  z R x ) )   =>    |-  ( ph  ->  E. x  e.  ~P  A ( F  |`  x ) : x -1-1-onto-> ran  F )
 
Theoremwessf1orn 37358* Given a function  F on a well ordered domain  A there exists a subset of  A such that  F restricted to such subset is injective and onto the range of  F (without using the axiom of choice). (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( ph  ->  F  Fn  A )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  R  We  A )   =>    |-  ( ph  ->  E. x  e.  ~P  A ( F  |`  x ) : x -1-1-onto-> ran  F )
 
Theoremfoelrnf 37359* Property of a surjective function. As foelrn 5993 but with less disjoint vars constraints. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x F   =>    |-  ( ( F : A -onto-> B  /\  C  e.  B )  ->  E. x  e.  A  C  =  ( F `  x ) )
 
Theoremnelrnres 37360 If  A is not in the range, it is not in the range of any restriction. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( -.  A  e.  ran  B  ->  -.  A  e.  ran  ( B  |`  C ) )
 
Theoremdisjrnmpt2 37361* Disjointness of the range of a function in map-to notation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  B )   =>    |-  (Disj  x  e.  A  B  -> Disj  y  e.  ran  F  y )
 
Theoremelrnmpt1sf 37362* Elementhood in an image set. Same as elrnmpt1s 5037, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/_ x C   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( x  =  D  ->  B  =  C )   =>    |-  ( ( D  e.  A  /\  C  e.  V )  ->  C  e.  ran  F )
 
Theoremfouniiun0 37363* Union expressed as an indexed union, when a map onto is given. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  ( F : A -onto-> ( B  u.  { (/) } )  ->  U. B  =  U_ x  e.  A  ( F `  x ) )
 
Theoremdisjf1o 37364* A bijection built from disjoint sets. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  V )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  C  =  { x  e.  A  |  B  =/=  (/) }   &    |-  D  =  ( ran  F  \  { (/)
 } )   =>    |-  ( ph  ->  ( F  |`  C ) : C -1-1-onto-> D )
 
Theoremfompt 37365* Express being onto for a mapping operation. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  F  =  ( x  e.  A  |->  C )   =>    |-  ( F : A -onto-> B 
 <->  ( A. x  e.  A  C  e.  B  /\  A. y  e.  B  E. x  e.  A  y  =  C )
 )
 
Theoremdisjinfi 37366* Only a finite number of disjoint sets can have a non empty intersection with a finite set  C (Contributed by Glauco Siliprandi, 17-Aug-2020.)
 |-  (
 ( ph  /\  x  e.  A )  ->  B  e.  V )   &    |-  ( ph  -> Disj  x  e.  A  B )   &    |-  ( ph  ->  C  e.  Fin )   =>    |-  ( ph  ->  { x  e.  A  |  ( B  i^i  C )  =/=  (/) }  e.  Fin )
 
Theoremfvovco 37367 Value of the composition of an operator, with a given function. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  F : X --> ( V  X.  W ) )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  ( ( O  o.  F ) `  Y )  =  ( ( 1st `  ( F `  Y ) ) O ( 2nd `  ( F `  Y ) ) ) )
 
Theoremssnnf1octb 37368* There exists a bijection between a subset of  NN and a given nonempty countable set. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  (
 ( A  ~<_  om  /\  A  =/=  (/) )  ->  E. f
 ( dom  f  C_  NN  /\  f : dom  f -1-1-onto-> A ) )
 
Theoremmapdm0 37369 The empty set is the only map with empty domain. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( A  e.  V  ->  ( A  ^m  (/) )  =  { (/) } )
 
Theoremnnf1oxpnn 37370 There is a bijection between the set of positive integers and the Cartesian product of the set of positive integers with itself. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  E. f  f : NN -1-1-onto-> ( NN  X.  NN )
 
Theoremrnmptssd 37371* The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  F/ x ph   &    |-  F  =  ( x  e.  A  |->  B )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  B  e.  C )   =>    |-  ( ph  ->  ran  F  C_  C )
 
