HomeHome Metamath Proof Explorer
Theorem List (p. 186 of 323)
< 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
(1-21500)
  Hilbert Space Explorer  Hilbert Space Explorer
(21501-23023)
  Users' Mathboxes  Users' Mathboxes
(23024-32227)
 

Theorem List for Metamath Proof Explorer - 18501-18600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Syntaxcomi 18501 Extend class notation with the loop space.
 class  Om 1
 
Syntaxcomn 18502 Extend class notation with the higher loop spaces.
 class  Om N
 
Syntaxcpi1 18503 Extend class notation with the fundamental group.
 class  pi 1
 
Syntaxcpin 18504 Extend class notation with the higher homotopy groups.
 class  pi N
 
Definitiondf-pco 18505* Define the concatenation of two paths in a topological space  J. For simplicity of definition, we define it on all paths, not just those whose endpoints line up. Definition of [Hatcher] p. 26. Hatcher denotes path concatenation with a square dot; other authors, such as Munkres, use a star. (Contributed by Jeff Madsen, 15-Jun-2010.)
 |- 
 *p  =  ( j  e.  Top  |->  ( f  e.  ( II  Cn  j ) ,  g  e.  ( II  Cn  j
 )  |->  ( x  e.  ( 0 [,] 1
 )  |->  if ( x  <_  ( 1  /  2
 ) ,  ( f `
  ( 2  x.  x ) ) ,  ( g `  (
 ( 2  x.  x )  -  1 ) ) ) ) ) )
 
Definitiondf-om1 18506* Define the loop space of a topological space, with a magma structure on it given by concatenation of loops. This structure is not a group, but the operation is compatible with homotopy, which allows the homotopy groups to be defined based on this operation. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |- 
 Om 1  =  ( j  e.  Top ,  y  e.  U. j  |->  {
 <. ( Base `  ndx ) ,  { f  e.  ( II  Cn  j )  |  ( ( f `  0 )  =  y  /\  ( f `  1
 )  =  y ) } >. ,  <. ( +g  ` 
 ndx ) ,  ( *p `  j ) >. , 
 <. (TopSet `  ndx ) ,  ( j  ^ k o  II ) >. } )
 
Definitiondf-omn 18507* Define the n-th iterated loop space of a topological space. Unlike  Om 1 this is actually a pointed topological space, which is to say a tuple of a topological space (a member of  TopSp, not  Top) and a point in the space. Higher loop spaces select the constant loop at the point from the lower loop space for the distinguished point. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |- 
 Om N  =  ( j  e.  Top ,  y  e.  U. j  |->  seq  0 ( ( ( x  e.  _V ,  p  e.  _V  |->  <. ( (
 TopOpen `  ( 1st `  x ) )  Om 1  ( 2nd `  x )
 ) ,  ( ( 0 [,] 1 )  X.  { ( 2nd `  x ) } ) >. )  o.  1st ) ,  <. { <. ( Base ` 
 ndx ) ,  U. j >. ,  <. (TopSet `  ndx ) ,  j >. } ,  y >. ) )
 
Definitiondf-pi1 18508* Define the fundamental group, whose operation is given by concatenation of homotopy classes of loops. Definition of [Hatcher] p. 26. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  pi 1  =  ( j  e.  Top ,  y  e.  U. j  |->  ( ( j  Om 1  y )  /.s  (  ~=ph  `  j )
 ) )
 
Definitiondf-pin 18509* Define the n-th homotopy group, which is formed by taking the  n-th loop space and forming the quotient under the relation of path homotopy equivalence in the base space of the  n-th loop space, which is the  n  -  1-th loop space. For  n  =  0, since this is not well-defined we replace this relation with the path-connectedness relation, so that the  0-th homotopy group is the set of path components of  X. (Since the  0-th loop space does not have a group operation, neither does the  0-th homotopy group, but the rest are genuine groups.) (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  pi N  =  ( j  e.  Top ,  p  e.  U. j  |->  ( n  e.  NN0  |->  ( ( 1st `  ( (
 j  Om N  p ) `
  n ) ) 
 /.s 
 if ( n  =  0 ,  { <. x ,  y >.  |  E. f  e.  ( II  Cn  j ) ( ( f `  0 )  =  x  /\  (
 f `  1 )  =  y ) } ,  (  ~=ph  `  ( TopOpen `  ( 1st `  ( ( j 
 Om N  p ) `
  ( n  -  1 ) ) ) ) ) ) ) ) )
 
Theorempcofval 18510* The value of the path concatenation function on a topological space. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
 |-  ( *p `  J )  =  ( f  e.  ( II  Cn  J ) ,  g  e.  ( II  Cn  J ) 
 |->  ( x  e.  (
 0 [,] 1 )  |->  if ( x  <_  (
 1  /  2 ) ,  ( f `  (
 2  x.  x ) ) ,  ( g `
  ( ( 2  x.  x )  -  1 ) ) ) ) )
 
Theorempcoval 18511* The concatenation of two paths. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Aug-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  ( F ( *p `  J ) G )  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
 1  /  2 ) ,  ( F `  (
 2  x.  x ) ) ,  ( G `
  ( ( 2  x.  x )  -  1 ) ) ) ) )
 
Theorempcovalg 18512 Evaluate the concatenation of two paths. (Contributed by Mario Carneiro, 7-Jun-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ( ph  /\  X  e.  ( 0 [,] 1 ) ) 
 ->  ( ( F ( *p `  J ) G ) `  X )  =  if ( X  <_  ( 1  / 
 2 ) ,  ( F `  ( 2  x.  X ) ) ,  ( G `  (
 ( 2  x.  X )  -  1 ) ) ) )
 
