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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremcnheibor 18401* Heine-Borel theorem for complex numbers. A subset of  CC is compact iff it is closed and bounded. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  T  =  ( Jt  X )   =>    |-  ( X  C_  CC  ->  ( T  e.  Comp  <->  ( X  e.  ( Clsd `  J )  /\  E. r  e.  RR  A. x  e.  X  ( abs `  x )  <_  r ) ) )
 
Theoremcnllycmp 18402 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. 𝑛Locally  Comp
 
Theoremrellycmp 18403 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e. 𝑛Locally  Comp
 
Theorembndth 18404* The Boundedness Theorem. A continuous function from a compact topological space to the reals is bounded (above). (Boundedness below is obtained by applying this theorem to  -u F.) (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  E. x  e.  RR  A. y  e.  X  ( F `  y )  <_  x )
 
Theoremevth 18405* The Extreme Value Theorem. A continuous function from a nonempty compact topological space to the reals attains its maximum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  y )  <_  ( F `
  x ) )
 
Theoremevth2 18406* The Extreme Value Theorem, minimum version. A continuous function from a nonempty compact topological space to the reals attains its minimum at some point in the domain. (Contributed by Mario Carneiro, 12-Aug-2014.)
 |-  X  =  U. J   &    |-  K  =  ( topGen `  ran  (,) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  X  =/=  (/) )   =>    |-  ( ph  ->  E. x  e.  X  A. y  e.  X  ( F `  x )  <_  ( F `
  y ) )
 
Theoremlebnumlem1 18407* Lemma for lebnum 18410. The function  F measures the sum of all of the distances to escape the sets of the cover. Since by assumption it is a cover, there is at least one set which covers a given point, and since it is open, the point is a positive distance from the edge of the set. Thus the sum is a strictly positive number. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  F  =  ( y  e.  X  |->  sum_ k  e.  U  sup ( ran  (  z  e.  ( X  \  k )  |->  ( y D z ) ) ,  RR* ,  `'  <  ) )   =>    |-  ( ph  ->  F : X --> RR+ )
 
Theoremlebnumlem2 18408* Lemma for lebnum 18410. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18308, the function  F is also continuous. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  F  =  ( y  e.  X  |->  sum_ k  e.  U  sup ( ran  (  z  e.  ( X  \  k )  |->  ( y D z ) ) ,  RR* ,  `'  <  ) )   &    |-  K  =  (
 topGen `  ran  (,) )   =>    |-  ( ph  ->  F  e.  ( J  Cn  K ) )
 
Theoremlebnumlem3 18409* Lemma for lebnum 18410. By the previous lemmas,  F is continuous and positive on a compact set, so it has a positive minimum  r. Then setting  d  =  r  /  # ( U ), since for each  u  e.  U we have  ball ( x ,  d )  C_  u iff  d  <_  d ( x ,  X  \  u ), if  -.  ball (
x ,  d ) 
C_  u for all  u then summing over  u yields  sum_ u  e.  U
d ( x ,  X  \  u )  =  F ( x )  <  sum_ u  e.  U d  =  r, in contradiction to the assumption that  r is the minimum of  F. (Contributed by Mario Carneiro, 14-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   &    |-  ( ph  ->  U  e.  Fin )   &    |-  ( ph  ->  -.  X  e.  U )   &    |-  F  =  ( y  e.  X  |->  sum_ k  e.  U  sup ( ran  (  z  e.  ( X  \  k )  |->  ( y D z ) ) ,  RR* ,  `'  <  ) )   &    |-  K  =  (
 topGen `  ran  (,) )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  X  E. u  e.  U  ( x ( ball `  D ) d )  C_  u )
 
Theoremlebnum 18410* The Lebesgue number lemma, or Lebesgue covering lemma. If  X is a compact metric space and  U is an open cover of  X, then there exists a positive real number 
d such that every ball of size  d (and every subset of a ball of size  d, including every subset of diameter less than  d) is a subset of some member of the cover. (Contributed by Mario Carneiro, 14-Feb-2015.) (Proof shortened by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U 
 C_  J )   &    |-  ( ph  ->  X  =  U. U )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  X  E. u  e.  U  ( x ( ball `  D ) d )  C_  u )
 