Theoremprojf1o 37372* A biijection from a set to a projection in a two dimensional space. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  F  =  ( x  e.  B  |->  <. A ,  x >. )   =>    |-  ( ph  ->  F : B -1-1-onto-> ( { A }  X.  B ) )
 
Theoremfvmap 37373 Function value for a member of a set exponentiation. (Contributed by Glauco Siliprandi, 21-Nov-2020.)
 |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  B  e.  W )   &    |-  ( ph  ->  F  e.  ( A  ^m  B ) )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  ( F `  C )  e.  A )
 
21.30.3  Ordering on real numbers - Real and complex numbers basic operations
 
Theoremsub2times 37374 Subtracting from a number, twice the number itself, gives negative the number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  CC  ->  ( A  -  ( 2  x.  A ) )  =  -u A )
 
Theoremxrltled 37375 'Less than' implies 'less than or equal to', for extended reals. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  A 
 <_  B )
 
Theoremabssubrp 37376 The distance of two distinct complex number is a strictly positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  A  =/=  B )  ->  ( abs `  ( A  -  B ) )  e.  RR+ )
 
Theoremelfzfzo 37377 Relationship between membership in a half open finite set of sequential integers and membership in a finite set of sequential intergers. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  ( M..^ N )  <->  ( A  e.  ( M ... N ) 
 /\  A  <  N ) )
 
Theoremoddfl 37378 Odd number representation by using the floor function. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( K  e.  ZZ  /\  ( K  mod  2
 )  =/=  0 )  ->  K  =  ( ( 2  x.  ( |_ `  ( K  /  2
 ) ) )  +  1 ) )
 
Theoremabscosbd 37379 Bound for the absolute value of the cosine of a real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR  ->  ( abs `  ( cos `  A ) )  <_ 
 1 )
 
Theoremmul13d 37380 Commutative/associative law that swaps the first and the third factor in a triple product. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( A  x.  ( B  x.  C ) )  =  ( C  x.  ( B  x.  A ) ) )
 
Theoremnegpilt0 37381 Negative  pi is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  -u pi  <  0
 
Theoremdstregt0 37382* A complex number  A that is not real, has a distance from the reals that is strictly larger than  0. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  ( CC  \  RR ) )   =>    |-  ( ph  ->  E. x  e.  RR+  A. y  e.  RR  x  <  ( abs `  ( A  -  y ) ) )
 
Theoremsubadd4b 37383 Rearrangement of 4 terms in a mixed addition and subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   =>    |-  ( ph  ->  (
 ( A  -  B )  +  ( C  -  D ) )  =  ( ( A  -  D )  +  ( C  -  B ) ) )
 
Theoremxrlttri5d 37384 Not equal and not larger implies smaller. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  A  =/=  B )   &    |-  ( ph  ->  -.  B  <  A )   =>    |-  ( ph  ->  A  <  B )
 
Theoremneglt 37385 The negative of a positive number is less than the number itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR+  ->  -u A  <  A )
 
Theoremzltlesub 37386 If an integer  N is smaller or equal to a real, and we subtract a quantity smaller than  1, then  N is smaller or equal to the result. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  N 
 <_  A )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  <  1
 )   &    |-  ( ph  ->  ( A  -  B )  e. 
 ZZ )   =>    |-  ( ph  ->  N  <_  ( A  -  B ) )
 
Theoremdivlt0gt0d 37387 The ratio of a negative numerator and a positive denominator is negative. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  A  <  0 )   =>    |-  ( ph  ->  ( A  /  B )  <  0 )
 
Theorem3rp 37388 3 is a positive real. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  3  e.  RR+
 
TheoremleimnltdOLD 37389 'Less than or equal to' implies 'not less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.) Obsolete as of 18-Jul-2020. Use lensymd 9730 instead, which is the same. (New usage is discouraged.) (Proof modification is discouraged.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  A 
 <_  B )   =>    |-  ( ph  ->  -.  B  <  A )
 