Theorempcoval1 18513 Evaluate the concatenation of two paths on the first half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ( ph  /\  X  e.  ( 0 [,] ( 1  / 
 2 ) ) ) 
 ->  ( ( F ( *p `  J ) G ) `  X )  =  ( F `  ( 2  x.  X ) ) )
 
Theorempco0 18514 The starting point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  ( ( F ( *p `  J ) G ) `
  0 )  =  ( F `  0
 ) )
 
Theorempco1 18515 The ending point of a path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  ( ( F ( *p `  J ) G ) `
  1 )  =  ( G `  1
 ) )
 
Theorempcoval2 18516 Evaluate the concatenation of two paths on the second half. (Contributed by Jeff Madsen, 15-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( G `  0 ) )   =>    |-  ( ( ph  /\  X  e.  ( ( 1  / 
 2 ) [,] 1
 ) )  ->  (
 ( F ( *p `  J ) G ) `
  X )  =  ( G `  (
 ( 2  x.  X )  -  1 ) ) )
 
Theorempcocn 18517 The concatenation of two paths is a path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( G `  0 ) )   =>    |-  ( ph  ->  ( F ( *p `  J ) G )  e.  ( II  Cn  J ) )
 
Theoremcopco 18518 The composition of a concatenation of paths with a continuous function. (Contributed by Mario Carneiro, 9-Jul-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( G `  0 ) )   &    |-  ( ph  ->  H  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( H  o.  ( F ( *p `  J ) G ) )  =  ( ( H  o.  F ) ( *p `  K ) ( H  o.  G ) ) )
 
Theorempcohtpylem 18519* Lemma for pcohtpy 18520. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  ( F `  1 )  =  ( G `  0
 ) )   &    |-  ( ph  ->  F (  ~=ph  `  J ) H )   &    |-  ( ph  ->  G (  ~=ph  `  J ) K )   &    |-  P  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  if ( x  <_  (
 1  /  2 ) ,  ( ( 2  x.  x ) M y ) ,  ( ( ( 2  x.  x )  -  1 ) N y ) ) )   &    |-  ( ph  ->  M  e.  ( F ( PHtpy `  J ) H ) )   &    |-  ( ph  ->  N  e.  ( G ( PHtpy `  J ) K ) )   =>    |-  ( ph  ->  P  e.  ( ( F ( *p `  J ) G ) ( PHtpy `  J ) ( H ( *p `  J ) K ) ) )
 
Theorempcohtpy 18520 Homotopy invariance of path concatenation. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 24-Feb-2015.)
 |-  ( ph  ->  ( F `  1 )  =  ( G `  0
 ) )   &    |-  ( ph  ->  F (  ~=ph  `  J ) H )   &    |-  ( ph  ->  G (  ~=ph  `  J ) K )   =>    |-  ( ph  ->  ( F ( *p `  J ) G ) (  ~=ph  `  J ) ( H ( *p `  J ) K ) )
 
Theorempcoptcl 18521 A constant function is a path from 
Y to itself. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 19-Mar-2015.)
 |-  P  =  ( ( 0 [,] 1 )  X.  { Y }
 )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  Y  e.  X )  ->  ( P  e.  ( II  Cn  J )  /\  ( P `  0 )  =  Y  /\  ( P `  1 )  =  Y ) )
 
Theorempcopt 18522 Concatenation with a point does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  P  =  ( ( 0 [,] 1 )  X.  { Y }
 )   =>    |-  ( ( F  e.  ( II  Cn  J ) 
 /\  ( F `  0 )  =  Y )  ->  ( P ( *p `  J ) F ) (  ~=ph  `  J ) F )
 
Theorempcopt2 18523 Concatenation with a point does not affect homotopy class. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  P  =  ( ( 0 [,] 1 )  X.  { Y }
 )   =>    |-  ( ( F  e.  ( II  Cn  J ) 
 /\  ( F `  1 )  =  Y )  ->  ( F ( *p `  J ) P ) (  ~=ph  `  J ) F )
 
Theorempcoass 18524* Order of concatenation does not affect homotopy class. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( G `  0 ) )   &    |-  ( ph  ->  ( G `  1 )  =  ( H `  0 ) )   &    |-  P  =  ( x  e.  (
 0 [,] 1 )  |->  if ( x  <_  (
 1  /  2 ) ,  if ( x  <_  ( 1  /  4
 ) ,  ( 2  x.  x ) ,  ( x  +  (
 1  /  4 )
 ) ) ,  (
 ( x  /  2
 )  +  ( 1 
 /  2 ) ) ) )   =>    |-  ( ph  ->  (
 ( F ( *p `  J ) G ) ( *p `  J ) H ) (  ~=ph  `  J ) ( F ( *p `  J ) ( G ( *p `  J ) H ) ) )
 
Theorempcorevcl 18525* Closure for a reversed path. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  G  =  ( x  e.  ( 0 [,] 1 )  |->  ( F `
  ( 1  -  x ) ) )   =>    |-  ( F  e.  ( II  Cn  J )  ->  ( G  e.  ( II  Cn  J )  /\  ( G `  0 )  =  ( F `  1 )  /\  ( G `
  1 )  =  ( F `  0
 ) ) )
 