Theoremxlebnum 18411* Generalize lebnum 18410 to extended metrics. (Contributed by Mario Carneiro, 5-Sep-2015.)
 |-  J  =  ( MetOpen `  D )   &    |-  ( ph  ->  D  e.  ( * Met `  X ) )   &    |-  ( ph  ->  J  e.  Comp )   &    |-  ( ph  ->  U  C_  J )   &    |-  ( ph  ->  X  =  U. U )   =>    |-  ( ph  ->  E. d  e.  RR+  A. x  e.  X  E. u  e.  U  ( x (
 ball `  D ) d )  C_  u )
 
Theoremlebnumii 18412* Specialize the Lebesgue number lemma lebnum 18410 to the unit interval. (Contributed by Mario Carneiro, 14-Feb-2015.)
 |-  ( ( U  C_  II  /\  ( 0 [,] 1 )  =  U. U )  ->  E. n  e.  NN  A. k  e.  ( 1 ... n ) E. u  e.  U  ( ( ( k  -  1 )  /  n ) [,] (
 k  /  n )
 )  C_  u )
 
11.3.10  Path homotopy
 
Syntaxchtpy 18413 Extend class notation with the class of homotopies between two continuous functions.
 class Htpy
 
Syntaxcphtpy 18414 Extend class notation with the class of path homotopies between two continuous functions.
 class  PHtpy
 
Syntaxcphtpc 18415 Extend class notation with the path homotopy relation.
 class  ~=ph
 
Definitiondf-htpy 18416* Define the function which takes topological spaces  X ,  Y and two continuous functions  F ,  G : X
--> Y and returns the class of homotopies from  F to  G. (Contributed by Mario Carneiro, 22-Feb-2015.)
 |- Htpy  =  ( x  e.  Top ,  y  e.  Top  |->  ( f  e.  ( x  Cn  y ) ,  g  e.  ( x  Cn  y
 )  |->  { h  e.  (
 ( x  tX  II )  Cn  y )  | 
 A. s  e.  U. x ( ( s h 0 )  =  ( f `  s
 )  /\  ( s h 1 )  =  ( g `  s
 ) ) } )
 )
 
Definitiondf-phtpy 18417* Define the class of path homotopies between two paths  F ,  G : II --> X; these are homotopies (in the sense of df-htpy 18416) which also preserve both endpoints of the paths throughout the homotopy. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |- 
 PHtpy  =  ( x  e.  Top  |->  ( f  e.  ( II  Cn  x ) ,  g  e.  ( II  Cn  x )  |->  { h  e.  (
 f ( II Htpy  x ) g )  | 
 A. s  e.  (
 0 [,] 1 ) ( ( 0 h s )  =  ( f `
  0 )  /\  ( 1 h s )  =  ( f `
  1 ) ) } ) )
 
Theoremishtpy 18418* Membership in the class of homotopies between two continuous functions. (Contributed by Mario Carneiro, 22-Feb-2015.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( H  e.  ( F ( J Htpy  K ) G )  <->  ( H  e.  ( ( J  tX  II )  Cn  K ) 
 /\  A. s  e.  X  ( ( s H 0 )  =  ( F `  s ) 
 /\  ( s H 1 )  =  ( G `  s ) ) ) ) )
 
Theoremhtpycn 18419 A homotopy is a continuous function. (Contributed by Mario Carneiro, 22-Feb-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( F ( J Htpy  K ) G )  C_  (
 ( J  tX  II )  Cn  K ) )
 
Theoremhtpyi 18420 A homotopy evaluated at its endpoints. (Contributed by Mario Carneiro, 22-Feb-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )   =>    |-  ( ( ph  /\  A  e.  X ) 
 ->  ( ( A H
 0 )  =  ( F `  A ) 
 /\  ( A H
 1 )  =  ( G `  A ) ) )
 
