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Theorem List for Metamath Proof Explorer - 25701-25800   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremheiborlem1 25701* Lemma for heibor 25711. We work with a fixed open cover  U throughout. The set  K is the set of all subsets of  X that admit no finite subcover of  U. (We wish to prove that  K is empty.) If a set  C has no finite subcover, then any finite cover of  C must contain a set that also has no finite subcover. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  B  e.  _V   =>    |-  ( ( A  e.  Fin  /\  C  C_  U_ x  e.  A  B  /\  C  e.  K )  ->  E. x  e.  A  B  e.  K )
 
Theoremheiborlem2 25702* Lemma for heibor 25711. Substitutions for the set  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  A  e.  _V   &    |-  C  e.  _V   =>    |-  ( A G C  <->  ( C  e.  NN0  /\  A  e.  ( F `  C )  /\  ( A B C )  e.  K ) )
 
Theoremheiborlem3 25703* Lemma for heibor 25711. Using countable choice ax-cc 7945, we have fixed in advance a collection of finite  2 ^ -u n nets  ( F `  n ) for  X (note that an  r-net is a set of points in  X whose  r -balls cover  X). The set  G is the subset of these points whose corresponding balls have no finite subcover (i.e. in the set  K). If the theorem was false, then  X would be in  K, and so some ball at each level would also be in  K. But we can say more than this; given a ball 
( y B n ) on level  n, since level  n  +  1 covers the space and thus also  (
y B n ), using heiborlem1 25701 there is a ball on the next level whose intersection with  ( y B n ) also has no finite subcover. Now since the set 
G is a countable union of finite sets, it is countable (which needs ax-cc 7945 via iunctb 8076), and so we can apply ax-cc 7945 to  G directly to get a function from  G to itself, which points from each ball in  K to a ball on the next level in  K, and such that the intersection between these balls is also in  K. (Contributed by Jeff Madsen, 18-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   =>    |-  ( ph  ->  E. g A. x  e.  G  ( ( g `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( g `  x ) B ( ( 2nd `  x )  +  1 ) ) )  e.  K ) )
 
Theoremheiborlem4 25704* Lemma for heibor 25711. Using the function  T constructed in heiborlem3 25703, construct an infinite path in  G. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   =>    |-  ( ( ph  /\  A  e.  NN0 )  ->  ( S `  A ) G A )
 
Theoremheiborlem5 25705* Lemma for heibor 25711. The function  M is a set of point-and-radius pairs suitable for application to caubl 18565. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   =>    |-  ( ph  ->  M : NN
 --> ( X  X.  RR+ )
 )
 
Theoremheiborlem6 25706* Lemma for heibor 25711. Since the sequence of balls connected by the function  T ensures that each ball nontrivially intersects with the next (since the empty set has a finite subcover, the intersection of any two successive balls in the sequence is nonempty), and each ball is half the size of the previous one, the distance between the centers is at most  3  /  2 times the size of the larger, and so if we expand each ball by a factor of  3 we get a nested sequence of balls. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   =>    |-  ( ph  ->  A. k  e. 
 NN  ( ( ball `  D ) `  ( M `  ( k  +  1 ) ) ) 
 C_  ( ( ball `  D ) `  ( M `  k ) ) )
 
Theoremheiborlem7 25707* Lemma for heibor 25711. Since the sizes of the balls decrease exponentially, the sequence converges to zero. (Contributed by Jeff Madsen, 23-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   =>    |-  A. r  e.  RR+  E. k  e.  NN  ( 2nd `  ( M `  k ) )  <  r
 
Theoremheiborlem8 25708* Lemma for heibor 25711. The previous lemmas establish that the sequence  M is Cauchy, so using completeness we now consider the convergent point 
Y. By assumption,  U is an open cover, so  Y is an element of some  Z  e.  U, and some ball centered at  Y is contained in  Z. But the sequence contains arbitrarily small balls close to  Y, so some element  ball ( M `  n ) of the sequence is contained in  Z. And finally we arrive at a contradiction, because  { Z } is a finite subcover of  U that covers  ball ( M `  n ), yet  ball ( M `  n )  e.  K. For convenience, we write this contradiction as 
ph  ->  ps where  ph is all the accumulated hypotheses and  ps is anything at all. (Contributed by Jeff Madsen, 22-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   &    |-  ( ph  ->  U  C_  J )   &    |-  Y  e.  _V   &    |-  ( ph  ->  Y  e.  Z )   &    |-  ( ph  ->  Z  e.  U )   &    |-  ( ph  ->  ( 1st  o.  M ) ( ~~> t `  J ) Y )   =>    |-  ( ph  ->  ps )
 