Theorempcorevlem 18526* Lemma for pcorev 18527. Prove continuity of the homotopy function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.)
 |-  G  =  ( x  e.  ( 0 [,] 1 )  |->  ( F `
  ( 1  -  x ) ) )   &    |-  P  =  ( (
 0 [,] 1 )  X.  { ( F `  1
 ) } )   &    |-  H  =  ( s  e.  (
 0 [,] 1 ) ,  t  e.  ( 0 [,] 1 )  |->  ( F `  if (
 s  <_  ( 1  /  2 ) ,  ( 1  -  (
 ( 1  -  t
 )  x.  ( 2  x.  s ) ) ) ,  ( 1  -  ( ( 1  -  t )  x.  ( 1  -  (
 ( 2  x.  s
 )  -  1 ) ) ) ) ) ) )   =>    |-  ( F  e.  ( II  Cn  J )  ->  ( H  e.  (
 ( G ( *p `  J ) F ) ( PHtpy `  J ) P )  /\  ( G ( *p `  J ) F ) (  ~=ph  `  J ) P ) )
 
Theorempcorev 18527* Concatenation with the reverse path. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 20-Dec-2013.)
 |-  G  =  ( x  e.  ( 0 [,] 1 )  |->  ( F `
  ( 1  -  x ) ) )   &    |-  P  =  ( (
 0 [,] 1 )  X.  { ( F `  1
 ) } )   =>    |-  ( F  e.  ( II  Cn  J ) 
 ->  ( G ( *p `  J ) F ) (  ~=ph  `  J ) P )
 
Theorempcorev2 18528* Concatenation with the reverse path. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  G  =  ( x  e.  ( 0 [,] 1 )  |->  ( F `
  ( 1  -  x ) ) )   &    |-  P  =  ( (
 0 [,] 1 )  X.  { ( F `  0
 ) } )   =>    |-  ( F  e.  ( II  Cn  J ) 
 ->  ( F ( *p `  J ) G ) (  ~=ph  `  J ) P )
 
Theorempcophtb 18529* The path homotopy equivalence relation on two paths  F ,  G with the same start and end point can be written in terms of the loop  F  -  G formed by concatenating  F with the inverse of  G. Thus, all the homotopy information in 
~=ph  `  J is available if we restrict our attention to closed loops, as in the definition of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  H  =  ( x  e.  ( 0 [,] 1 )  |->  ( G `
  ( 1  -  x ) ) )   &    |-  P  =  ( (
 0 [,] 1 )  X.  { ( F `  0
 ) } )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  0 )  =  ( G `  0 ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( G `  1 ) )   =>    |-  ( ph  ->  ( ( F ( *p `  J ) H ) (  ~=ph  `  J ) P  <->  F (  ~=ph  `  J ) G ) )
 
Theoremom1val 18530* The definition of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  B  =  { f  e.  ( II  Cn  J )  |  ( ( f `  0 )  =  Y  /\  ( f `  1
 )  =  Y ) } )   &    |-  ( ph  ->  .+  =  ( *p `  J ) )   &    |-  ( ph  ->  K  =  ( J  ^ k o  II )
 )   &    |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  O  =  { <. ( Base `  ndx ) ,  B >. , 
 <. ( +g  `  ndx ) ,  .+  >. ,  <. (TopSet `  ndx ) ,  K >. } )
 
Theoremom1bas 18531* The base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  O ) )   =>    |-  ( ph  ->  B  =  { f  e.  ( II  Cn  J )  |  ( ( f `  0 )  =  Y  /\  ( f `  1
 )  =  Y ) } )
 
Theoremom1elbas 18532 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  O ) )   =>    |-  ( ph  ->  ( F  e.  B  <->  ( F  e.  ( II  Cn  J ) 
 /\  ( F `  0 )  =  Y  /\  ( F `  1
 )  =  Y ) ) )
 
Theoremom1addcl 18533 Closure of the group operation of the loop space. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  O ) )   &    |-  ( ph  ->  H  e.  B )   &    |-  ( ph  ->  K  e.  B )   =>    |-  ( ph  ->  ( H ( *p `  J ) K )  e.  B )
 
Theoremom1plusg 18534 The group operation (which isn't much more than a magma) of the loop space. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  ( *p `  J )  =  ( +g  `  O ) )
 
Theoremom1tset 18535 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  ( J  ^ k o  II )  =  (TopSet `  O ) )
 
Theoremom1opn 18536 The topology of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  O  =  ( J 
 Om 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  K  =  (
 TopOpen `  O )   &    |-  ( ph  ->  B  =  (
 Base `  O ) )   =>    |-  ( ph  ->  K  =  ( ( J  ^ k o  II )t  B ) )
 
Theorempi1val 18537 The definition of the fundamental group. (Contributed by Mario Carneiro, 11-Feb-2015.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  O  =  ( J  Om 1  Y )   =>    |-  ( ph  ->  G  =  ( O  /.s  (  ~=ph  `  J ) ) )
 
Theorempi1bas 18538 The base set of the fundamental group of a topological space at a given base point. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  O  =  ( J  Om 1  Y )   &    |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  K  =  ( Base `  O )
 )   =>    |-  ( ph  ->  B  =  ( K /. (  ~=ph  `  J ) ) )
 
Theorempi1blem 18539 Lemma for pi1buni 18540. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  O  =  ( J  Om 1  Y )   &    |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  K  =  ( Base `  O )
 )   =>    |-  ( ph  ->  (
 ( (  ~=ph  `  J ) " K )  C_  K  /\  K  C_  ( II  Cn  J ) ) )
 
Theorempi1buni 18540 Another way to write the loop space base in terms of the base of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  O  =  ( J  Om 1  Y )   &    |-  ( ph  ->  B  =  ( Base `  G )
 )   &    |-  ( ph  ->  K  =  ( Base `  O )
 )   =>    |-  ( ph  ->  U. B  =  K )
 