Theoremishtpyd 18421* Deduction for membership in the class of homotopies. (Contributed by Mario Carneiro, 22-Feb-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  H  e.  (
 ( J  tX  II )  Cn  K ) )   &    |-  ( ( ph  /\  s  e.  X )  ->  (
 s H 0 )  =  ( F `  s ) )   &    |-  (
 ( ph  /\  s  e.  X )  ->  (
 s H 1 )  =  ( G `  s ) )   =>    |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )
 
Theoremhtpycom 18422* Given a homotopy from  F to  G, produce a homotopy from  G to  F. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  J  e.  (TopOn `  X )
 )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  M  =  ( x  e.  X ,  y  e.  (
 0 [,] 1 )  |->  ( x H ( 1  -  y ) ) )   &    |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )   =>    |-  ( ph  ->  M  e.  ( G ( J Htpy 
 K ) F ) )
 
Theoremhtpyid 18423* A homotopy from a function to itself. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  G  =  ( x  e.  X ,  y  e.  ( 0 [,] 1
 )  |->  ( F `  x ) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  G  e.  ( F ( J Htpy  K ) F ) )
 
Theoremhtpyco1 18424* Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  N  =  ( x  e.  X ,  y  e.  ( 0 [,] 1
 )  |->  ( ( P `
  x ) H y ) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  P  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  F  e.  ( K  Cn  L ) )   &    |-  ( ph  ->  G  e.  ( K  Cn  L ) )   &    |-  ( ph  ->  H  e.  ( F ( K Htpy  L ) G ) )   =>    |-  ( ph  ->  N  e.  ( ( F  o.  P ) ( J Htpy  L ) ( G  o.  P ) ) )
 
Theoremhtpyco2 18425 Compose a homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  P  e.  ( K  Cn  L ) )   &    |-  ( ph  ->  H  e.  ( F ( J Htpy  K ) G ) )   =>    |-  ( ph  ->  ( P  o.  H )  e.  ( ( P  o.  F ) ( J Htpy  L ) ( P  o.  G ) ) )
 
Theoremhtpycc 18426* Concatenate two homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
 |-  N  =  ( x  e.  X ,  y  e.  ( 0 [,] 1
 )  |->  if ( y  <_  ( 1  /  2
 ) ,  ( x L ( 2  x.  y ) ) ,  ( x M ( ( 2  x.  y
 )  -  1 ) ) ) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ph  ->  F  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  G  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  H  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  L  e.  ( F ( J Htpy  K ) G ) )   &    |-  ( ph  ->  M  e.  ( G ( J Htpy  K ) H ) )   =>    |-  ( ph  ->  N  e.  ( F ( J Htpy  K ) H ) )
 
Theoremisphtpy 18427* Membership in the class of path homotopies between two continuous functions. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  ( H  e.  ( F ( PHtpy `  J ) G )  <->  ( H  e.  ( F ( II Htpy  J ) G )  /\  A. s  e.  ( 0 [,] 1 ) ( ( 0 H s )  =  ( F `  0 )  /\  ( 1 H s )  =  ( F `  1
 ) ) ) ) )
 
Theoremphtpyhtpy 18428 A path homotopy is a homotopy. (Contributed by Mario Carneiro, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  ( F ( PHtpy `  J ) G )  C_  ( F ( II Htpy  J ) G ) )
 
Theoremphtpycn 18429 A path homotopy is a continuous function. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  ( F ( PHtpy `  J ) G )  C_  (
 ( II  tX  II )  Cn  J ) )
 
Theoremphtpyi 18430 Membership in the class of path homotopies between two continuous functions. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )   =>    |-  ( ( ph  /\  A  e.  ( 0 [,] 1 ) ) 
 ->  ( ( 0 H A )  =  ( F `  0 ) 
 /\  ( 1 H A )  =  ( F `  1 ) ) )
 
Theoremphtpy01 18431 Two path-homotopic paths have the same start and end point. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )   =>    |-  ( ph  ->  ( ( F `  0
 )  =  ( G `
  0 )  /\  ( F `  1 )  =  ( G `  1 ) ) )
 