Theoremheiborlem9 25709* Lemma for heibor 25711. Discharge the hypotheses of heiborlem8 25708 by applying caubl 18565 to get a convergent point and adding the open cover assumption. (Contributed by Jeff Madsen, 20-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   &    |-  ( ph  ->  A. x  e.  G  ( ( T `  x ) G ( ( 2nd `  x )  +  1 )  /\  ( ( B `  x )  i^i  ( ( T `
  x ) B ( ( 2nd `  x )  +  1 )
 ) )  e.  K ) )   &    |-  ( ph  ->  C G 0 )   &    |-  S  =  seq  0 ( T ,  ( m  e. 
 NN0  |->  if ( m  =  0 ,  C ,  ( m  -  1
 ) ) ) )   &    |-  M  =  ( n  e.  NN  |->  <. ( S `  n ) ,  (
 3  /  ( 2 ^ n ) ) >. )   &    |-  ( ph  ->  U  C_  J )   &    |-  ( ph  ->  U. U  =  X )   =>    |-  ( ph  ->  ps )
 
Theoremheiborlem10 25710* Lemma for heibor 25711. The last remaining piece of the proof is to find an element  C such that  C G 0, i.e. 
C is an element of  ( F ` 
0 ) that has no finite subcover, which is true by heiborlem1 25701, since  ( F `  0 ) is a finite cover of  X, which has no finite subcover. Thus the rest of the proof follows to a contradiction, and thus there must be a finite subcover of  U that covers  X, i.e.  X is compact. (Contributed by Jeff Madsen, 22-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   &    |-  K  =  { u  |  -.  E. v  e.  ( ~P U  i^i  Fin ) u  C_  U. v }   &    |-  G  =  { <. y ,  n >.  |  ( n  e. 
 NN0  /\  y  e.  ( F `  n )  /\  ( y B n )  e.  K ) }   &    |-  B  =  ( z  e.  X ,  m  e.  NN0  |->  ( z ( ball `  D )
 ( 1  /  (
 2 ^ m ) ) ) )   &    |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  F : NN0
 --> ( ~P X  i^i  Fin ) )   &    |-  ( ph  ->  A. n  e.  NN0  X  =  U_ y  e.  ( F `  n ) ( y B n ) )   =>    |-  ( ( ph  /\  ( U  C_  J  /\  U. J  =  U. U ) )  ->  E. v  e.  ( ~P U  i^i  Fin ) U. J  =  U. v )
 
Theoremheibor 25711 Generalized Heine-Borel Theorem. A metric space is compact iff it is complete and totally bounded. See heibor1 25700 and heiborlem1 25701 for a description of the proof. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 28-Jan-2014.)
 |-  J  =  ( MetOpen `  D )   =>    |-  (
 ( D  e.  ( Met `  X )  /\  J  e.  Comp )  <->  ( D  e.  ( CMet `  X )  /\  D  e.  ( TotBnd `  X ) ) )
 
16.13.11  Banach Fixed Point Theorem
 
Theorembfplem1 25712* Lemma for bfp 25714. The sequence  G, which simply starts from any point in the space and iterates  F, satisfies the property that the distance from  G ( n ) to  G ( n  + 
1 ) decreases by at least  K after each step. Thus the total distance from any  G ( i ) to  G ( j ) is bounded by a geometric series, and the sequence is Cauchy. Therefore, it converges to a point  ( ( ~~> t `  J
) `  G ) since the space is complete. (Contributed by Jeff Madsen, 17-Jun-2014.)
 |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  K  e.  RR+ )   &    |-  ( ph  ->  K  <  1 )   &    |-  ( ph  ->  F : X --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( F `  x ) D ( F `  y ) )  <_  ( K  x.  ( x D y ) ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  A  e.  X )   &    |-  G  =  seq  1
 ( ( F  o.  1st ) ,  ( NN 
 X.  { A } )
 )   =>    |-  ( ph  ->  G (
 ~~> t `  J ) ( ( ~~> t `  J ) `  G ) )
 
