HomeHome Metamath Proof Explorer
Theorem List (p. 276 of 378)
< 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
(1-25813)
  Hilbert Space Explorer  Hilbert Space Explorer
(25814-27338)
  Users' Mathboxes  Users' Mathboxes
(27339-37797)
 

Theorem List for Metamath Proof Explorer - 27501-27600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremdf2ndres 27501* 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 )
 
Theorem1stpreima 27502 The preimage by  1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  ( A  C_  B  ->  ( `' ( 1st  |`  ( B  X.  C ) )
 " A )  =  ( A  X.  C ) )
 
Theorem2ndpreima 27503 The preimage by  2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  ( A  C_  C  ->  ( `' ( 2nd  |`  ( B  X.  C ) )
 " A )  =  ( B  X.  A ) )
 
Theoremcurry2ima 27504* 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 ) } )
 
21.3.4.7  Supremum - misc additions
 
Theoremsupssd 27505* 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 ) )
 
21.3.4.8  Finite Sets
 
Theoremunifi3 27506 If a union is finite, then all its elements are finite. See unifi 7811. (Contributed by Thierry Arnoux, 27-Aug-2017.)
 |-  ( U. A  e.  Fin  ->  A  C_  Fin )
 
21.3.4.9  Countable Sets
 
Theoremnnct 27507  NN is countable (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  NN  ~<_  om
 
Theoremfict 27508 A finite set is countable (weaker version of isfinite 8072). (Contributed by Thierry Arnoux, 27-Mar-2018.)
 |-  ( A  e.  Fin  ->  A  ~<_  om )
 
Theoremctex 27509 A countable set is a set (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  A  e.  _V )
 
Theoremssct 27510 Any subset of a countable set is countable (Contributed by Thierry Arnoux, 31-Jan-2017.)
 |-  (
 ( A  C_  B  /\  B  ~<_  om )  ->  A  ~<_  om )
 
Theoremxpct 27511 The cartesian product of two countable sets is countable. (Contributed by Thierry Arnoux, 24-Sep-2017.)
 |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( A  X.  B )  ~<_  om )
 
Theoremsnct 27512 A singleton is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  ( A  e.  V  ->  { A }  ~<_  om )
 
Theoremprct 27513 An unordered pair is countable (Contributed by Thierry Arnoux, 16-Sep-2016.)
 |-  (
 ( A  e.  V  /\  B  e.  W ) 
 ->  { A ,  B } 
 ~<_  om )
 
Theoremfnct 27514 If the domain of a function is countable, the function is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  (
 ( F  Fn  A  /\  A  ~<_  om )  ->  F  ~<_  om )
 
Theoremdmct 27515 The domain of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  dom  A  ~<_  om )
 
Theoremcnvct 27516 If a set is countable, so is its converse. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  `' A  ~<_  om )
 
Theoremrnct 27517 The range of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ran  A  ~<_  om )
 
Theoremfimact 27518 The image by a function of a countable set is countable. (Contributed by Thierry Arnoux, 27-Mar-2018.)
 |-  (
 ( A  ~<_  om  /\  Fun 
 F )  ->  ( F " A )  ~<_  om )
 
Theoremmptct 27519* A countable mapping set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremmpt2cti 27520* An operation is countable if both its domains are countable. (Contributed by Thierry Arnoux, 17-Sep-2017.)
 |-  A. x  e.  A  A. y  e.  B  C  e.  V   =>    |-  (
 ( A  ~<_  om  /\  B 
 ~<_  om )  ->  ( x  e.  A ,  y  e.  B  |->  C )  ~<_ 
 om )
 
Theoremabrexct 27521* An image set of a countable set is countable. (Contributed by Thierry Arnoux, 29-Dec-2016.)
 |-  ( A 
 ~<_  om  ->  { y  |  E. x  e.  A  y  =  B }  ~<_  om )
 
Theoremmptctf 27522 A countable mapping set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  ( x  e.  A  |->  B )  ~<_  om )
 
Theoremabrexctf 27523* An image set of a countable set is countable, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Thierry Arnoux, 8-Mar-2017.)
 |-  F/_ x A   =>    |-  ( A  ~<_  om  ->  { y  |  E. x  e.  A  y  =  B } 
 ~<_  om )
 