Theorempi1bas2 18541 The base set of the fundamental group, written self-referentially. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  G ) )   =>    |-  ( ph  ->  B  =  ( U. B /. (  ~=ph  `  J )
 ) )
 
Theorempi1eluni 18542 Elementhood in the base set of the loop space. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  G ) )   =>    |-  ( ph  ->  ( F  e.  U. B  <->  ( F  e.  ( II  Cn  J ) 
 /\  ( F `  0 )  =  Y  /\  ( F `  1
 )  =  Y ) ) )
 
Theorempi1bas3 18543 The base set of the fundamental group. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  G ) )   &    |-  R  =  ( (  ~=ph  `  J )  i^i  ( U. B  X.  U. B ) )   =>    |-  ( ph  ->  B  =  ( U. B /. R ) )
 
Theorempi1cpbl 18544 The group operation, loop concatenation, is compatible with homotopy equivalence. (Contributed by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  B  =  ( Base `  G ) )   &    |-  R  =  ( (  ~=ph  `  J )  i^i  ( U. B  X.  U. B ) )   &    |-  O  =  ( J  Om 1  Y )   &    |-  .+  =  ( +g  `  O )   =>    |-  ( ph  ->  ( ( M R N  /\  P R Q ) 
 ->  ( M  .+  P ) R ( N  .+  Q ) ) )
 
Theoremelpi1 18545* The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   =>    |-  ( ph  ->  ( F  e.  B  <->  E. f  e.  ( II  Cn  J ) ( ( ( f `  0 )  =  Y  /\  ( f `  1
 )  =  Y ) 
 /\  F  =  [
 f ] (  ~=ph  `  J ) ) ) )
 
Theoremelpi1i 18546 The elements of the fundamental group. (Contributed by Jeff Madsen, 19-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  0 )  =  Y )   &    |-  ( ph  ->  ( F `  1 )  =  Y )   =>    |-  ( ph  ->  [ F ] (  ~=ph  `  J )  e.  B )
 
Theorempi1addf 18547 The group operation of  pi 1 is a binary operation. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  .+  =  ( +g  `  G )   =>    |-  ( ph  ->  .+  : ( B  X.  B ) --> B )
 
Theorempi1addval 18548 The concatenation of two path-homotopy classes in the fundamental group. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Jul-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  .+  =  ( +g  `  G )   &    |-  ( ph  ->  M  e.  U. B )   &    |-  ( ph  ->  N  e.  U. B )   =>    |-  ( ph  ->  ( [ M ] (  ~=ph  `  J )  .+  [ N ]
 (  ~=ph  `  J )
 )  =  [ ( M ( *p `  J ) N ) ] (  ~=ph  `  J ) )
 
Theorempi1grplem 18549 Lemma for pi1grp 18550. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 10-Aug-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  B  =  ( Base `  G )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  .0.  =  ( ( 0 [,] 1 )  X.  { Y } )   =>    |-  ( ph  ->  ( G  e.  Grp  /\  [  .0.  ] (  ~=ph  `  J )  =  ( 0g `  G ) ) )
 
Theorempi1grp 18550 The fundamental group is a group. Proposition 1.3 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 19-Jun-2010.) (Proof shortened by Mario Carneiro, 8-Jun-2014.) (Revised by Mario Carneiro, 10-Aug-2015.)
 |-  G  =  ( J  pi 1  Y )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  Y  e.  X )  ->  G  e.  Grp )
 
Theorempi1id 18551 The identity element of the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  .0.  =  ( ( 0 [,] 1 )  X.  { Y } )   =>    |-  ( ( J  e.  (TopOn `  X )  /\  Y  e.  X )  ->  [  .0.  ] (  ~=ph  `  J )  =  ( 0g `  G ) )
 
Theorempi1inv 18552* An inverse in the fundamental group. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 10-Aug-2015.)
 |-  G  =  ( J  pi 1  Y )   &    |-  N  =  ( inv g `
  G )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  Y  e.  X )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  0 )  =  Y )   &    |-  ( ph  ->  ( F `  1 )  =  Y )   &    |-  I  =  ( x  e.  ( 0 [,] 1 )  |->  ( F `
  ( 1  -  x ) ) )   =>    |-  ( ph  ->  ( N ` 
 [ F ] (  ~=ph  `  J ) )  =  [ I ] (  ~=ph  `  J ) )
 
Theorempi1xfrf 18553* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  ( F `
  0 ) )   &    |-  Q  =  ( J  pi 1  ( F `  1 ) )   &    |-  B  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p `  J ) F ) ) ]
 (  ~=ph  `  J ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  I  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( I `  0 ) )   &    |-  ( ph  ->  ( I `  1 )  =  ( F `  0 ) )   =>    |-  ( ph  ->  G : B --> ( Base `  Q ) )
 
Theorempi1xfrval 18554* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 12-Feb-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  ( F `
  0 ) )   &    |-  Q  =  ( J  pi 1  ( F `  1 ) )   &    |-  B  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p `  J ) F ) ) ]
 (  ~=ph  `  J ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  I  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  ( F `  1 )  =  ( I `  0 ) )   &    |-  ( ph  ->  ( I `  1 )  =  ( F `  0 ) )   &    |-  ( ph  ->  A  e.  U. B )   =>    |-  ( ph  ->  ( G `  [ A ]
 (  ~=ph  `  J )
 )  =  [ ( I ( *p `  J ) ( A ( *p `  J ) F ) ) ]
 (  ~=ph  `  J )
 )
 