Theorembfplem2 25713* Lemma for bfp 25714. Using the point found in bfplem1 25712, we show that this convergent point is a fixed point of  F. Since for any positive  x, the sequence  G is in  B ( x  /  2 ,  P ) for all  k  e.  (
ZZ>= `  j ) (where  P  =  ( ( ~~> t `  J ) `  G
)), we have  D ( G ( j  +  1 ) ,  F ( P ) )  <_  D ( G ( j ) ,  P
)  <  x  / 
2 and  D ( G ( j  +  1 ) ,  P )  <  x  /  2, so  F ( P ) is in every neighborhood of  P and  P is a fixed point of  F. (Contributed by Jeff Madsen, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  K  e.  RR+ )   &    |-  ( ph  ->  K  <  1 )   &    |-  ( ph  ->  F : X --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( F `  x ) D ( F `  y ) )  <_  ( K  x.  ( x D y ) ) )   &    |-  J  =  (
 MetOpen `  D )   &    |-  ( ph  ->  A  e.  X )   &    |-  G  =  seq  1
 ( ( F  o.  1st ) ,  ( NN 
 X.  { A } )
 )   =>    |-  ( ph  ->  E. z  e.  X  ( F `  z )  =  z
 )
 
Theorembfp 25714* Banach fixed point theorem, also known as contraction mapping theorem. A contraction on a complete metric space has a unique fixed point. We show existence in the lemmas, and uniqueness here - if  F has two fixed points, then the distance between them is less than  K times itself, a contradiction. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  ( ph  ->  D  e.  ( CMet `  X ) )   &    |-  ( ph  ->  X  =/=  (/) )   &    |-  ( ph  ->  K  e.  RR+ )   &    |-  ( ph  ->  K  <  1 )   &    |-  ( ph  ->  F : X --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( ( F `  x ) D ( F `  y ) )  <_  ( K  x.  ( x D y ) ) )   =>    |-  ( ph  ->  E! z  e.  X  ( F `  z )  =  z )
 
16.13.12  Euclidean space
 
Syntaxcrrn 25715 Extend class notation with the n-dimensional Euclidean space.
 class  Rn
 
Definitiondf-rrn 25716* Define n-dimensional Euclidean space as a metric space with the standard Euclidean norm given by the quadratic mean. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  Rn  =  ( i  e.  Fin  |->  ( x  e.  ( RR  ^m  i ) ,  y  e.  ( RR 
 ^m  i )  |->  ( sqr `  sum_ k  e.  i  ( ( ( x `  k )  -  ( y `  k ) ) ^
 2 ) ) ) )
 
Theoremrrnval 25717* The n-dimensional Euclidean space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  ( Rn `  I )  =  ( x  e.  X ,  y  e.  X  |->  ( sqr `  sum_ k  e.  I  ( (
 ( x `  k
 )  -  ( y `
  k ) ) ^ 2 ) ) ) )
 
Theoremrrnmval 25718* The value of the Euclidean metric. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( ( I  e. 
 Fin  /\  F  e.  X  /\  G  e.  X ) 
 ->  ( F ( Rn `  I ) G )  =  ( sqr `  sum_ k  e.  I  ( (
 ( F `  k
 )  -  ( G `
  k ) ) ^ 2 ) ) )
 
Theoremrrnmet 25719 Euclidean space is a metric space. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 5-Jun-2014.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( Met `  X ) )
 
Theoremrrndstprj1 25720 The distance between two points in Euclidean space is greater than the distance between the projections onto one coordinate. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( ( I  e.  Fin  /\  A  e.  I )  /\  ( F  e.  X  /\  G  e.  X ) )  ->  ( ( F `  A ) M ( G `  A ) )  <_  ( F ( Rn `  I ) G ) )
 
Theoremrrndstprj2 25721* Bound on the distance between two points in Euclidean space given bounds on the distances in each coordinate. This theorem and rrndstprj1 25720 can be used to show that the supremum norm and Euclidean norm are equivalent. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   =>    |-  ( ( ( I  e.  ( Fin  \  { (/)
 } )  /\  F  e.  X  /\  G  e.  X )  /\  ( R  e.  RR+  /\  A. n  e.  I  ( ( F `  n ) M ( G `  n ) )  <  R ) )  ->  ( F ( Rn `  I ) G )  <  ( R  x.  ( sqr `  ( # `
  I ) ) ) )
 
Theoremrrncmslem 25722* Lemma for rrncms 25723. (Contributed by Jeff Madsen, 6-Jun-2014.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( RR  X.  RR ) )   &    |-  J  =  (
 MetOpen `  ( Rn `  I
 ) )   &    |-  ( ph  ->  I  e.  Fin )   &    |-  ( ph  ->  F  e.  ( Cau `  ( Rn `  I
 ) ) )   &    |-  ( ph  ->  F : NN --> X )   &    |-  P  =  ( m  e.  I  |->  (  ~~>  `  ( t  e.  NN  |->  ( ( F `  t ) `  m ) ) ) )   =>    |-  ( ph  ->  F  e.  dom  ( ~~> t `  J ) )
 