Theoremcnvoprab 27524* The converse of a class abstraction of nested ordered pairs. (Contributed by Thierry Arnoux, 17-Aug-2017.)
 |-  F/ x ps   &    |-  F/ y ps   &    |-  ( a  =  <. x ,  y >.  ->  ( ps 
 <-> 
 ph ) )   &    |-  ( ps  ->  a  e.  ( _V  X.  _V ) )   =>    |-  `' { <. <. x ,  y >. ,  z >.  |  ph }  =  { <. z ,  a >.  |  ps }
 
Theoremf1od2 27525* Describe an implicit one-to-one onto function of two variables. (Contributed by Thierry Arnoux, 17-Aug-2017.)
 |-  F  =  ( x  e.  A ,  y  e.  B  |->  C )   &    |-  ( ( ph  /\  ( x  e.  A  /\  y  e.  B ) )  ->  C  e.  W )   &    |-  ( ( ph  /\  z  e.  D ) 
 ->  ( I  e.  X  /\  J  e.  Y ) )   &    |-  ( ph  ->  ( ( ( x  e.  A  /\  y  e.  B )  /\  z  =  C )  <->  ( z  e.  D  /\  ( x  =  I  /\  y  =  J ) ) ) )   =>    |-  ( ph  ->  F : ( A  X.  B ) -1-1-onto-> D )
 
Theoremfcobij 27526* Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)
 |-  ( ph  ->  G : S -1-1-onto-> T )   &    |-  ( ph  ->  R  e.  U )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  T  e.  W )   =>    |-  ( ph  ->  (
 f  e.  ( S 
 ^m  R )  |->  ( G  o.  f ) ) : ( S 
 ^m  R ) -1-1-onto-> ( T 
 ^m  R ) )
 
Theoremfcobijfs 27527* Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfienOLD 8141. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
 |-  ( ph  ->  G : S -1-1-onto-> T )   &    |-  ( ph  ->  R  e.  U )   &    |-  ( ph  ->  S  e.  V )   &    |-  ( ph  ->  T  e.  W )   &    |-  ( ph  ->  O  e.  S )   &    |-  Q  =  ( G `  O )   &    |-  X  =  { g  e.  ( S  ^m  R )  |  g finSupp  O }   &    |-  Y  =  { h  e.  ( T  ^m  R )  |  h finSupp  Q }   =>    |-  ( ph  ->  (
 f  e.  X  |->  ( G  o.  f ) ) : X -1-1-onto-> Y )
 
Theoremsuppss3 27528* Deduce a function's support's inclusion in another function's support. (Contributed by Thierry Arnoux, 7-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
 |-  G  =  ( x  e.  A  |->  B )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  Z  e.  W )   &    |-  ( ph  ->  F  Fn  A )   &    |-  ( ( ph  /\  x  e.  A  /\  ( F `  x )  =  Z )  ->  B  =  Z )   =>    |-  ( ph  ->  ( G supp  Z )  C_  ( F supp  Z ) )
 
Theoremffs2 27529 Rewrite a function's support based with its range rather than the universal class. See also frnsuppeq 6915. (Contributed by Thierry Arnoux, 27-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
 |-  C  =  ( B  \  { Z } )   =>    |-  ( ( A  e.  V  /\  Z  e.  W  /\  F : A --> B ) 
 ->  ( F supp  Z )  =  ( `' F " C ) )
 
Theoremffsrn 27530 The range of a finitely supported function is finite. (Contributed by Thierry Arnoux, 27-Aug-2017.)
 |-  ( ph  ->  Z  e.  W )   &    |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ( F supp  Z )  e. 
 Fin )   =>    |-  ( ph  ->  ran  F  e.  Fin )
 
Theoremresf1o 27531* Restriction of functions to a superset of their support creates a bijection. (Contributed by Thierry Arnoux, 12-Sep-2017.)
 |-  X  =  { f  e.  ( B  ^m  A )  |  ( `' f "
 ( B  \  { Z } ) )  C_  C }   &    |-  F  =  ( f  e.  X  |->  ( f  |`  C )
 )   =>    |-  ( ( ( A  e.  V  /\  B  e.  W  /\  C  C_  A )  /\  Z  e.  B )  ->  F : X
 -1-1-onto-> ( B  ^m  C ) )
 
Theoremmaprnin 27532* Restricting the range of the mapping operator (Contributed by Thierry Arnoux, 30-Aug-2017.)
 |-  A  e.  _V   &    |-  B  e.  _V   =>    |-  (
 ( B  i^i  C )  ^m  A )  =  { f  e.  ( B  ^m  A )  | 
 ran  f  C_  C }
 