Theorempi1xfr 18555* Given a path  F and its inverse  I between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  P  =  ( J  pi 1  ( F `
  0 ) )   &    |-  Q  =  ( J  pi 1  ( F `  1 ) )   &    |-  B  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p `  J ) F ) ) ]
 (  ~=ph  `  J ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  I  =  ( x  e.  (
 0 [,] 1 )  |->  ( F `  ( 1  -  x ) ) )   =>    |-  ( ph  ->  G  e.  ( P  GrpHom  Q ) )
 
Theorempi1xfrcnvlem 18556* Given a path  F between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  ( F `
  0 ) )   &    |-  Q  =  ( J  pi 1  ( F `  1 ) )   &    |-  B  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p `  J ) F ) ) ]
 (  ~=ph  `  J ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  I  =  ( x  e.  (
 0 [,] 1 )  |->  ( F `  ( 1  -  x ) ) )   &    |-  H  =  ran  ( h  e.  U. ( Base `  Q )  |->  <. [ h ] (  ~=ph  `  J ) ,  [
 ( F ( *p `  J ) ( h ( *p `  J ) I ) ) ]
 (  ~=ph  `  J ) >. )   =>    |-  ( ph  ->  `' G  C_  H )
 
Theorempi1xfrcnv 18557* Given a path  F between two basepoints, there is an induced group homomorphism on the fundamental groups. (Contributed by Mario Carneiro, 12-Feb-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  ( F `
  0 ) )   &    |-  Q  =  ( J  pi 1  ( F `  1 ) )   &    |-  B  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p `  J ) F ) ) ]
 (  ~=ph  `  J ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  I  =  ( x  e.  (
 0 [,] 1 )  |->  ( F `  ( 1  -  x ) ) )   &    |-  H  =  ran  ( h  e.  U. ( Base `  Q )  |->  <. [ h ] (  ~=ph  `  J ) ,  [
 ( F ( *p `  J ) ( h ( *p `  J ) I ) ) ]
 (  ~=ph  `  J ) >. )   =>    |-  ( ph  ->  ( `' G  =  H  /\  `' G  e.  ( Q  GrpHom  P ) ) )
 
Theorempi1xfrgim 18558* The mapping  G between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.)
 |-  P  =  ( J  pi 1  ( F `
  0 ) )   &    |-  Q  =  ( J  pi 1  ( F `  1 ) )   &    |-  B  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. B  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( I ( *p `  J ) ( g ( *p `  J ) F ) ) ]
 (  ~=ph  `  J ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  I  =  ( x  e.  (
 0 [,] 1 )  |->  ( F `  ( 1  -  x ) ) )   =>    |-  ( ph  ->  G  e.  ( P GrpIso  Q )
 )
 
Theorempi1cof 18559* Functionality of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  A )   &    |-  Q  =  ( K  pi 1  B )   &    |-  V  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. V  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( F  o.  g ) ]
 (  ~=ph  `  K ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  ( F `  A )  =  B )   =>    |-  ( ph  ->  G : V --> ( Base `  Q ) )
 
Theorempi1coval 18560* The value of the loop transfer function on the equivalence class of a path. (Contributed by Mario Carneiro, 10-Aug-2015.) (Proof shortened by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  A )   &    |-  Q  =  ( K  pi 1  B )   &    |-  V  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. V  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( F  o.  g ) ]
 (  ~=ph  `  K ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  ( F `  A )  =  B )   =>    |-  ( ( ph  /\  T  e.  U. V )  ->  ( G `  [ T ] (  ~=ph  `  J ) )  =  [
 ( F  o.  T ) ] (  ~=ph  `  K ) )
 
Theorempi1coghm 18561* The mapping  G between fundamental groups is a group homomorphism. (Contributed by Mario Carneiro, 10-Aug-2015.) (Revised by Mario Carneiro, 23-Dec-2016.)
 |-  P  =  ( J  pi 1  A )   &    |-  Q  =  ( K  pi 1  B )   &    |-  V  =  ( Base `  P )   &    |-  G  =  ran  ( g  e. 
 U. V  |->  <. [ g ] (  ~=ph  `  J ) ,  [ ( F  o.  g ) ]
 (  ~=ph  `  K ) >. )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  A  e.  X )   &    |-  ( ph  ->  ( F `  A )  =  B )   =>    |-  ( ph  ->  G  e.  ( P  GrpHom  Q ) )
 
11.4  Complex metric vector spaces
 
11.4.1  Complex left modules
 
Syntaxcclm 18562 Complex module.
 class CMod
 
Definitiondf-clm 18563* Define a complex module, which is just a left module over a subring of  CC, which allows us to use conventional addition, multiplication, etc. in the left module theorems. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |- CMod  =  { w  e.  LMod  | 
 [. (Scalar `  w )  /  f ]. [. ( Base `  f )  /  k ]. ( f  =  (flds  k )  /\  k  e.  (SubRing ` fld ) ) }
 
Theoremisclm 18564 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  <->  ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) ) )
 
Theoremclmsca 18565 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  ->  F  =  (flds  K ) )
 
Theoremclmsubrg 18566 A complex module is a left module over a subring of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  ->  K  e.  (SubRing ` fld ) )
 
Theoremclmlmod 18567 A complex module is a left module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  ( W  e. CMod  ->  W  e.  LMod )
 
Theoremclmgrp 18568 A complex module is an additive group. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  ( W  e. CMod  ->  W  e.  Grp )
 
Theoremclmabl 18569 A complex module is an abelian group. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  ( W  e. CMod  ->  W  e.  Abel )
 
Theoremclmrng 18570 The scalar ring of a complex module is a ring. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  F  e.  Ring )
 
Theoremclmfgrp 18571 The scalar ring of a complex module is a group. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  F  e.  Grp )
 