Theoremrrncms 25723 Euclidean space is complete. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  ( Rn `  I )  e.  ( CMet `  X ) )
 
Theoremrepwsmet 25724 The supremum metric on  RR ^ I is a metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
 |-  Y  =  ( (flds  RR )  ^s  I )   &    |-  D  =  (
 dist `  Y )   &    |-  X  =  ( RR  ^m  I
 )   =>    |-  ( I  e.  Fin  ->  D  e.  ( Met `  X ) )
 
Theoremrrnequiv 25725 The supremum metric on  RR ^ I is equivalent to the  Rn metric. (Contributed by Jeff Madsen, 15-Sep-2015.)
 |-  Y  =  ( (flds  RR )  ^s  I )   &    |-  D  =  (
 dist `  Y )   &    |-  X  =  ( RR  ^m  I
 )   &    |-  ( ph  ->  I  e.  Fin )   =>    |-  ( ( ph  /\  ( F  e.  X  /\  G  e.  X )
 )  ->  ( ( F D G )  <_  ( F ( Rn `  I
 ) G )  /\  ( F ( Rn `  I
 ) G )  <_  ( ( sqr `  ( # `
  I ) )  x.  ( F D G ) ) ) )
 
Theoremrrntotbnd 25726 A set in Euclidean space is totally bounded iff its is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 16-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( Rn `  I )  |`  ( Y  X.  Y ) )   =>    |-  ( I  e.  Fin  ->  ( M  e.  ( TotBnd `
  Y )  <->  M  e.  ( Bnd `  Y ) ) )
 
Theoremrrnheibor 25727 Heine-Borel theorem for Euclidean space. A subset of Euclidean space is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  X  =  ( RR  ^m  I
 )   &    |-  M  =  ( ( Rn `  I )  |`  ( Y  X.  Y ) )   &    |-  T  =  (
 MetOpen `  M )   &    |-  U  =  ( MetOpen `  ( Rn `  I ) )   =>    |-  ( ( I  e.  Fin  /\  Y  C_  X )  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U )  /\  M  e.  ( Bnd `  Y ) ) ) )
 
16.13.13  Intervals (continued)
 
Theoremismrer1 25728* An isometry between  RR and  RR ^ 1. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  R  =  ( ( abs  o.  -  )  |`  ( RR  X. 
 RR ) )   &    |-  F  =  ( x  e.  RR  |->  ( { A }  X.  { x } ) )   =>    |-  ( A  e.  V  ->  F  e.  ( R 
 Ismty  ( Rn `  { A } ) ) )
 
Theoremreheibor 25729 Heine-Borel theorem for real numbers. A subset of  RR is compact iff it is closed and bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  M  =  ( ( abs  o.  -  )  |`  ( Y  X.  Y ) )   &    |-  T  =  ( MetOpen `  M )   &    |-  U  =  ( topGen `  ran  (,) )   =>    |-  ( Y  C_  RR  ->  ( T  e.  Comp  <->  ( Y  e.  ( Clsd `  U )  /\  M  e.  ( Bnd `  Y ) ) ) )
 
Theoremiccbnd 25730 A closed interval in  RR is bounded. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 22-Sep-2015.)
 |-  J  =  ( A [,] B )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( J  X.  J ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  M  e.  ( Bnd `  J ) )
 
TheoremicccmpALT 25731 A closed interval in  RR is compact. Alternate proof of icccmp 18162 using the Heine-Borel theorem heibor 25711. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 14-Aug-2014.)
 |-  J  =  ( A [,] B )   &    |-  M  =  ( ( abs  o.  -  )  |`  ( J  X.  J ) )   &    |-  T  =  (
 MetOpen `  M )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  T  e.  Comp )
 
16.13.14  Groups and related structures
 
Theoremexidcl 25732 Closure of the binary operation of a magma with identity. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  X  =  ran  G   =>    |-  ( ( G  e.  ( Magma  i^i  ExId  )  /\  A  e.  X  /\  B  e.  X )  ->  ( A G B )  e.  X )
 
Theoremexidreslem 25733* Lemma for exidres 25734 and exidresid 25735. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  ( Magma  i^i  ExId  ) 
 /\  Y  C_  X  /\  U  e.  Y ) 
 ->  ( U  e.  dom  dom 
 H  /\  A. x  e. 
 dom  dom  H ( ( U H x )  =  x  /\  ( x H U )  =  x ) ) )
 