Theoremfpwrelmapffslem 27533* Lemma for fpwrelmapffs 27535. For this theorem, the sets  A and  B could be infinite, but the relation  R itself is finite. (Contributed by Thierry Arnoux, 1-Sep-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  ( ph  ->  F : A --> ~P B )   &    |-  ( ph  ->  R  =  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( F `  x ) ) } )   =>    |-  ( ph  ->  ( R  e.  Fin  <->  ( ran  F  C_  Fin  /\  ( F supp  (/) )  e. 
 Fin ) ) )
 
Theoremfpwrelmap 27534* Define a canonical mapping between functions from  A into subsets of  B and the relations with domain  A and range within  B. Note that the same relation is used in axdc2lem 8831 and marypha2lem1 7897. (Contributed by Thierry Arnoux, 28-Aug-2017.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  M  =  ( f  e.  ( ~P B  ^m  A ) 
 |->  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( f `  x ) ) } )   =>    |-  M : ( ~P B  ^m  A ) -1-1-onto-> ~P ( A  X.  B )
 
Theoremfpwrelmapffs 27535* Define a canonical mapping between finite relations (finite subsets of a cartesian product) and functions with finite support into finite subsets (Contributed by Thierry Arnoux, 28-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)
 |-  A  e.  _V   &    |-  B  e.  _V   &    |-  M  =  ( f  e.  ( ~P B  ^m  A ) 
 |->  { <. x ,  y >.  |  ( x  e.  A  /\  y  e.  ( f `  x ) ) } )   &    |-  S  =  { f  e.  (
 ( ~P B  i^i  Fin )  ^m  A )  |  ( f supp  (/) )  e. 
 Fin }   =>    |-  ( M  |`  S ) : S -1-1-onto-> ( ~P ( A  X.  B )  i^i 
 Fin )
 
21.3.5  Real and Complex Numbers
 
21.3.5.1  Complex operations - misc. additions
 
Theoremaddeq0 27536 Two complex which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  (
 ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A  +  B )  =  0  <->  A  =  -u B ) )
 
Theoremsubeqxfrd 27537 Transfer two terms of a subtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( A  -  B )  =  ( C  -  D ) )   =>    |-  ( ph  ->  ( A  -  C )  =  ( B  -  D ) )
 
Theoremaddeqxfrd 27538 Transfer two terms of an addtraction in an equality. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  A  e.  CC )   &    |-  ( ph  ->  B  e.  CC )   &    |-  ( ph  ->  C  e.  CC )   &    |-  ( ph  ->  D  e.  CC )   &    |-  ( ph  ->  ( A  +  B )  =  ( C  +  D ) )   =>    |-  ( ph  ->  ( A  -  D )  =  ( C  -  B ) )
 
Theoremznsqcld 27539 Squaring of non-zero relative numbers. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  ( ph  ->  N  e.  ZZ )   &    |-  ( ph  ->  N  =/=  0 )   =>    |-  ( ph  ->  ( N ^ 2 )  e. 
 NN )
 
Theoremnn0sqeq1 27540 Integer square one. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  (
 ( N  e.  NN0  /\  ( N ^ 2
 )  =  1 ) 
 ->  N  =  1 )
 
21.3.5.2  Ordering on reals - misc additions
 
Theoremlt2addrd 27541* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand 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 ) )
 
Theoremnegelrp 27542 Elementhood of a negation in the positive real numbers (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( A  e.  RR  ->  (
 -u A  e.  RR+  <->  A  <  0 ) )
 
Theoremmul2lt0rlt0 27543 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( A  x.  B )  <  0 )   =>    |-  ( ( ph  /\  B  <  0 ) 
 ->  0  <  A )
 
Theoremmul2lt0rgt0 27544 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( A  x.  B )  <  0 )   =>    |-  ( ( ph  /\  0  <  B ) 
 ->  A  <  0 )
 
Theoremmul2lt0llt0 27545 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( A  x.  B )  <  0 )   =>    |-  ( ( ph  /\  A  <  0 ) 
 ->  0  <  B )
 
Theoremmul2lt0lgt0 27546 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 2-Oct-2018.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  ( A  x.  B )  <  0 )   =>    |-  ( ( ph  /\  0  <  A ) 
 ->  B  <  0 )
 
Theoremmul2lt0bi 27547 If the result of a multiplication is strictly negative, then multiplicands are of different signs. (Contributed by Thierry Arnoux, 19-Sep-2018.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   =>    |-  ( ph  ->  (
 ( A  x.  B )  <  0  <->  ( ( A  <  0  /\  0  <  B )  \/  (
 0  <  A  /\  B  <  0 ) ) ) )
 
