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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiocopnst 18401 A half-open interval ending at  B is open in the closed interval from  A to  B. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Dec-2013.)
 |-  J  =  ( MetOpen `  ( ( abs  o.  -  )  |`  ( ( A [,] B )  X.  ( A [,] B ) ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( C  e.  ( A [,) B )  ->  ( C (,] B )  e.  J ) )
 
Theoremicchmeo 18402* The natural bijection from  [ 0 ,  1 ] to an arbitrary nontrivial closed interval  [ A ,  B ] is a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  F  =  ( x  e.  (
 0 [,] 1 )  |->  ( ( x  x.  B )  +  ( (
 1  -  x )  x.  A ) ) )   =>    |-  ( ( A  e.  RR  /\  B  e.  RR  /\  A  <  B ) 
 ->  F  e.  ( II 
 Homeo  ( Jt  ( A [,] B ) ) ) )
 
Theoremicopnfcnv 18403* Define a bijection from  [ 0 ,  1 ) to  [ 0 , 
+oo ). (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,) 1 )  |->  ( x 
 /  ( 1  -  x ) ) )   =>    |-  ( F : ( 0 [,) 1 ) -1-1-onto-> ( 0 [,)  +oo )  /\  `' F  =  ( y  e.  ( 0 [,)  +oo )  |->  ( y  /  ( 1  +  y
 ) ) ) )
 
Theoremicopnfhmeo 18404* The defined bijection from  [ 0 ,  1 ) to  [ 0 , 
+oo ) is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,) 1 )  |->  ( x 
 /  ( 1  -  x ) ) )   &    |-  J  =  ( TopOpen ` fld )   =>    |-  ( F  Isom  <  ,  <  ( ( 0 [,) 1
 ) ,  ( 0 [,)  +oo ) )  /\  F  e.  ( ( Jt  ( 0 [,) 1
 ) )  Homeo  ( Jt  ( 0 [,)  +oo )
 ) ) )
 
Theoremiccpnfcnv 18405* Define a bijection from  [ 0 ,  1 ] to  [ 0 , 
+oo ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) ) )   =>    |-  ( F : ( 0 [,] 1 ) -1-1-onto-> ( 0 [,]  +oo )  /\  `' F  =  ( y  e.  ( 0 [,]  +oo )  |->  if ( y  = 
 +oo ,  1 ,  ( y  /  (
 1  +  y ) ) ) ) )
 
Theoremiccpnfhmeo 18406 The defined bijection from  [ 0 ,  1 ] to  [ 0 , 
+oo ] is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 8-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) ) )   &    |-  K  =  ( (ordTop ` 
 <_  )t  ( 0 [,]  +oo ) )   =>    |-  ( F  Isom  <  ,  <  ( ( 0 [,] 1 ) ,  ( 0 [,]  +oo ) )  /\  F  e.  ( II  Homeo  K ) )
 
Theoremxrhmeo 18407* The bijection from  [ -u 1 ,  1 ] to the extended reals is an order isomorphism and a homeomorphism. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  F  =  ( x  e.  ( 0 [,] 1 )  |->  if ( x  =  1 ,  +oo ,  ( x  /  ( 1  -  x ) ) ) )   &    |-  G  =  ( y  e.  ( -u 1 [,] 1
 )  |->  if ( 0  <_  y ,  ( F `  y ) ,  - e
 ( F `  -u y
 ) ) )   &    |-  J  =  ( TopOpen ` fld )   &    |-  K  =  (ordTop `  <_  )   =>    |-  ( G  Isom  <  ,  <  ( ( -u 1 [,] 1 ) , 
 RR* )  /\  G  e.  ( ( Jt  ( -u 1 [,] 1 ) ) 
 Homeo  (ordTop `  <_  ) ) )
 
Theoremxrhmph 18408 The extended reals are homeomorphic to the interval  [
0 ,  1 ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  ~=  (ordTop `  <_  )
 
Theoremxrcmp 18409 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18275), this means that  RR* is a compactification of  RR. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Comp
 
Theoremxrcon 18410 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Con
 
Theoremicccvx 18411 A linear combination of two reals lies in the interval between them. Equivalently, a closed interval is a convex set. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( C  e.  ( A [,] B )  /\  D  e.  ( A [,] B ) 
 /\  T  e.  (
 0 [,] 1 ) ) 
 ->  ( ( ( 1  -  T )  x.  C )  +  ( T  x.  D ) )  e.  ( A [,] B ) ) )
 