Theoremexidres 25734 The restriction of a binary operation with identity to a subset containing the identity has an identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( G  e.  ( Magma  i^i  ExId  ) 
 /\  Y  C_  X  /\  U  e.  Y ) 
 ->  H  e.  ExId  )
 
Theoremexidresid 25735 The restriction of a binary operation with identity to a subset containing the identity has the same identity element. (Contributed by Jeff Madsen, 8-Jun-2010.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  H  =  ( G  |`  ( Y  X.  Y ) )   =>    |-  ( ( ( G  e.  ( Magma  i^i  ExId  )  /\  Y  C_  X  /\  U  e.  Y )  /\  H  e.  Magma ) 
 ->  (GId `  H )  =  U )
 
Theoremablo4pnp 25736 A commutative/associative law for Abelian groups. (Contributed by Jeff Madsen, 11-Jun-2010.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  AbelOp  /\  ( ( A  e.  X  /\  B  e.  X )  /\  ( C  e.  X  /\  F  e.  X ) ) )  ->  ( ( A G B ) D ( C G F ) )  =  ( ( A D C ) G ( B D F ) ) )
 
Theoremgrpoeqdivid 25737 Two group elements are equal iff their quotient is the identity. (Contributed by Jeff Madsen, 6-Jan-2011.)
 |-  X  =  ran  G   &    |-  U  =  (GId `  G )   &    |-  D  =  ( 
 /g  `  G )   =>    |-  (
 ( G  e.  GrpOp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  =  B  <->  ( A D B )  =  U ) )
 
Theoremghomf 25738 Mapping property of a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.)
 |-  X  =  ran  G   &    |-  W  =  ran  H   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  F : X
 --> W )
 
Theoremghomco 25739 The composition of two group homomorphisms is a group homomorphism. (Contributed by Jeff Madsen, 1-Dec-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
 |-  (
 ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  K  e.  GrpOp )  /\  ( S  e.  ( G GrpOpHom  H )  /\  T  e.  ( H GrpOpHom  K ) ) )  ->  ( T  o.  S )  e.  ( G GrpOpHom  K ) )
 
Theoremghomdiv 25740 Group homomorphisms preserve division. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  X  =  ran  G   &    |-  D  =  ( 
 /g  `  G )   &    |-  C  =  (  /g  `  H )   =>    |-  ( ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A D B ) )  =  (
 ( F `  A ) C ( F `  B ) ) )
 
Theoremgrpokerinj 25741 A group homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  X  =  ran  G   &    |-  W  =  (GId `  G )   &    |-  Y  =  ran  H   &    |-  U  =  (GId `  H )   =>    |-  ( ( G  e.  GrpOp  /\  H  e.  GrpOp  /\  F  e.  ( G GrpOpHom  H )
 )  ->  ( F : X -1-1-> Y  <->  ( `' F " { U } )  =  { W } )
 )
 
16.13.15  Rings
 
Theoremrngonegcl 25742 A ring is closed under negation. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( N `  A )  e.  X )
 
Theoremrngoaddneg1 25743 Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( A G ( N `  A ) )  =  Z )
 
Theoremrngoaddneg2 25744 Adding the negative in a ring gives zero. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  Z  =  (GId `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( ( N `  A ) G A )  =  Z )
 
Theoremrngosub 25745 Subtraction in a ring, in terms of addition and negation. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( A G ( N `  B ) ) )
 
Theoremrngonegmn1l 25746 Negation in a ring is the same as left multiplication by  -u 1. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( ( N `  U ) H A ) )
 
Theoremrngonegmn1r 25747 Negation in a ring is the same as right multiplication by  -u 1. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X ) 
 ->  ( N `  A )  =  ( A H ( N `  U ) ) )
 
Theoremrngoneglmul 25748 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( ( N `
  A ) H B ) )
 
Theoremrngonegrmul 25749 Negation of a product in a ring. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  N  =  ( inv `  G )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( N `  ( A H B ) )  =  ( A H ( N `  B ) ) )
 
Theoremrngosubdi 25750 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( A H ( B D C ) )  =  ( ( A H B ) D ( A H C ) ) )
 
Theoremrngosubdir 25751 Ring multiplication distributes over subtraction. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  D  =  (  /g  `  G )   =>    |-  ( ( R  e.  RingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A D B ) H C )  =  ( ( A H C ) D ( B H C ) ) )
 
