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Theorem List for Metamath Proof Explorer - 24201-24300   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremxrstos 24201 The extended real numbers form a toset. (Contributed by Thierry Arnoux, 15-Feb-2018.)
 |-  RR* s  e. Toset
 
Theoremxrsclat 24202 The extended real numbers form a complete lattice. (Contributed by Thierry Arnoux, 15-Feb-2018.)
 |-  RR* s  e.  CLat
 
Theoremxrsp0 24203 The poset 0 of the extended real numbers is minus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  -oo  =  ( 0. `  RR* s )
 
Theoremxrsp1 24204 The poset 1 of the extended real numbers is plus infinity. (Contributed by Thierry Arnoux, 18-Feb-2018.)
 |-  +oo  =  ( 1. `  RR* s )
 
Theoremressmulgnn 24205 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 12-Jun-2017.)
 |-  H  =  ( Gs  A )   &    |-  A  C_  ( Base `  G )   &    |-  .*  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   =>    |-  ( ( N  e.  NN  /\  X  e.  A )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
 
Theoremressmulgnn0 24206 Values for the group multiple function in a restricted structure (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  H  =  ( Gs  A )   &    |-  A  C_  ( Base `  G )   &    |-  .*  =  (.g `  G )   &    |-  I  =  ( inv g `  G )   &    |-  ( 0g `  G )  =  ( 0g `  H )   =>    |-  ( ( N  e.  NN0  /\  X  e.  A )  ->  ( N (.g `  H ) X )  =  ( N  .*  X ) )
 
19.3.6.5  The extended non-negative real numbers monoid
 
Theoremxrge0base 24207 The base of the extended non-negative real numbers. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  (
 0 [,]  +oo )  =  ( Base `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge00 24208 The zero of the extended non-negative real numbers monoid. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  0  =  ( 0g `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge0plusg 24209 The additive law of the extended non-negative real numbers monoid is the addition in the extended real numbers. (Contributed by Thierry Arnoux, 20-Mar-2017.)
 |-  + e  =  ( +g  `  ( RR* ss  ( 0 [,]  +oo ) ) )
 
Theoremxrge0mulgnn0 24210 The group multiple function in the extended non-negative real numbers. (Contributed by Thierry Arnoux, 14-Jun-2017.)
 |-  (
 ( A  e.  NN0  /\  B  e.  ( 0 [,]  +oo ) )  ->  ( A (.g `  ( RR* ss  ( 0 [,]  +oo ) ) ) B )  =  ( A x e B ) )
 
Theoremxrge0addass 24211 Associativity of extended non-negative real addition. (Contributed by Thierry Arnoux, 8-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  ( ( A + e B ) + e C )  =  ( A + e ( B + e C ) ) )
 
Theoremxrge0neqmnf 24212 An extended non-negative real cannot be minus infinity. (Contributed by Thierry Arnoux, 9-Jun-2017.)
 |-  ( A  e.  ( 0 [,]  +oo )  ->  A  =/=  -oo )
 
Theoremxrge0nre 24213 An extended real which is not a real is plus infinity. (Contributed by Thierry Arnoux, 16-Oct-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  -.  A  e.  RR )  ->  A  =  +oo )
 
Theoremxrge0addgt0 24214 The sum of nonnegative and positive numbers is positive. See addgtge0 9516 (Contributed by Thierry Arnoux, 6-Jul-2017.)
 |-  (
 ( ( A  e.  ( 0 [,]  +oo )  /\  B  e.  (
 0 [,]  +oo ) ) 
 /\  0  <  A )  ->  0  <  ( A + e B ) )
 
Theoremxrge0adddir 24215 Distributivity of extended non-negative real multiplication over addition. (Contributed by Thierry Arnoux, 30-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  C  e.  ( 0 [,]  +oo ) )  ->  ( ( A + e B ) x e C )  =  ( ( A x e C ) + e ( B x e C ) ) )
 
Theoremxrge0npcan 24216 Extended non-negative real version of npcan 9314. (Contributed by Thierry Arnoux, 9-Jun-2017.)
 |-  (
 ( A  e.  (
 0 [,]  +oo )  /\  B  e.  ( 0 [,]  +oo )  /\  B  <_  A )  ->  (
 ( A + e  - e B ) + e B )  =  A )
 