Theoremclm0 18572 The zero of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  0  =  ( 0g `  F ) )
 
Theoremclm1 18573 The identity of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  1  =  ( 1r `  F ) )
 
Theoremclmadd 18574 The addition of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  +  =  ( +g  `  F ) )
 
Theoremclmmul 18575 The multiplication of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  x.  =  ( .r `  F ) )
 
Theoremclmcj 18576 The conjugation of the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( W  e. CMod  ->  *  =  ( * r `
  F ) )
 
Theoremisclmi 18577 Reverse direction of isclm 18564. (Contributed by Mario Carneiro, 30-Oct-2015.)
 |-  F  =  (Scalar `  W )   =>    |-  ( ( W  e.  LMod  /\  F  =  (flds  K )  /\  K  e.  (SubRing ` fld ) )  ->  W  e. CMod )
 
Theoremclmzss 18578 The scalar ring of a complex module contains the integers. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  ->  ZZ  C_  K )
 
Theoremclmsscn 18579 The scalar ring of a complex module is a subset of the complexes. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( W  e. CMod  ->  K 
 C_  CC )
 
Theoremclmsub 18580 Subtraction in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  A  e.  K  /\  B  e.  K )  ->  ( A  -  B )  =  ( A ( -g `  F ) B ) )
 
Theoremclmneg 18581 Negation in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  A  e.  K ) 
 ->  -u A  =  ( ( inv g `  F ) `  A ) )
 
Theoremclmabs 18582 Norm in the scalar ring of a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  A  e.  K ) 
 ->  ( abs `  A )  =  ( ( norm `  F ) `  A ) )
 
Theoremclmacl 18583 Closure of ring addition for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  +  Y )  e.  K )
 
Theoremclmmcl 18584 Closure of ring multiplication for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  x.  Y )  e.  K )
 
Theoremclmsubcl 18585 Closure of ring subtraction for a complex module. (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   =>    |-  ( ( W  e. CMod  /\  X  e.  K  /\  Y  e.  K )  ->  ( X  -  Y )  e.  K )
 
Theoremlmhmclm 18586 The domain of a linear operator is a complex module iff the range is. (Contributed by Mario Carneiro, 21-Oct-2015.)
 |-  ( F  e.  ( S LMHom  T )  ->  ( S  e. CMod  <->  T  e. CMod ) )
 
Theoremclmvsass 18587 Associative law for scalar product. (lmodvsass 15656 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   =>    |-  (
 ( W  e. CMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
 )  ->  ( ( Q  x.  R )  .x.  X )  =  ( Q 
 .x.  ( R  .x.  X ) ) )
 
Theoremclmvsdir 18588 Distributive law for scalar product. (lmodvsdir 15654 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  K  =  ( Base `  F )   &    |-  .+  =  ( +g  `  W )   =>    |-  (
 ( W  e. CMod  /\  ( Q  e.  K  /\  R  e.  K  /\  X  e.  V )
 )  ->  ( ( Q  +  R )  .x.  X )  =  ( ( Q  .x.  X )  .+  ( R  .x.  X ) ) )
 
Theoremclmvs1 18589 Scalar product with ring unit. (lmodvs1 15660 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e. CMod  /\  X  e.  V )  ->  ( 1 
 .x.  X )  =  X )
 
Theoremclm0vs 18590 Zero times a vector is the zero vector. Equation 1a of [Kreyszig] p. 51. (lmod0vs 15665 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  .0.  =  ( 0g `  W )   =>    |-  ( ( W  e. CMod  /\  X  e.  V ) 
 ->  ( 0  .x.  X )  =  .0.  )
 
Theoremclmvneg1 18591 Minus 1 times a vector is the negative of the vector. Equation 2 of [Kreyszig] p. 51. (lmodvneg1 15669 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  N  =  ( inv g `  W )   &    |-  F  =  (Scalar `  W )   &    |- 
 .x.  =  ( .s `  W )   =>    |-  ( ( W  e. CMod  /\  X  e.  V ) 
 ->  ( -u 1  .x.  X )  =  ( N `  X ) )
 
Theoremclmvsneg 18592 Multiplication of a vector by a negated scalar. (lmodvsneg 15671 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  B  =  ( Base `  W )   &    |-  F  =  (Scalar `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  N  =  ( inv g `  W )   &    |-  K  =  (
 Base `  F )   &    |-  ( ph  ->  W  e. CMod )   &    |-  ( ph  ->  X  e.  B )   &    |-  ( ph  ->  R  e.  K )   =>    |-  ( ph  ->  ( N `  ( R  .x.  X ) )  =  (
 -u R  .x.  X ) )
 
Theoremclmmulg 18593 The group multiple function matches the scalar multiplication function. (Contributed by Mario Carneiro, 15-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .xb  =  (.g `  W )   &    |-  .x.  =  ( .s `  W )   =>    |-  ( ( W  e. CMod  /\  A  e.  ZZ  /\  B  e.  V )  ->  ( A  .xb  B )  =  ( A 
 .x.  B ) )
 
Theoremclmsubdir 18594 Scalar multiplication distributive law for subtraction. (lmodsubdir 15685 analog.) (Contributed by Mario Carneiro, 16-Oct-2015.)
 |-  V  =  ( Base `  W )   &    |-  .x.  =  ( .s `  W )   &    |-  F  =  (Scalar `  W )   &    |-  K  =  ( Base `  F )   &    |-  .-  =  ( -g `  W )   &    |-  ( ph  ->  W  e. CMod )   &    |-  ( ph  ->  A  e.  K )   &    |-  ( ph  ->  B  e.  K )   &    |-  ( ph  ->  X  e.  V )   =>    |-  ( ph  ->  (
 ( A  -  B )  .x.  X )  =  ( ( A  .x.  X )  .-  ( B  .x.  X ) ) )
 