Theoremzerdivemp1x 25752* In a unitary ring a left invertible element is not a zero divisor. Generalization of zerdivemp1 24602 by Frederic Line. (Contributed by Jeff Madsen, 18-Apr-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( ( R  e.  RingOps  /\  A  e.  X  /\  E. a  e.  X  ( a H A )  =  U )  ->  ( B  e.  X  ->  ( ( A H B )  =  Z  ->  B  =  Z ) ) )
 
Theoremisdrngo1 25753 The predicate "is a division ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   =>    |-  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( H  |`  ( ( X  \  { Z } )  X.  ( X  \  { Z }
 ) ) )  e. 
 GrpOp ) )
 
Theoremdivrngcl 25754 The product of two nonzero elements of a division ring is nonzero. (Contributed by Jeff Madsen, 9-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   =>    |-  ( ( R  e.  DivRingOps  /\  A  e.  ( X 
 \  { Z }
 )  /\  B  e.  ( X  \  { Z } ) )  ->  ( A H B )  e.  ( X  \  { Z } ) )
 
Theoremisdrngo2 25755* A division ring is a ring in which  1  =/=  0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z }
 ) E. y  e.  ( X  \  { Z } ) ( y H x )  =  U ) ) )
 
Theoremisdrngo3 25756* A division ring is a ring in which  1  =/=  0 and every nonzero element is invertible. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  Z  =  (GId `  G )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   =>    |-  ( R  e.  DivRingOps  <->  ( R  e.  RingOps  /\  ( U  =/=  Z  /\  A. x  e.  ( X  \  { Z }
 ) E. y  e.  X  ( y H x )  =  U ) ) )
 
16.13.16  Ring homomorphisms
 
Syntaxcrnghom 25757 Extend class notation with the class of ring homomorphisms.
 class  RngHom
 
Syntaxcrngiso 25758 Extend class notation with the class of ring isomorphisms.
 class  RngIso
 
Syntaxcrisc 25759 Extend class notation with the ring isomorphism relation.
 class  ~=r
 
Definitiondf-rngohom 25760* Define the function which gives the set of ring homomorphisms between two given rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  RngHom  =  ( r  e.  RingOps ,  s  e. 
 RingOps 
 |->  { f  e.  ( ran  ( 1st `  s
 )  ^m  ran  ( 1st `  r ) )  |  ( ( f `  (GId `  ( 2nd `  r
 ) ) )  =  (GId `  ( 2nd `  s ) )  /\  A. x  e.  ran  ( 1st `  r ) A. y  e.  ran  ( 1st `  r ) ( ( f `  ( x ( 1st `  r
 ) y ) )  =  ( ( f `
  x ) ( 1st `  s )
 ( f `  y
 ) )  /\  (
 f `  ( x ( 2nd `  r )
 y ) )  =  ( ( f `  x ) ( 2nd `  s ) ( f `
  y ) ) ) ) } )
 
Theoremrngohomval 25761* The set of ring homomorphisms. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 22-Sep-2015.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 2nd `  S )   &    |-  Y  =  ran  J   &    |-  V  =  (GId `  K )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngHom  S )  =  { f  e.  ( Y  ^m  X )  |  ( (
 f `  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( f `
  ( x G y ) )  =  ( ( f `  x ) J ( f `  y ) )  /\  ( f `
  ( x H y ) )  =  ( ( f `  x ) K ( f `  y ) ) ) ) }
 )
 
Theoremisrngohom 25762* The predicate "is a ring homomorphism from  R to  S." (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   &    |-  U  =  (GId `  H )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 2nd `  S )   &    |-  Y  =  ran  J   &    |-  V  =  (GId `  K )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngHom  S )  <->  ( F : X
 --> Y  /\  ( F `
  U )  =  V  /\  A. x  e.  X  A. y  e.  X  ( ( F `
  ( x G y ) )  =  ( ( F `  x ) J ( F `  y ) )  /\  ( F `
  ( x H y ) )  =  ( ( F `  x ) K ( F `  y ) ) ) ) ) )
 
Theoremrngohomf 25763 A ring homomorphism is a function. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F : X
 --> Y )
 
Theoremrngohomcl 25764 Closure law for a ring homomorphism. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  A  e.  X )  ->  ( F `
  A )  e.  Y )
 
Theoremrngohom1 25765 A ring homomorphism preserves  1. (Contributed by Jeff Madsen, 24-Jun-2011.)
 |-  H  =  ( 2nd `  R )   &    |-  U  =  (GId `  H )   &    |-  K  =  ( 2nd `  S )   &    |-  V  =  (GId `  K )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  U )  =  V )
 