Theoremfsumrp0cl 24217* Closure of a finite sum of positive integers. (Contributed by Thierry Arnoux, 25-Jun-2017.)
 |-  ( ph  ->  A  e.  Fin )   &    |-  ( ( ph  /\  k  e.  A )  ->  B  e.  ( 0 [,)  +oo ) )   =>    |-  ( ph  ->  sum_ k  e.  A  B  e.  (
 0 [,)  +oo ) )
 
19.3.7  Algebra
 
19.3.7.1  Finitely supported group sums - misc additions
 
Theoremsumpr 24218* A sum over a pair is the sum of the elements. (Contributed by Thierry Arnoux, 12-Dec-2016.)
 |-  (
 k  =  A  ->  C  =  D )   &    |-  (
 k  =  B  ->  C  =  E )   &    |-  ( ph  ->  ( D  e.  CC  /\  E  e.  CC ) )   &    |-  ( ph  ->  ( A  e.  V  /\  B  e.  W )
 )   &    |-  ( ph  ->  A  =/=  B )   =>    |-  ( ph  ->  sum_ k  e.  { A ,  B } C  =  ( D  +  E )
 )
 
Theoremgsumsn2 24219* Group sum of a singleton. (Contributed by Thierry Arnoux, 30-Jan-2017.)
 |-  B  =  ( Base `  G )   &    |-  G  e.  Mnd   &    |-  ( ( ph  /\  k  =  M )  ->  A  =  C )   &    |-  ( ph  ->  M  e.  V )   &    |-  ( ph  ->  C  e.  B )   =>    |-  ( ph  ->  ( G  gsumg  ( k  e.  { M }  |->  A ) )  =  C )
 
Theoremgsumpropd2lem 24220* Lemma for gsumpropd2 24221 (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  (
 ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) )  ->  ( s
 ( +g  `  G ) t )  e.  ( Base `  G ) )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   &    |-  A  =  ( `' F "
 ( _V  \  {
 s  e.  ( Base `  G )  |  A. t  e.  ( Base `  G ) ( ( s ( +g  `  G ) t )  =  t  /\  ( t ( +g  `  G ) s )  =  t ) } )
 )   &    |-  B  =  ( `' F " ( _V  \  { s  e.  ( Base `  H )  | 
 A. t  e.  ( Base `  H ) ( ( s ( +g  `  H ) t )  =  t  /\  (
 t ( +g  `  H ) s )  =  t ) } )
 )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumpropd2 24221* A stronger version of gsumpropd 14776, working for magma, where only the closure of the addition operation on a common base is required. (Contributed by Thierry Arnoux, 28-Jun-2017.)
 |-  ( ph  ->  F  e.  V )   &    |-  ( ph  ->  G  e.  W )   &    |-  ( ph  ->  H  e.  X )   &    |-  ( ph  ->  ( Base `  G )  =  ( Base `  H ) )   &    |-  (
 ( ph  /\  ( s  e.  ( Base `  G )  /\  t  e.  ( Base `  G ) ) )  ->  ( s
 ( +g  `  G ) t )  e.  ( Base `  G ) )   &    |-  ( ( ph  /\  (
 s  e.  ( Base `  G )  /\  t  e.  ( Base `  G )
 ) )  ->  (
 s ( +g  `  G ) t )  =  ( s ( +g  `  H ) t ) )   &    |-  ( ph  ->  Fun 
 F )   &    |-  ( ph  ->  ran 
 F  C_  ( Base `  G ) )   =>    |-  ( ph  ->  ( G  gsumg 
 F )  =  ( H  gsumg 
 F ) )
 
Theoremgsumconstf 24222* Sum of a constant series (Contributed by Thierry Arnoux, 5-Jul-2017.)
 |-  F/_ k X   &    |-  B  =  ( Base `  G )   &    |-  .x.  =  (.g `  G )   =>    |-  ( ( G  e.  Mnd  /\  A  e.  Fin  /\  X  e.  B )  ->  ( G  gsumg  ( k  e.  A  |->  X ) )  =  ( ( # `  A )  .x.  X ) )
 
Theoremxrge0tsmsd 24223* Any finite or infinite sum in the nonnegative extended reals is uniquely convergent to the supremum of all finite sums. (Contributed by Mario Carneiro, 13-Sep-2015.) (Revised by Thierry Arnoux, 30-Jan-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  S  =  sup ( ran  ( s  e.  ( ~P A  i^i  Fin )  |->  ( G  gsumg  ( F  |`  s ) ) ) ,  RR* ,  <  ) )   =>    |-  ( ph  ->  ( G tsums  F )  =  { S } )
 