Theoremisphtpyd 18432* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  ( F ( II Htpy  J ) G ) )   &    |-  (
 ( ph  /\  s  e.  ( 0 [,] 1
 ) )  ->  (
 0 H s )  =  ( F `  0 ) )   &    |-  (
 ( ph  /\  s  e.  ( 0 [,] 1
 ) )  ->  (
 1 H s )  =  ( F `  1 ) )   =>    |-  ( ph  ->  H  e.  ( F (
 PHtpy `  J ) G ) )
 
Theoremisphtpy2d 18433* Deduction for membership in the class of path homotopies. (Contributed by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  (
 ( II  tX  II )  Cn  J ) )   &    |-  ( ( ph  /\  s  e.  ( 0 [,] 1
 ) )  ->  (
 s H 0 )  =  ( F `  s ) )   &    |-  (
 ( ph  /\  s  e.  ( 0 [,] 1
 ) )  ->  (
 s H 1 )  =  ( G `  s ) )   &    |-  (
 ( ph  /\  s  e.  ( 0 [,] 1
 ) )  ->  (
 0 H s )  =  ( F `  0 ) )   &    |-  (
 ( ph  /\  s  e.  ( 0 [,] 1
 ) )  ->  (
 1 H s )  =  ( F `  1 ) )   =>    |-  ( ph  ->  H  e.  ( F (
 PHtpy `  J ) G ) )
 
Theoremphtpycom 18434* Given a homotopy from  F to  G, produce a homotopy from  G to  F. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  K  =  ( x  e.  (
 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( x H ( 1  -  y ) ) )   &    |-  ( ph  ->  H  e.  ( F (
 PHtpy `  J ) G ) )   =>    |-  ( ph  ->  K  e.  ( G ( PHtpy `  J ) F ) )
 
Theoremphtpyid 18435* A homotopy from a path to itself. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 23-Feb-2015.)
 |-  G  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1
 )  |->  ( F `  x ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   =>    |-  ( ph  ->  G  e.  ( F ( PHtpy `  J ) F ) )
 
Theoremphtpyco2 18436 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 10-Mar-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  P  e.  ( J  Cn  K ) )   &    |-  ( ph  ->  H  e.  ( F ( PHtpy `  J ) G ) )   =>    |-  ( ph  ->  ( P  o.  H )  e.  ( ( P  o.  F ) (
 PHtpy `  K ) ( P  o.  G ) ) )
 
Theoremphtpycc 18437* Concatenate two path homotopies. (Contributed by Jeff Madsen, 2-Sep-2009.) (Proof shortened by Mario Carneiro, 7-Jun-2014.)
 |-  M  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1
 )  |->  if ( y  <_  ( 1  /  2
 ) ,  ( x K ( 2  x.  y ) ) ,  ( x L ( ( 2  x.  y
 )  -  1 ) ) ) )   &    |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  H  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  K  e.  ( F ( PHtpy `  J ) G ) )   &    |-  ( ph  ->  L  e.  ( G ( PHtpy `  J ) H ) )   =>    |-  ( ph  ->  M  e.  ( F (
 PHtpy `  J ) H ) )
 
Definitiondf-phtpc 18438* Define the function which takes a topology and returns the path homotopy relation on that topology. Definition of [Hatcher] p. 25. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Jun-2014.)
 |-  ~=ph  =  ( x  e. 
 Top  |->  { <. f ,  g >.  |  ( { f ,  g }  C_  ( II  Cn  x )  /\  ( f ( PHtpy `  x ) g )  =/=  (/) ) } )
 
Theoremphtpcrel 18439 The path homotopy relation is a relation. (Contributed by Mario Carneiro, 7-Jun-2014.) (Revised by Mario Carneiro, 7-Aug-2014.)
 |- 
 Rel  (  ~=ph  `  J )
 
Theoremisphtpc 18440 The relation "is path homotopic to". (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Sep-2015.)
 |-  ( F (  ~=ph  `  J ) G  <->  ( F  e.  ( II  Cn  J ) 
 /\  G  e.  ( II  Cn  J )  /\  ( F ( PHtpy `  J ) G )  =/=  (/) ) )
 