Theoremrngohomadd 25766 Ring homomorphisms preserve addition. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   =>    |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R 
 RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A G B ) )  =  (
 ( F `  A ) J ( F `  B ) ) )
 
Theoremrngohommul 25767 Ring homomorphisms preserve multiplication. (Contributed by Jeff Madsen, 3-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  H  =  ( 2nd `  R )   &    |-  K  =  ( 2nd `  S )   =>    |-  ( ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R 
 RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A H B ) )  =  (
 ( F `  A ) K ( F `  B ) ) )
 
Theoremrngogrphom 25768 A ring homomorphism is a group homomorphism. (Contributed by Jeff Madsen, 2-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  J  =  ( 1st `  S )   =>    |-  ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  F  e.  ( G GrpOpHom  J ) )
 
Theoremrngohom0 25769 A ring homomorphism preserves  0. (Contributed by Jeff Madsen, 2-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  Z  =  (GId `  G )   &    |-  J  =  ( 1st `  S )   &    |-  W  =  (GId `  J )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F `  Z )  =  W )
 
Theoremrngohomsub 25770 Ring homomorphisms preserve subtraction. (Contributed by Jeff Madsen, 15-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  H  =  (  /g  `  G )   &    |-  J  =  ( 1st `  S )   &    |-  K  =  ( 
 /g  `  J )   =>    |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  /\  ( A  e.  X  /\  B  e.  X ) )  ->  ( F `  ( A H B ) )  =  ( ( F `
  A ) K ( F `  B ) ) )
 
Theoremrngohomco 25771 The composition of two ring homomorphisms is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e. 
 RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  G  e.  ( S  RngHom  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngHom  T ) )
 
Theoremrngokerinj 25772 A ring homomorphism is injective if and only if its kernel is zero. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  W  =  (GId `  G )   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   &    |-  Z  =  (GId `  J )   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngHom  S ) )  ->  ( F : X -1-1-> Y  <->  ( `' F " { Z } )  =  { W } )
 )
 
Definitiondf-rngoiso 25773* Define the function which gives the set of ring isomorphisms between two given rings. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  RngIso  =  ( r  e.  RingOps ,  s  e. 
 RingOps 
 |->  { f  e.  (
 r  RngHom  s )  |  f : ran  ( 1st `  r ) -1-1-onto-> ran  ( 1st `  s ) }
 )
 
Theoremrngoisoval 25774* The set of ring isomorphisms. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  RngIso  S )  =  { f  e.  ( R  RngHom  S )  |  f : X -1-1-onto-> Y } )
 
Theoremisrngoiso 25775 The predicate "is a ring isomorphism between  R and 
S." (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( F  e.  ( R  RngIso  S )  <->  ( F  e.  ( R  RngHom  S ) 
 /\  F : X -1-1-onto-> Y ) ) )
 
Theoremrngoiso1o 25776 A ring isomorphism is a bijection. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F : X
 -1-1-onto-> Y )
 
Theoremrngoisohom 25777 A ring isomorphism is a ring homomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  F  e.  ( R  RngHom  S ) )
 
Theoremrngoisocnv 25778 The inverse of a ring isomorphism is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  `' F  e.  ( S  RngIso  R ) )
 
Theoremrngoisoco 25779 The composition of two ring isomorphisms is a ring isomorphism. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( ( R  e.  RingOps  /\  S  e.  RingOps  /\  T  e. 
 RingOps )  /\  ( F  e.  ( R  RngIso  S )  /\  G  e.  ( S  RngIso  T ) ) )  ->  ( G  o.  F )  e.  ( R  RngIso  T ) )
 
Definitiondf-risc 25780* Define the ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  ~=r  =  { <. r ,  s >.  |  ( ( r  e.  RingOps  /\  s  e.  RingOps )  /\  E. f  f  e.  ( r  RngIso  s ) ) }
 
Theoremisriscg 25781* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  A  /\  S  e.  B ) 
 ->  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) ) )
 
Theoremisrisc 25782* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  R  e.  _V   &    |-  S  e.  _V   =>    |-  ( R  ~=r  S  <->  ( ( R  e.  RingOps  /\  S  e.  RingOps )  /\  E. f  f  e.  ( R  RngIso  S ) ) )
 
Theoremrisc 25783* The ring isomorphism relation. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps )  ->  ( R  ~=r  S  <->  E. f  f  e.  ( R  RngIso  S ) ) )
 
Theoremrisci 25784 Determine that two rings are isomorphic. (Contributed by Jeff Madsen, 16-Jun-2011.)
 |-  (
 ( R  e.  RingOps  /\  S  e.  RingOps  /\  F  e.  ( R  RngIso  S ) )  ->  R  ~=r  S )
 