21.3.5.3  Extended reals - misc additions
 
Theoremxgepnf 27548 An extended real which is greater than plus infinity is plus infinity. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  ( A  e.  RR*  ->  ( +oo  <_  A  <->  A  = +oo ) )
 
Theoremxlemnf 27549 An extended real which is less than minus infinity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  ( A  e.  RR*  ->  ( A  <_ -oo  <->  A  = -oo ) )
 
Theoremxrlelttric 27550 Trichotomy law for extended reals. (Contributed by Thierry Arnoux, 12-Sep-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( A  <_  B  \/  B  <  A ) )
 
Theoremxaddeq0 27551 Two extended reals which add up to zero are each other's negatives. (Contributed by Thierry Arnoux, 13-Jun-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR* )  ->  ( ( A +e B )  =  0  <->  A  =  -e B ) )
 
Theoreminfxrmnf 27552 The infinimum of a set of extended reals containing minus infnity is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  (
 ( A  C_  RR*  /\ -oo  e.  A )  ->  sup ( A ,  RR* ,  `'  <  )  = -oo )
 
Theoremxrinfm 27553 The extended real numbers are unbounded below. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  sup ( RR* ,  RR* ,  `'  <  )  = -oo
 
Theoremle2halvesd 27554 A sum is less than the whole if each term is less than half. (Contributed by Thierry Arnoux, 29-Nov-2017.)
 |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  A  <_  ( C  /  2 ) )   &    |-  ( ph  ->  B  <_  ( C  /  2 ) )   =>    |-  ( ph  ->  ( A  +  B )  <_  C )
 
Theoremxraddge02 27555 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 27556* If the right-hand side of a 'less than' relationship is an addition, then we can express the left-hand 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 27557 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*
 ,  <  ) )
 
Theoremxrge0infss 27558* Any subset of nonnegative extended reals has an infimum. (Contributed by Thierry Arnoux, 16-Sep-2019.)
 |-  ( A  C_  ( 0 [,] +oo )  ->  E. x  e.  ( 0 [,] +oo ) ( A. y  e.  A  -.  x `'  <  y  /\  A. y  e.  ( 0 [,] +oo ) ( y `' 
 <  x  ->  E. z  e.  A  y `'  <  z ) ) )
 
Theoremxrge0infssd 27559 Inequality deduction for infimum of a nonnegative extended real subset. (Contributed by Thierry Arnoux, 16-Sep-2019.)
 |-  ( ph  ->  C  C_  B )   &    |-  ( ph  ->  B  C_  ( 0 [,] +oo ) )   =>    |-  ( ph  ->  sup ( B ,  ( 0 [,] +oo ) ,  `'  <  )  <_  sup ( C ,  ( 0 [,] +oo ) ,  `'  <  ) )
 
Theoremxrofsup 27560 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 27561 The supremum of a nonempty 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 27562 The topology of the extended reals is Hausdorff. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  (ordTop ` 
 <_  )  e.  Haus
 
21.3.5.4  Real number intervals - misc additions
 
Theoremjoiniooico 27563 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 ) ) )
 
Theoremubico 27564 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 27565 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 27566 Relate elementhood to a closed interval with elementhood to the same closed-below, open-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 27567 Relate elementhood to a closed-below, open-above interval with elementhood to the same open 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 27568 Intersection between two open-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 27569 Set difference of a half-open interval in the extended reals. (Contributed by Thierry Arnoux, 1-Aug-2017.)
 |-  A  e.  RR*   =>    |-  ( RR*  \  ( A [,] +oo ) )  =  ( -oo [,) A )
 
Theoremiocinif 27570 Relate intersection of two open-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 27571 The difference between two open 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 ) )
 
Theoremdifico 27572 The difference between two closed-below, open-above intervals sharing the same upper bound (Contributed by Thierry Arnoux, 13-Oct-2017.)
 |-  (
 ( ( A  e.  RR*  /\  B  e.  RR*  /\  C  e.  RR* )  /\  ( A  <_  B  /\  B  <_  C ) )  ->  ( ( A [,) C )  \  ( B [,) C ) )  =  ( A [,) B ) )
 
21.3.5.5  Finite intervals of integers - misc additions
 
Theoremfzssnn 27573 Finite sets of sequential integers starting from a natural are a subset of the positive integers. (Contributed by Thierry Arnoux, 4-Aug-2017.)
 |-  ( M  e.  NN  ->  ( M ... N ) 
 C_  NN )
 