Theoremoprpiece1res1 18412* Restriction to the first part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <_  B   &    |-  R  e.  _V   &    |-  S  e.  _V   &    |-  K  e.  ( A [,] B )   &    |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )   &    |-  G  =  ( x  e.  ( A [,] K ) ,  y  e.  C  |->  R )   =>    |-  ( F  |`  ( ( A [,] K )  X.  C ) )  =  G
 
Theoremoprpiece1res2 18413* Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)
 |-  A  e.  RR   &    |-  B  e.  RR   &    |-  A  <_  B   &    |-  R  e.  _V   &    |-  S  e.  _V   &    |-  K  e.  ( A [,] B )   &    |-  F  =  ( x  e.  ( A [,] B ) ,  y  e.  C  |->  if ( x  <_  K ,  R ,  S ) )   &    |-  ( x  =  K  ->  R  =  P )   &    |-  ( x  =  K  ->  S  =  Q )   &    |-  (
 y  e.  C  ->  P  =  Q )   &    |-  G  =  ( x  e.  ( K [,] B ) ,  y  e.  C  |->  S )   =>    |-  ( F  |`  ( ( K [,] B )  X.  C ) )  =  G
 
Theoremcnrehmeo 18414* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 10317 is in fact a homeomorphism of the usual topologies on these sets. (It is also an isometry, if  ( RR  X.  RR ) is metrized with the l<SUP>2</SUP> norm.) (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y
 ) ) )   &    |-  J  =  ( topGen `  ran  (,) )   &    |-  K  =  ( TopOpen ` fld )   =>    |-  F  e.  ( ( J  tX  J )  Homeo  K )
 
Theoremcnheiborlem 18415* Lemma for cnheibor 18416. (Contributed by Mario Carneiro, 14-Sep-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  T  =  ( Jt  X )   &    |-  F  =  ( x  e.  RR ,  y  e.  RR  |->  ( x  +  ( _i  x.  y ) ) )   &    |-  Y  =  ( F " ( ( -u R [,] R )  X.  ( -u R [,] R ) ) )   =>    |-  ( ( X  e.  ( Clsd `  J )  /\  ( R  e.  RR  /\ 
 A. z  e.  X  ( abs `  z )  <_  R ) )  ->  T  e.  Comp )
 
Theoremcnheibor 18416* 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 18417 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. 𝑛Locally  Comp
 
Theoremrellycmp 18418 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e. 𝑛Locally  Comp
 
Theorembndth 18419* 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 18420* 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 18421* 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 18422* Lemma for lebnum 18425. 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 18423* Lemma for lebnum 18425. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18323, 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 18424* Lemma for lebnum 18425. 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 18425* 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 18426* Generalize lebnum 18425 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 18427* Specialize the Lebesgue number lemma lebnum 18425 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 18428 Extend class notation with the class of homotopies between two continuous functions.
 class Htpy
 
Syntaxcphtpy 18429 Extend class notation with the class of path homotopies between two continuous functions.
 class  PHtpy
 
Syntaxcphtpc 18430 Extend class notation with the path homotopy relation.
 class  ~=ph
 
Definitiondf-htpy 18431* 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 18432* Define the class of path homotopies between two paths  F ,  G : II --> X; these are homotopies (in the sense of df-htpy 18431) 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 18433* 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 18434 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 18435 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 18436* 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 18437* 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 18438* 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 18439* 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 18440 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 18441* 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 18442* 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 18443 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 18444 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 18445 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 18446 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 18447* 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 18448* 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 18449* 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 18450* 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 18451 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 18452* 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 18453* 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 18454 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 18455 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 18456 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 18457 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 18458* Lemma for reparpht 18459. (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 18459 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 18460 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 18461 Extend class notation with the concatenation operation for paths in a topological space.
 class  *p
 
Syntaxcomi 18462 Extend class notation with the loop space.
 class  Om 1
 
Syntaxcomn 18463 Extend class notation with the higher loop spaces.
 class  Om N
 
Syntaxcpi1 18464 Extend class notation with the fundamental group.
 class  pi 1
 
Syntaxcpin 18465 Extend class notation with the higher homotopy groups.
 class  pi N
 
Definitiondf-pco 18466* 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 18467* 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 18468* 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 18469* 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 18470* 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 18471* 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 18472* 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 18473 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 18474 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 18475 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 18476 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 18477 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 18478 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 18479 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 18480* Lemma for pcohtpy 18481. (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 18481 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 18482 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 18483 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 18484 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 18485* 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 18486* 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 18487* Lemma for pcorev 18488. 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 18488* 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 18489* 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 18490* 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 18491* 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 18492* 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 18493 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 18494 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 18495 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 18496 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 18497 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 18498 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 18499 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 18500 Lemma for pi1buni 18501. (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 ) ) )
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