Theoremxrge0tsmsbi 24224 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 23-Jun-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   =>    |-  ( ph  ->  ( C  e.  ( G tsums  F )  <->  C  =  U. ( G tsums  F ) ) )
 
Theoremxrge0tsmseq 24225 Any limit of a finite or infinite sum in the nonnegative extended reals is the union of the sets limits, since this set is a singleton. (Contributed by Thierry Arnoux, 24-Mar-2017.)
 |-  G  =  ( RR* ss  ( 0 [,]  +oo ) )   &    |-  ( ph  ->  A  e.  V )   &    |-  ( ph  ->  F : A --> ( 0 [,]  +oo ) )   &    |-  ( ph  ->  C  e.  ( G tsums  F ) )   =>    |-  ( ph  ->  C  =  U. ( G tsums  F ) )
 
19.3.7.2  Rings - misc additions
 
Theoremdvrdir 24226 Distributive law for the division operation of a ring. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  ./  =  (/r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  U )
 )  ->  ( ( X  .+  Y )  ./  Z )  =  (
 ( X  ./  Z )  .+  ( Y  ./  Z ) ) )
 
Theoremrdivmuldivd 24227 Multiplication of two ratios. Theorem I.14 of [Apostol] p. 18. (Contributed by Thierry Arnoux, 30-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  .+  =  ( +g  `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   &    |-  ( ph  ->  R  e.  CRing )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  Y  e.  U )   &    |-  ( ph  ->  Z  e.  B )   &    |-  ( ph  ->  W  e.  U )   =>    |-  ( ph  ->  (
 ( X  ./  Y )  .x.  ( Z  ./  W ) )  =  ( ( X  .x.  Z )  ./  ( Y  .x.  W ) ) )
 
Theoremrnginvval 24228* The ring inverse expressed in terms of multiplication. (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  B  =  ( Base `  R )   &    |-  .*  =  ( .r `  R )   &    |- 
 .1.  =  ( 1r `  R )   &    |-  N  =  (
 invr `  R )   &    |-  U  =  (Unit `  R )   =>    |-  (
 ( R  e.  Ring  /\  X  e.  U ) 
 ->  ( N `  X )  =  ( iota_ y  e.  U ( y  .*  X )  =  .1.  ) )
 
Theoremdvrcan5 24229 Cancellation law for common factor in ratio. (divcan5 9716 analog.) (Contributed by Thierry Arnoux, 26-Oct-2016.)
 |-  B  =  ( Base `  R )   &    |-  U  =  (Unit `  R )   &    |-  ./  =  (/r `  R )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( R  e.  Ring  /\  ( X  e.  B  /\  Y  e.  U  /\  Z  e.  U ) )  ->  ( ( X  .x.  Z )  ./  ( Y  .x.  Z ) )  =  ( X  ./  Y ) )
 
Theoremsubrgchr 24230 If  A is a subring of  R, then they have the same characteristic. (Contributed by Thierry Arnoux, 24-Feb-2018.)
 |-  ( A  e.  (SubRing `  R )  ->  (chr `  ( Rs  A ) )  =  (chr `  R )
 )
 
19.3.7.3  Ordered groups
 
Syntaxcogrp 24231 Extend class notation with the class of all ordered groups.
 class oGrp
 
Definitiondf-ogrp 24232* Define class of all ordered groups. An ordered group is a group with a total ordering compatible with its operation. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |- oGrp  =  {
 g  e.  Grp  |  [. ( Base `  g )  /  v ]. [. ( +g  `  g )  /  p ]. [. ( le `  g )  /  l ]. ( g  e. Toset  /\  A. a  e.  v  A. b  e.  v  A. c  e.  v  (
 a l b  ->  ( a p c ) l ( b p c ) ) ) }
 
19.3.7.4  Ordered fields
 
Syntaxcofld 24233 Extend class notation with the class of all ordered fields.
 class oField
 