Theoremnndiffz1 27574 Upper set of the positive integers. (Contributed by Thierry Arnoux, 22-Aug-2017.)
 |-  ( N  e.  NN0  ->  ( NN  \  ( 1 ...
 N ) )  =  ( ZZ>= `  ( N  +  1 ) ) )
 
Theoremssnnssfz 27575* 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 ) )
 
Theoremfzspl 27576 Split the last element of a finite set of sequential integers. (more generic than fzsuc 11738) (Contributed by Thierry Arnoux, 7-Nov-2016.)
 |-  ( N  e.  ( ZZ>= `  M )  ->  ( M
 ... N )  =  ( ( M ... ( N  -  1
 ) )  u.  { N } ) )
 
Theoremfzsplit3 27577 Split a finite interval of integers into two parts. (Contributed by Thierry Arnoux, 2-May-2017.)
 |-  ( K  e.  ( M ... N )  ->  ( M ... N )  =  ( ( M ... ( K  -  1
 ) )  u.  ( K ... N ) ) )
 
Theorembcm1n 27578 The proportion of one binomial coefficient to another with  N decreased by 1. (Contributed by Thierry Arnoux, 9-Nov-2016.)
 |-  (
 ( K  e.  (
 0 ... ( N  -  1 ) )  /\  N  e.  NN )  ->  ( ( ( N  -  1 )  _C  K )  /  ( N  _C  K ) )  =  ( ( N  -  K )  /  N ) )
 
21.3.5.6  Half-open integer ranges - misc additions
 
Theoremiundisjfi 27579* Rewrite a countable union as a disjoint union, finite version. Cf. iundisj 21936 (Contributed by Thierry Arnoux, 15-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |-  U_ n  e.  ( 1..^ N ) A  =  U_ n  e.  ( 1..^ N ) ( A 
 \  U_ k  e.  (
 1..^ n ) B )
 
Theoremiundisj2fi 27580* A disjoint union is disjoint, finite version. Cf. iundisj2 21937 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   =>    |- Disj  n  e.  ( 1..^ N ) ( A  \  U_ k  e.  ( 1..^ n ) B )
 
Theoremiundisjcnt 27581* Rewrite a countable union as a disjoint union. (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  ->  U_ n  e.  N  A  =  U_ n  e.  N  ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
Theoremiundisj2cnt 27582* A countable disjoint union is disjoint. Cf. iundisj2 21937 (Contributed by Thierry Arnoux, 16-Feb-2017.)
 |-  F/_ n B   &    |-  ( n  =  k 
 ->  A  =  B )   &    |-  ( ph  ->  ( N  =  NN  \/  N  =  ( 1..^ M ) ) )   =>    |-  ( ph  -> Disj  n  e.  N  ( A  \  U_ k  e.  ( 1..^ n ) B ) )
 
21.3.5.7  The ` # ` (finite set size) function - misc additions
 
Theoremhashunif 27583* The cardinality of a disjoint finite union of finite sets. Cf. hashuni 13620 (Contributed by Thierry Arnoux, 17-Feb-2017.)
 |-  F/ x ph   &    |-  ( ph  ->  A  e.  Fin )   &    |-  ( ph  ->  A  C_  Fin )   &    |-  ( ph  -> Disj  x  e.  A  x )   =>    |-  ( ph  ->  ( # `
  U. A )  = 
 sum_ x  e.  A  ( # `  x ) )
 
Theoremishashinf 27584* Any set that is not finite contains subsets of arbitrarily large finite cardinality. Cf. isinf 7735 (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  ( -.  A  e.  Fin  ->  A. n  e.  NN  E. x  e.  ~P  A ( # `  x )  =  n )
 
21.3.5.8  The greatest common divisor operator - misc. add
 
Theoremgcdnncl 27585 Closure of the  gcd operator. (Contributed by Thierry Arnoux, 2-Feb-2020.)
 |-  (
 ( M  e.  NN  /\  N  e.  NN )  ->  ( M  gcd  N )  e.  NN )
 
Theoremnumdenneg 27586 Numerator and denominator of the negative (Contributed by Thierry Arnoux, 27-Oct-2017.)
 |-  ( Q  e.  QQ  ->  ( (numer `  -u Q )  =  -u (numer `  Q )  /\  (denom `  -u Q )  =  (denom `  Q ) ) )
 