Theoremzlmclm 18595 The  ZZ-module operation turns an arbitrary abelian group into a complex module. (Contributed by Mario Carneiro, 30-Oct-2015.)
 |-  W  =  ( ZMod `  G )   =>    |-  ( G  e.  Abel  <->  W  e. CMod )
 
Theoremclmzlmvsca 18596 The scalar product of a complex module matches the scalar product of the derived  ZZ-module, which implies, together with zlmbas 16474 and zlmplusg 16475, that any module over  ZZ is structure-equivalent to the canonical  ZZ-module  ZMod `  G. (Contributed by Mario Carneiro, 30-Oct-2015.)
 |-  W  =  ( ZMod `  G )   &    |-  X  =  (
 Base `  G )   =>    |-  ( ( G  e. CMod  /\  ( A  e.  ZZ  /\  B  e.  X ) )  ->  ( A ( .s `  G ) B )  =  ( A ( .s `  W ) B ) )
 
Theoremnmoleub2lem 18597* Lemma for nmoleub2a 18600 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ( ph  /\ 
 A. x  e.  V  ( ps  ->  ( ( M `  ( F `  x ) )  /  R )  <_  A ) )  ->  0  <_  A )   &    |-  ( ( ( ( ph  /\  A. x  e.  V  ( ps  ->  ( ( M `
  ( F `  x ) )  /  R )  <_  A ) )  /\  A  e.  RR )  /\  ( y  e.  V  /\  y  =/=  ( 0g `  S ) ) )  ->  ( M `  ( F `
  y ) ) 
 <_  ( A  x.  ( L `  y ) ) )   &    |-  ( ( ph  /\  x  e.  V ) 
 ->  ( ps  ->  ( L `  x )  <_  R ) )   =>    |-  ( ph  ->  ( ( N `  F )  <_  A  <->  A. x  e.  V  ( ps  ->  ( ( M `  ( F `  x ) )  /  R )  <_  A ) ) )
 
Theoremnmoleub2lem3 18598* Lemma for nmoleub2a 18600 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   &    |-  .x.  =  ( .s `  S )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  0  <_  A )   &    |-  ( ph  ->  B  e.  V )   &    |-  ( ph  ->  B  =/=  ( 0g `  S ) )   &    |-  ( ph  ->  ( (
 r  .x.  B )  e.  V  ->  ( ( L `  ( r  .x.  B ) )  <  R  ->  ( ( M `  ( F `  ( r 
 .x.  B ) ) ) 
 /  R )  <_  A ) ) )   &    |-  ( ph  ->  -.  ( M `  ( F `  B ) )  <_  ( A  x.  ( L `  B ) ) )   =>    |- 
 -.  ph
 
Theoremnmoleub2lem2 18599* Lemma for nmoleub2a 18600 and similar theorems. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   &    |-  ( ( ( L `  x )  e.  RR  /\  R  e.  RR )  ->  (
 ( L `  x ) O R  ->  ( L `  x )  <_  R ) )   &    |-  (
 ( ( L `  x )  e.  RR  /\  R  e.  RR )  ->  ( ( L `  x )  <  R  ->  ( L `  x ) O R ) )   =>    |-  ( ph  ->  ( ( N `  F )  <_  A 
 <-> 
 A. x  e.  V  ( ( L `  x ) O R  ->  ( ( M `  ( F `  x ) )  /  R ) 
 <_  A ) ) )
 
Theoremnmoleub2a 18600* The operator norm is the supremum of the value of a linear operator in the closed unit ball. (Contributed by Mario Carneiro, 19-Oct-2015.)
 |-  N  =  ( S
 normOp T )   &    |-  V  =  (
 Base `  S )   &    |-  L  =  ( norm `  S )   &    |-  M  =  ( norm `  T )   &    |-  G  =  (Scalar `  S )   &    |-  K  =  ( Base `  G )   &    |-  ( ph  ->  S  e.  (NrmMod  i^i CMod ) )   &    |-  ( ph  ->  T  e.  (NrmMod  i^i CMod )
 )   &    |-  ( ph  ->  F  e.  ( S LMHom  T ) )   &    |-  ( ph  ->  A  e.  RR* )   &    |-  ( ph  ->  R  e.  RR+ )   &    |-  ( ph  ->  QQ  C_  K )   =>    |-  ( ph  ->  (
 ( N `  F )  <_  A  <->  A. x  e.  V  ( ( L `  x )  <_  R  ->  ( ( M `  ( F `  x ) ) 
 /  R )  <_  A ) ) )
    < Previous  Next >