Definitiondf-ofld 24234* Define class of all ordered fields. An ordered field is a field with a total ordering compatible with the operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
 |- oField  =  {
 f  e. Field  |  [. ( Base `  f )  /  v ]. [. ( +g  `  f )  /  p ].
 [. ( 0g `  f )  /  z ]. [. ( .r `  f )  /  t ]. [. ( le `  f
 )  /  l ]. ( f  e. Toset  /\  A. a  e.  v  A. b  e.  v  ( A. c  e.  v  ( a l b 
 ->  ( a p c ) l ( b p c ) ) 
 /\  ( ( z l a  /\  z
 l b )  ->  z l ( a t b ) ) ) ) }
 
Theoremisofld 24235* An ordered field is a field with a total ordering compatible with the operations. (Contributed by Thierry Arnoux, 18-Jan-2018.)
 |-  B  =  ( Base `  F )   &    |-  .+  =  ( +g  `  F )   &    |-  .0.  =  ( 0g `  F )   &    |- 
 .x.  =  ( .r `  F )   &    |-  .<_  =  ( le `  F )   =>    |-  ( F  e. oField  <->  ( F  e. Field  /\  F  e. Toset  /\  A. a  e.  B  A. b  e.  B  ( A. c  e.  B  ( a  .<_  b 
 ->  ( a  .+  c
 )  .<_  ( b  .+  c ) )  /\  ( (  .0.  .<_  a  /\  .0.  .<_  b )  ->  .0.  .<_  ( a  .x.  b ) ) ) ) )
 
Theoremofldfld 24236 An ordered field is a field. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  ( F  e. oField  ->  F  e. Field )
 
Theoremofldtos 24237 An ordered field is a totally ordered set. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  ( F  e. oField  ->  F  e. Toset )
 
Theoremofldadd 24238 In an ordered field, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  B  =  ( Base `  F )   &    |-  .<_  =  ( le `  F )   &    |- 
 .+  =  ( +g  `  F )   =>    |-  ( ( F  e. oField  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( X  .+  Z ) 
 .<_  ( Y  .+  Z ) )
 
Theoremofldmul 24239 In an ordered field, the ordering is compatible with the ring multiplication operation. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  B  =  ( Base `  F )   &    |-  .<_  =  ( le `  F )   &    |- 
 .0.  =  ( 0g `  F )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( F  e. oField  /\  ( X  e.  B  /\  .0.  .<_  X ) 
 /\  ( Y  e.  B  /\  .0.  .<_  Y ) )  ->  .0.  .<_  ( X 
 .x.  Y ) )
 
Theoremofldsqr 24240 In an ordered field, all squares are positive. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  B  =  ( Base `  F )   &    |-  .<_  =  ( le `  F )   &    |- 
 .0.  =  ( 0g `  F )   &    |-  .x.  =  ( .r `  F )   =>    |-  ( ( F  e. oField  /\  X  e.  B )  ->  .0.  .<_  ( X 
 .x.  X ) )
 
Theoremofldaddlt 24241 In an ordered field, the ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
 |-  B  =  ( Base `  F )   &    |-  .<  =  ( lt `  F )   &    |- 
 .+  =  ( +g  `  F )   =>    |-  ( ( F  e. oField  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z )  .<  ( Y  .+  Z ) )
 
Theoremofld0le1 24242 In an ordered field, the ring unit is positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  .0.  =  ( 0g `  F )   &    |- 
 .1.  =  ( 1r `  F )   &    |-  .<_  =  ( le `  F )   =>    |-  ( F  e. oField  ->  .0. 
 .<_  .1.  )
 
Theoremofldlt1 24243 In an ordered field, the ring unit is strictly positive. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  .0.  =  ( 0g `  F )   &    |- 
 .1.  =  ( 1r `  F )   &    |-  .<  =  ( lt `  F )   =>    |-  ( F  e. oField  ->  .0.  .<  .1.  )
 
Theoremofldchr 24244 The characteristic of an ordered field is zero. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  ( F  e. oField  ->  (chr `  F )  =  0
 )
 
Theoremsubofld 24245 Every subfield of an ordered field is also an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  (
 ( F  e. oField  /\  ( Fs  A )  e. Field )  ->  ( Fs  A )  e. oField )
 
19.3.7.5  The Archimedean property for generic algebraic structures
 
Syntaxcinftm 24246 Class notation for the infinitesimal relation.
 class <<<
 
Syntaxcarchi 24247 Class notation for the Archimedean property.
 class Archi
 
Definitiondf-inftm 24248* Define the relation " x is infinitesimal with respect to  y " for a structure  w. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |- <<<  =  ( w  e.  _V  |->  {
 <. x ,  y >.  |  ( ( x  e.  ( Base `  w )  /\  y  e.  ( Base `  w ) ) 
 /\  ( ( 0g
 `  w ) ( lt `  w ) x  /\  A. n  e.  NN  ( n (.g `  w ) x ) ( lt `  w ) y ) ) } )
 