Theoremdivnumden2 27587 Calculate the reduced form of a quotient using  gcd. This version extends divnumden 14263 for the negative integers. (Contributed by Thierry Arnoux, 25-Oct-2017.)
 |-  (
 ( A  e.  ZZ  /\  B  e.  ZZ  /\  -u B  e.  NN )  ->  ( (numer `  ( A  /  B ) )  =  -u ( A  /  ( A  gcd  B ) )  /\  (denom `  ( A  /  B ) )  =  -u ( B  /  ( A  gcd  B ) ) ) )
 
21.3.5.9  Integers
 
Theoremnnindf 27588* Principle of Mathematical Induction, using a bound-variable hypothesis instead of distinct variables. (Contributed by Thierry Arnoux, 6-May-2018.)
 |-  F/ y ph   &    |-  ( x  =  1  ->  ( ph  <->  ps ) )   &    |-  ( x  =  y  ->  ( ph  <->  ch ) )   &    |-  ( x  =  ( y  +  1 )  ->  ( ph  <->  th ) )   &    |-  ( x  =  A  ->  ( ph  <->  ta ) )   &    |-  ps   &    |-  ( y  e. 
 NN  ->  ( ch  ->  th ) )   =>    |-  ( A  e.  NN  ->  ta )
 
Theoremnn0min 27589* Extracting the minimum positive integer for which a property  ch does not hold. This uses substitutions similar to nn0ind 10966. (Contributed by Thierry Arnoux, 6-May-2018.)
 |-  ( n  =  0  ->  ( ps  <->  ch ) )   &    |-  ( n  =  m  ->  ( ps  <->  th ) )   &    |-  ( n  =  ( m  +  1 )  ->  ( ps  <->  ta ) )   &    |-  ( ph  ->  -.  ch )   &    |-  ( ph  ->  E. n  e.  NN  ps )   =>    |-  ( ph  ->  E. m  e.  NN0  ( -.  th  /\ 
 ta ) )
 
Theoremltesubnnd 27590 Subtracting an integer number from another number decreases it. See ltsubrpd 11295 (Contributed by Thierry Arnoux, 18-Apr-2017.)
 |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  N  e.  NN )   =>    |-  ( ph  ->  (
 ( M  +  1 )  -  N ) 
 <_  M )
 
21.3.5.10  Division in the extended real number system
 
Syntaxcxdiv 27591 Extend class notation to include division of extended reals.
 class /𝑒
 
Definitiondf-xdiv 27592* Define division over extended real numbers. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |- /𝑒  =  ( x  e.  RR* ,  y  e.  ( RR  \  {
 0 } )  |->  (
 iota_ z  e.  RR*  (
 y xe z )  =  x ) )
 
Theoremxdivval 27593* Value of division: the (unique) element  x such that  ( B  x.  x )  =  A. This is meaningful only when  B is nonzero. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  (
 iota_ x  e.  RR*  ( B xe x )  =  A ) )
 
Theoremxrecex 27594* Existence of reciprocal of nonzero real number. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  A  =/=  0 ) 
 ->  E. x  e.  RR  ( A xe x )  =  1 )
 
Theoremxmulcand 27595 Cancellation law for extended multiplication. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR* )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  C  =/=  0
 )   =>    |-  ( ph  ->  (
 ( C xe A )  =  ( C xe B )  <->  A  =  B )
 )
 
Theoremxreceu 27596* Existential uniqueness of reciprocals. Theorem I.8 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 17-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  E! x  e.  RR*  ( B xe x )  =  A )
 
Theoremxdivcld 27597 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  B  e.  RR )   &    |-  ( ph  ->  B  =/=  0 )   =>    |-  ( ph  ->  ( A /𝑒 
 B )  e.  RR* )
 
Theoremxdivcl 27598 Closure law for the extended division. (Contributed by Thierry Arnoux, 15-Mar-2017.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  e.  RR* )
 
Theoremxdivmul 27599 Relationship between division and multiplication. (Contributed by Thierry Arnoux, 24-Dec-2016.)
 |-  (
 ( A  e.  RR*  /\  B  e.  RR*  /\  ( C  e.  RR  /\  C  =/=  0 ) )  ->  ( ( A /𝑒  C )  =  B  <->  ( C xe B )  =  A ) )
 
Theoremrexdiv 27600 The extended real division operation when both arguments are real. (Contributed by Thierry Arnoux, 18-Dec-2016.)
 |-  (
 ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 )  ->  ( A /𝑒  B )  =  ( A  /  B ) )
    < 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 37701-37797
  Copyright terms: Public domain < Previous  Next >