Page List
Jump to page: Contents  1 1-100 2 101-200 3 201-300 4 301-400 5 401-500 6 501-600 7 601-700 8 701-800 9 801-900 10 901-1000 11 1001-1100 12 1101-1200 13 1201-1300 14 1301-1400 15 1401-1500 16 1501-1600 17 1601-1700 18 1701-1800 19 1801-1900 20 1901-2000 21 2001-2100 22 2101-2200 23 2201-2300 24 2301-2400 25 2401-2500 26 2501-2600 27 2601-2700 28 2701-2800 29 2801-2900 30 2901-3000 31 3001-3100 32 3101-3200 33 3201-3300 34 3301-3400 35 3401-3500 36 3501-3600 37 3601-3700 38 3701-3800 39 3801-3900 40 3901-4000 41 4001-4100 42 4101-4200 43 4201-4300 44 4301-4400 45 4401-4500 46 4501-4600 47 4601-4700 48 4701-4800 49 4801-4900 50 4901-5000 51 5001-5100 52 5101-5200 53 5201-5300 54 5301-5400 55 5401-5500 56 5501-5600 57 5601-5700 58 5701-5800 59 5801-5900 60 5901-6000 61 6001-6100 62 6101-6200 63 6201-6300 64 6301-6400 65 6401-6500 66 6501-6600 67 6601-6700 68 6701-6800 69 6801-6900 70 6901-7000 71 7001-7100 72 7101-7200 73 7201-7300 74 7301-7400 75 7401-7500 76 7501-7600 77 7601-7700 78 7701-7800 79 7801-7900 80 7901-8000 81 8001-8100 82 8101-8200 83 8201-8300 84 8301-8400 85 8401-8500 86 8501-8600 87 8601-8700 88 8701-8800 89 8801-8900 90 8901-9000 91 9001-9100 92 9101-9200 93 9201-9300 94 9301-9400 95 9401-9500 96 9501-9600 97 9601-9700 98 9701-9800 99 9801-9900 100 9901-10000 101 10001-10100 102 10101-10200 103 10201-10300 104 10301-10400 105 10401-10500 106 10501-10600 107 10601-10700 108 10701-10800 109 10801-10900 110 10901-11000 111 11001-11100 112 11101-11200 113 11201-11300 114 11301-11400 115 11401-11500 116 11501-11600 117 11601-11700 118 11701-11800 119 11801-11900 120 11901-12000 121 12001-12100 122 12101-12200 123 12201-12300 124 12301-12400 125 12401-12500 126 12501-12600 127 12601-12700 128 12701-12800 129 12801-12900 130 12901-13000 131 13001-13100 132 13101-13200 133 13201-13300 134 13301-13400 135 13401-13500 136 13501-13600 137 13601-13700 138 13701-13800 139 13801-13900 140 13901-14000 141 14001-14100 142 14101-14200 143 14201-14300 144 14301-14400 145 14401-14500 146 14501-14600 147 14601-14700 148 14701-14800 149 14801-14900 150 14901-15000 151 15001-15100 152 15101-15200 153 15201-15300 154 15301-15400 155 15401-15500 156 15501-15600 157 15601-15700 158 15701-15800 159 15801-15900 160 15901-16000 161 16001-16100 162 16101-16200 163 16201-16300 164 16301-16400 165 16401-16500 166 16501-16600 167 16601-16700 168 16701-16800 169 16801-16900 170 16901-17000 171 17001-17100 172 17101-17200 173 17201-17300 174 17301-17400 175 17401-17500 176 17501-17600 177 17601-17700 178 17701-17800 179 17801-17900 180 17901-18000 181 18001-18100 182 18101-18200 183 18201-18300 184 18301-18400 185 18401-18500 186 18501-18600 187 18601-18700 188 18701-18800 189 18801-18900 190 18901-19000 191 19001-19100 192 19101-19200 193 19201-19300 194 19301-19400 195 19401-19500 196 19501-19600 197 19601-19700 198 19701-19800 199 19801-19900 200 19901-20000 201 20001-20100 202 20101-20200 203 20201-20300 204 20301-20400 205 20401-20500 206 20501-20600 207 20601-20700 208 20701-20800 209 20801-20900 210 20901-21000 211 21001-21100 212 21101-21200 213 21201-21300 214 21301-21400 215 21401-21500 216 21501-21600 217 21601-21700 218 21701-21800 219 21801-21900 220 21901-22000 221 22001-22100 222 22101-22200 223 22201-22300 224 22301-22400 225 22401-22500 226 22501-22600 227 22601-22700 228 22701-22800 229 22801-22900 230 22901-23000 231 23001-23100 232 23101-23200 233 23201-23300 234 23301-23400 235 23401-23500 236 23501-23600 237 23601-23700 238 23701-23800 239 23801-23900 240 23901-24000 241 24001-24100 242 24101-24200 243 24201-24300 244 24301-24400 245 24401-24500 246 24501-24600 247 24601-24700 248 24701-24800 249 24801-24900 250 24901-25000 251 25001-25100 252 25101-25200 253 25201-25300 254 25301-25400 255 25401-25500 256 25501-25600 257 25601-25700 258 25701-25800 259 25801-25900 260 25901-26000 261 26001-26100 262 26101-26200 263 26201-26300 264 26301-26400 265 26401-26500 266 26501-26600 267 26601-26700 268 26701-26800 269 26801-26900 270 26901-27000 271 27001-27100 272 27101-27200 273 27201-27300 274 27301-27400 275 27401-27500 276 27501-27600 277 27601-27700 278 27701-27800 279 27801-27900 280 27901-28000 281 28001-28100 282 28101-28200 283 28201-28300 284 28301-28400 285 28401-28500 286 28501-28600 287 28601-28700 288 28701-28800 289 28801-28900 290 28901-29000 291 29001-29100 292 29101-29200 293 29201-29300 294 29301-29400 295 29401-29500 296 29501-29600 297 29601-29700 298 29701-29800 299 29801-29900 300 29901-30000 301 30001-30100 302 30101-30200 303 30201-30300 304 30301-30400 305 30401-30500 306 30501-30600 307 30601-30700 308 30701-30800 309 30801-30900 310 30901-31000 311 31001-31100 312 31101-31200 313 31201-31300 314 31301-31400 315 31401-31500 316 31501-31600 317 31601-31700 318 31701-31800 319 31801-31900 320 31901-32000 321 32001-32100 322 32101-32200 323 32201-32227
  Copyright terms: Public domain < Previous  Next >