Definitiondf-archi 24249 A structure said to be Archimedean if it has no infinitesimal elements. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |- Archi  =  { w  e.  _V  |  (<<< `  w )  =  (/) }
 
Theoreminftmrel 24250 The infinitesimal relation for a structure  W (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   =>    |-  ( W  e.  V  ->  (<<< `  W )  C_  ( B  X.  B ) )
 
Theoremisinftm 24251* Express  x is infinitesimal with respect to  y for a structure  W. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  (.g `  W )   &    |- 
 .<  =  ( lt `  W )   =>    |-  ( ( W  e.  V  /\  X  e.  B  /\  Y  e.  B ) 
 ->  ( X (<<< `  W ) Y  <->  (  .0.  .<  X  /\  A. n  e.  NN  ( n  .x.  X )  .<  Y ) ) )
 
Theoremisarchi 24252* Express the predicate " W is Archimedean ". (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .<  =  (<<< `  W )   =>    |-  ( W  e.  V  ->  ( W  e. Archi  <->  A. x  e.  B  A. y  e.  B  -.  x  .<  y ) )
 
Theorempnfinf 24253 Plus infinity is an infinite for the completed real line, as any real number is infinitesimal compared to it. (Contributed by Thierry Arnoux, 1-Feb-2018.)
 |-  ( A  e.  RR+  ->  A (<<<
 `  RR* s )  +oo )
 
Theoremxrnarchi 24254 The completed real line is not Archimedean. (Contributed by Thierry Arnoux, 1-Feb-2018.)
 |-  -.  RR*
 s  e. Archi
 
Theoremisarchi2 24255* Alternative way to express the predicate " W is Archimedean ", for Tosets. (Contributed by Thierry Arnoux, 30-Jan-2018.)
 |-  B  =  ( Base `  W )   &    |-  .0.  =  ( 0g `  W )   &    |- 
 .x.  =  (.g `  W )   &    |- 
 .<_  =  ( le `  W )   &    |- 
 .<  =  ( lt `  W )   =>    |-  ( ( W  e. Toset  /\  W  e.  Grp )  ->  ( W  e. Archi  <->  A. x  e.  B  A. y  e.  B  (  .0.  .<  x  ->  E. n  e.  NN  y  .<_  ( n 
 .x.  x ) ) ) )
 
19.3.7.6  Ring homomorphisms - misc additions
 
Theoremrhmdvdsr 24256 A ring homomorphism preserves the divisibility relation. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  X  =  ( Base `  R )   &    |-  .||  =  (
 ||r `  R )   &    |-  ./  =  ( ||r `  S )   =>    |-  ( ( ( F  e.  ( R RingHom  S ) 
 /\  A  e.  X  /\  B  e.  X ) 
 /\  A  .||  B ) 
 ->  ( F `  A )  ./  ( F `  B ) )
 
Theoremrhmopp 24257 A ring homomorphism is also a ring homomorphism for the opposite rings. (Contributed by Thierry Arnoux, 27-Oct-2017.)
 |-  ( F  e.  ( R RingHom  S )  ->  F  e.  ( (oppr `  R ) RingHom  (oppr `  S ) ) )
 
Theoremelrhmunit 24258 Ring homomorphisms preserve unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  (
 ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
 )  ->  ( F `  A )  e.  (Unit `  S ) )
 
Theoremrhmdvd 24259 A ring homomorphism preserves ratios. (Contributed by Thierry Arnoux, 22-Oct-2017.)
 |-  U  =  (Unit `  S )   &    |-  X  =  ( Base `  R )   &    |-  ./  =  (/r `  S )   &    |-  .x.  =  ( .r `  R )   =>    |-  ( ( F  e.  ( R RingHom  S ) 
 /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X ) 
 /\  ( ( F `
  B )  e.  U  /\  ( F `
  C )  e.  U ) )  ->  ( ( F `  A )  ./  ( F `
  B ) )  =  ( ( F `
  ( A  .x.  C ) )  ./  ( F `  ( B  .x.  C ) ) ) )
 