Theoremphtpcer 18441 Path homotopy is an equivalence relation. Proposition 1.2 of [Hatcher] p. 26. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 6-Jul-2015.)
 |-  (  ~=ph  `  J )  Er  ( II  Cn  J )
 
Theoremphtpc01 18442 Path homotopic paths have the same endpoints. (Contributed by Mario Carneiro, 24-Feb-2015.)
 |-  ( F (  ~=ph  `  J ) G  ->  ( ( F `  0
 )  =  ( G `
  0 )  /\  ( F `  1 )  =  ( G `  1 ) ) )
 
Theoremreparphti 18443* Lemma for reparpht 18444. (Contributed by NM, 15-Jun-2010.) (Revised by Mario Carneiro, 7-Jun-2014.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  II ) )   &    |-  ( ph  ->  ( G `  0 )  =  0 )   &    |-  ( ph  ->  ( G `  1 )  =  1
 )   &    |-  H  =  ( x  e.  ( 0 [,] 1 ) ,  y  e.  ( 0 [,] 1
 )  |->  ( F `  ( ( ( 1  -  y )  x.  ( G `  x ) )  +  (
 y  x.  x ) ) ) )   =>    |-  ( ph  ->  H  e.  ( ( F  o.  G ) (
 PHtpy `  J ) F ) )
 
Theoremreparpht 18444 Reparametrization lemma. The reparametrization of a path by any continuous map  G : II --> II with  G
( 0 )  =  0 and  G ( 1 )  =  1 is path-homotopic to the original path. (Contributed by Jeff Madsen, 15-Jun-2010.) (Revised by Mario Carneiro, 23-Feb-2015.)
 |-  ( ph  ->  F  e.  ( II  Cn  J ) )   &    |-  ( ph  ->  G  e.  ( II  Cn  II ) )   &    |-  ( ph  ->  ( G `  0 )  =  0 )   &    |-  ( ph  ->  ( G `  1 )  =  1
 )   =>    |-  ( ph  ->  ( F  o.  G ) ( 
 ~=ph  `  J ) F )
 
Theoremphtpcco2 18445 Compose a path homotopy with a continuous map. (Contributed by Mario Carneiro, 6-Jul-2015.)
 |-  ( ph  ->  F (  ~=ph  `  J ) G )   &    |-  ( ph  ->  P  e.  ( J  Cn  K ) )   =>    |-  ( ph  ->  ( P  o.  F ) (  ~=ph  `  K )
 ( P  o.  G ) )
 
11.3.11  The fundamental group
 
Syntaxcpco 18446 Extend class notation with the concatenation operation for paths in a topological space.
 class  *p
 
Syntaxcomi 18447 Extend class notation with the loop space.
 class  Om 1
 
Syntaxcomn 18448 Extend class notation with the higher loop spaces.
 class  Om N
 
Syntaxcpi1 18449 Extend class notation with the fundamental group.
 class  pi 1
 
Syntaxcpin 18450 Extend class notation with the higher homotopy groups.
 class  pi N
 
Definitiondf-pco 18451* 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 18452* 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 18453* 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 18454* 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 18455* 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 18456* 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 18457* 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 18458 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 18459 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 18460 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 18461 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 18462 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 18463 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 18464 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 18465* Lemma for pcohtpy 18466. (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 18466 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 18467 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 18468 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 18469 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 18470* 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 18471* 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 18472* Lemma for pcorev 18473. 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 18473* 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 18474* 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 18475* 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 18476* 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 18477* 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 18478 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 18479 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 18480 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 18481 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 18482 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 18483 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 18484 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 18485 Lemma for pi1buni 18486. (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 18486 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 18487 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 18488 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 18489 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 18490 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 18491* 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 18492 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 18493 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 18494 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 18495 Lemma for pi1grp 18496. (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 18496 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 18497 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 18498* 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 18499* 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 18500* 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 )
 )
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