Theoremlt2addmuld 37390 If two real numbers are less than a third real number, the sum of the two real numbers is less than twice the third real number. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <  C )   &    |-  ( ph  ->  B  <  C )   =>    |-  ( ph  ->  ( A  +  B )  <  ( 2  x.  C ) )
 
Theoremsubsub23d 37391 Swap subtrahend and result of subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   =>    |-  ( ph  ->  ( ( A  -  B )  =  C  <->  ( A  -  C )  =  B ) )
 
Theorem2timesgt 37392 Double of a positive real is larger than the real itself. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( A  e.  RR+  ->  A  <  ( 2  x.  A ) )
 
Theoremreopn 37393 The reals are open with respect to the standard topology. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  RR  e.  ( topGen `  ran  (,) )
 
Theoremelfzop1le2 37394 A member in a half-open integer interval plus 1 is less or equal than the upper bound. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( K  e.  ( M..^ N )  ->  ( K  +  1 )  <_  N )
 
Theoremsub31 37395 Swap the first and third terms in a double subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  -  ( B  -  C ) )  =  ( C  -  ( B  -  A ) ) )
 
Theoremnnne1ge2 37396 A positive integer which is not 1 is greater than or equal to 2. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( N  e.  NN  /\  N  =/=  1 ) 
 ->  2  <_  N )
 
Theoremlefldiveq 37397 A closed enough, smaller real  C has the same floor of  A when both are divided by  B. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  ( ( A  -  ( A  mod  B ) ) [,] A ) )   =>    |-  ( ph  ->  ( |_ `  ( A  /  B ) )  =  ( |_ `  ( C  /  B ) ) )
 
Theoremnegsubdi3d 37398 Distribution of negative over subtraction. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   =>    |-  ( ph  ->  -u ( A  -  B )  =  ( -u A  -  -u B ) )
 
Theoremltaddneg 37399 Adding a negative number to another number decreases it. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR )  ->  ( A  <  0  <->  ( B  +  A )  <  B ) )
 
Theoremltdiv2dd 37400 Division of a positive number by both sides of 'less than'. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
 |-  ( ph  ->  A  e.  RR+ )   &    |-  ( ph  ->  B  e.  RR+ )   &    |-  ( ph  ->  C  e.  RR+ )   &    |-  ( ph  ->  A  <  B )   =>    |-  ( ph  ->  ( C  /  B )  <  ( C  /  A ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32300 324 32301-32400 325 32401-32500 326 32501-32600 327 32601-32700 328 32701-32800 329 32801-32900 330 32901-33000 331 33001-33100 332 33101-33200 333 33201-33300 334 33301-33400 335 33401-33500 336 33501-33600 337 33601-33700 338 33701-33800 339 33801-33900 340 33901-34000 341 34001-34100 342 34101-34200 343 34201-34300 344 34301-34400 345 34401-34500 346 34501-34600 347 34601-34700 348 34701-34800 349 34801-34900 350 34901-35000 351 35001-35100 352 35101-35200 353 35201-35300 354 35301-35400 355 35401-35500 356 35501-35600 357 35601-35700 358 35701-35800 359 35801-35900 360 35901-36000 361 36001-36100 362 36101-36200 363 36201-36300 364 36301-36400 365 36401-36500 366 36501-36600 367 36601-36700 368 36701-36800 369 36801-36900 370 36901-37000 371 37001-37100 372 37101-37200 373 37201-37300 374 37301-37400 375 37401-37500 376 37501-37600 377 37601-37700 378 37701-37800 379 37801-37900 380 37901-38000 381 38001-38100 382 38101-38200 383 38201-38300 384 38301-38400 385 38401-38500 386 38501-38600 387 38601-38700 388 38701-38800 389 38801-38900 390 38901-39000 391 39001-39100 392 39101-39200 393 39201-39300 394 39301-39400 395 39401-39500 396 39501-39600 397 39601-39700 398 39701-39800 399 39801-39900 400 39901-40000 401 40001-40100 402 40101-40127
  Copyright terms: Public domain < Previous  Next >