Theoremrhmunitinv 24260 Ring homomorphisms preserve the inverse of unit elements. (Contributed by Thierry Arnoux, 23-Oct-2017.)
 |-  (
 ( F  e.  ( R RingHom  S )  /\  A  e.  (Unit `  R )
 )  ->  ( F `  ( ( invr `  R ) `  A ) )  =  ( ( invr `  S ) `  ( F `  A ) ) )
 
Theoremkerunit 24261 If a unit element lies in the kernel of a ring homomorphism, then  0  = 
1, i.e. the target ring is the zero ring. (Contributed by Thierry Arnoux, 24-Oct-2017.)
 |-  U  =  (Unit `  R )   &    |-  .0.  =  ( 0g `  S )   &    |- 
 .1.  =  ( 1r `  S )   =>    |-  ( ( F  e.  ( R RingHom  S )  /\  ( U  i^i  ( `' F " {  .0.  } ) )  =/=  (/) )  ->  .1.  =  .0.  )
 
Theoremkerf1hrm 24262 A ring homomorphism  F is injective if and only if its kernel is the singleton  { N }. (Contributed by Thierry Arnoux, 27-Oct-2017.)
 |-  A  =  ( Base `  R )   &    |-  B  =  ( Base `  S )   &    |-  N  =  ( 0g `  R )   &    |- 
 .0.  =  ( 0g `  S )   =>    |-  ( F  e.  ( R RingHom  S )  ->  ( F : A -1-1-> B  <->  ( `' F " {  .0.  } )  =  { N } )
 )
 
19.3.7.7  The ring of integers
 
Theoremzzsbase 24263 The base of the ring of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.)
 |-  Z  =  (flds  ZZ )   =>    |- 
 ZZ  =  ( Base `  Z )
 
Theoremzzsplusg 24264 The addition operation of the ring of integers. (Contributed by Thierry Arnoux, 8-Nov-2017.)
 |-  Z  =  (flds  ZZ )   =>    |- 
 +  =  ( +g  `  Z )
 
Theoremzzsmulg 24265 The multiplication (group power) opereation of the group of integers. (Contributed by Thierry Arnoux, 31-Oct-2017.)
 |-  Z  =  (flds  ZZ )   =>    |-  ( ( A  e.  ZZ  /\  B  e.  ZZ )  ->  ( A (.g `  Z ) B )  =  ( A  x.  B ) )
 
Theoremzzsmulr 24266 The multiplication operation of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  Z  =  (flds  ZZ )   =>    |- 
 x.  =  ( .r
 `  Z )
 
Theoremzzs0 24267 The neutral element of the ring of integers. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  Z  =  (flds  ZZ )   =>    |-  0  =  ( 0g
 `  Z )
 
Theoremzzs1 24268 The multiplicative neutral element of the ring of integers (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  Z  =  (flds  ZZ )   =>    |-  1  =  ( 1r
 `  Z )
 
19.3.7.8  The ordered field of reals
 
Theoremrebase 24269 The base of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |- 
 RR  =  ( Base `  R )
 
Theoremremulg 24270 The multiplication (group power) operation of the group of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |-  ( ( N  e.  ZZ  /\  A  e.  RR )  ->  ( N (.g `  R ) A )  =  ( N  x.  A ) )
 
Theoremreplusg 24271 The addition operation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  R  =  (flds  RR )   =>    |- 
 +  =  ( +g  `  R )
 
Theoremremulr 24272 The multiplication operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |- 
 x.  =  ( .r
 `  R )
 
Theoremre0g 24273 The neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |-  0  =  ( 0g
 `  R )
 
Theoremre1r 24274 The multiplicative neutral element of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |-  1  =  ( 1r
 `  R )
 
Theoremrele2 24275 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  R  =  (flds  RR )   =>    |- 
 <_  =  ( le `  R )
 
Theoremrelt 24276 The ordering relation of the field of reals. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  R  =  (flds  RR )   =>    |- 
 <  =  ( lt `  R )
 
Theoremredvr 24277 The division operation of the field of reals. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  B  =/=  0 ) 
 ->  ( A (/r `  R ) B )  =  ( A  /  B ) )
 
Theoremretos 24278 The real numbers are a totally ordered set. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  R  =  (flds  RR )   =>    |-  R  e. Toset
 