Theoremriscer 25785 Ring isomorphism is an equivalence relation. (Contributed by Jeff Madsen, 16-Jun-2011.) (Revised by Mario Carneiro, 12-Aug-2015.)
 |-  ~=r  Er  dom  ~=r
 
16.13.17  Commutative rings
 
Syntaxccring 25786 Extend class notation with the class of commutative rings.
 class CRingOps
 
Definitiondf-crngo 25787 Define the class of commutative rings. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |- CRingOps  =  (
 RingOps  i^i  Com2 )
 
Theoremiscrngo 25788 The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  ( R  e. CRingOps  <->  ( R  e.  RingOps  /\  R  e.  Com2 )
 )
 
Theoremiscrngo2 25789* The predicate "is a commutative ring". (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( R  e. CRingOps  <->  ( R  e.  RingOps  /\ 
 A. x  e.  X  A. y  e.  X  ( x H y )  =  ( y H x ) ) )
 
Theoremiscringd 25790* Conditions that determine a commutative ring. (Contributed by Jeff Madsen, 20-Jun-2011.) (Revised by Mario Carneiro, 23-Dec-2013.)
 |-  ( ph  ->  G  e.  AbelOp )   &    |-  ( ph  ->  X  =  ran  G )   &    |-  ( ph  ->  H : ( X  X.  X ) --> X )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X )
 )  ->  ( ( x H y ) H z )  =  ( x H ( y H z ) ) )   &    |-  ( ( ph  /\  ( x  e.  X  /\  y  e.  X  /\  z  e.  X ) )  ->  ( x H ( y G z ) )  =  ( ( x H y ) G ( x H z ) ) )   &    |-  ( ph  ->  U  e.  X )   &    |-  (
 ( ph  /\  y  e.  X )  ->  (
 y H U )  =  y )   &    |-  (
 ( ph  /\  ( x  e.  X  /\  y  e.  X ) )  ->  ( x H y )  =  ( y H x ) )   =>    |-  ( ph  ->  <. G ,  H >.  e. CRingOps )
 
Theoremcrngorngo 25791 A commutative ring is a ring. (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( R  e. CRingOps  ->  R  e.  RingOps )
 
Theoremcrngocom 25792 The multiplication operation of a commutative ring is commutative. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  A  e.  X  /\  B  e.  X )  ->  ( A H B )  =  ( B H A ) )
 
Theoremcrngm23 25793 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
 )  ->  ( ( A H B ) H C )  =  ( ( A H C ) H B ) )
 
Theoremcrngm4 25794 Commutative/associative law for commutative rings. (Contributed by Jeff Madsen, 19-Jun-2010.)
 |-  G  =  ( 1st `  R )   &    |-  H  =  ( 2nd `  R )   &    |-  X  =  ran  G   =>    |-  ( ( R  e. CRingOps  /\  ( A  e.  X  /\  B  e.  X ) 
 /\  ( C  e.  X  /\  D  e.  X ) )  ->  ( ( A H B ) H ( C H D ) )  =  ( ( A H C ) H ( B H D ) ) )
 
Theoremfldcrng 25795 A field is a commutative ring. (Contributed by Jeff Madsen, 8-Jun-2010.)
 |-  ( K  e.  Fld  ->  K  e. CRingOps )
 
Theoremisfld2 25796 The predicate "is a field". (Contributed by Jeff Madsen, 10-Jun-2010.)
 |-  ( K  e.  Fld  <->  ( K  e.  DivRingOps  /\  K  e. CRingOps ) )
 
Theoremcrngohomfo 25797 The image of a homomorphism from a commutative ring is commutative. (Contributed by Jeff Madsen, 4-Jan-2011.)
 |-  G  =  ( 1st `  R )   &    |-  X  =  ran  G   &    |-  J  =  ( 1st `  S )   &    |-  Y  =  ran  J   =>    |-  (
 ( ( R  e. CRingOps  /\  S  e.  RingOps )  /\  ( F  e.  ( R  RngHom  S )  /\  F : X -onto-> Y ) )  ->  S  e. CRingOps )
 
16.13.18  Ideals
 
Syntaxcidl 25798 Extend class notation with the class of ideals.
 class  Idl
 
Syntaxcpridl 25799 Extend class notation with the class of prime ideals.
 class  PrIdl
 
Syntaxcmaxidl 25800 Extend class notation with the class of maximal ideals.
 class  MaxIdl
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