Theoremrefld 24279 The real numbers form a field. (Contributed by Thierry Arnoux, 1-Nov-2017.)
 |-  R  =  (flds  RR )   =>    |-  R  e. Field
 
Theoremreofld 24280 The real numbers form an ordered field. (Contributed by Thierry Arnoux, 21-Jan-2018.)
 |-  R  =  (flds  RR )   =>    |-  R  e. oField
 
19.3.8  Topology
 
19.3.8.1  Pseudometrics
 
Syntaxcmetid 24281 Extend class notation with the class of metric identifications.
 class ~Met
 
Syntaxcpstm 24282 Extend class notation with the metric induced by a pseudometric.
 class pstoMet
 
Definitiondf-metid 24283* Define the metric identification relation for a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |- ~Met  =  ( d  e.  U. ran PsMet  |->  {
 <. x ,  y >.  |  ( ( x  e. 
 dom  dom  d  /\  y  e.  dom  dom  d )  /\  ( x d y )  =  0 ) } )
 
Definitiondf-pstm 24284* Define the metric induced by a pseudometric. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |- pstoMet  =  ( d  e.  U. ran PsMet  |->  ( a  e.  ( dom 
 dom  d /. (~Met `  d ) ) ,  b  e.  ( dom 
 dom  d /. (~Met `  d ) )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x d y ) } ) )
 
Theoremmetidval 24285* Value of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  (~Met `  D )  =  { <. x ,  y >.  |  ( ( x  e.  X  /\  y  e.  X )  /\  ( x D y )  =  0 ) } )
 
Theoremmetidss 24286 As a relation, the metric identification is a subset of a cross product. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  (~Met `  D )  C_  ( X  X.  X ) )
 
Theoremmetidv 24287  A and  B identify by the metric  D if their distance is zero. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  (
 ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X )
 )  ->  ( A (~Met `  D ) B  <-> 
 ( A D B )  =  0 )
 )
 
Theoremmetideq 24288 Basic property of the metric identification relation. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  (
 ( D  e.  (PsMet `  X )  /\  ( A (~Met `  D ) B  /\  E (~Met `  D ) F ) )  ->  ( A D E )  =  ( B D F ) )
 
Theoremmetider 24289 The metric identification is an equivalence relation. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( D  e.  (PsMet `  X )  ->  (~Met `  D )  Er  X )
 
Theorempstmval 24290* Value of the metric induced by a pseudometric  D. (Contributed by Thierry Arnoux, 7-Feb-2018.)
 |-  .~  =  (~Met `  D )   =>    |-  ( D  e.  (PsMet `  X )  ->  (pstoMet `  D )  =  ( a  e.  ( X /.  .~  ) ,  b  e.  ( X
 /.  .~  )  |->  U. { z  |  E. x  e.  a  E. y  e.  b  z  =  ( x D y ) } ) )
 
Theorempstmfval 24291 Function value of the metric induced by a pseudometric  D (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  .~  =  (~Met `  D )   =>    |-  (
 ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( [ A ]  .~  (pstoMet `  D ) [ B ]  .~  )  =  ( A D B ) )
 
Theorempstmxmet 24292 The metric induced by a pseudometric is a full-fledged metric on the equivalence classes of the metric identification. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  .~  =  (~Met `  D )   =>    |-  ( D  e.  (PsMet `  X )  ->  (pstoMet `  D )  e.  ( * Met `  ( X /.  .~  ) ) )
 
19.3.8.2  Continuity - misc additions
 
Theoremhauseqcn 24293 In a Hausdorff topology, two continuous functions which agree on a dense set agree everywhere. (Contributed by Thierry Arnoux, 28-Dec-2017.)
 |-  X  =  U. J   &    |-  ( ph  ->  K  e.  Haus )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  ( F  |`  A )  =  ( G  |`  A )
 )   &    |-  ( ph  ->  A  C_  X )   &    |-  ( ph  ->  ( ( cls `  J ) `  A )  =  X )   =>    |-  ( ph  ->  F  =  G )
 
19.3.8.3  Topology of the closed unit
 
Theoremunitsscn 24294 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 24295 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 24296 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 24297 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 24298 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 24299 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 24300 The closed unit monoid is a topological monoid. (Contributed by Thierry Arnoux, 25-Mar-2017.)
 |-  I  =  ( (mulGrp ` fld )s  ( 0 [,] 1
 ) )   =>    |-  I  e. TopMnd
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