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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremimcncf 18401 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 18402 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 18403* Multiplication by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 30-Apr-2014.)
 |-  F  =  ( x  e.  CC  |->  ( A  x.  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( CC
 -cn-> CC ) )
 
Theoremdivccncf 18404* Division by a constant is continuous. (Contributed by Paul Chapman, 28-Nov-2007.)
 |-  F  =  ( x  e.  CC  |->  ( x 
 /  A ) )   =>    |-  ( ( A  e.  CC  /\  A  =/=  0
 )  ->  F  e.  ( CC -cn-> CC ) )
 
Theoremcncfco 18405 The composition of two continuous maps on complex numbers is also continuous. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  G  e.  ( B -cn-> C ) )   =>    |-  ( ph  ->  ( G  o.  F )  e.  ( A -cn-> C ) )
 
Theoremcncfmet 18406 Relate complex function continuity to metric space continuity. (Contributed by Paul Chapman, 26-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  C  =  ( ( abs  o.  -  )  |`  ( A  X.  A ) )   &    |-  D  =  ( ( abs  o.  -  )  |`  ( B  X.  B ) )   &    |-  J  =  ( MetOpen `  C )   &    |-  K  =  ( MetOpen `  D )   =>    |-  (
 ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( J  Cn  K ) )
 
Theoremcncfcn 18407 Relate complex function continuity to topological continuity. (Contributed by Mario Carneiro, 17-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  A )   &    |-  L  =  ( Jt  B )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  ( K  Cn  L ) )
 
Theoremcncfcn1 18408 Relate complex function continuity to topological continuity. (Contributed by Paul Chapman, 28-Nov-2007.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  ( CC -cn-> CC )  =  ( J  Cn  J )
 
Theoremcncfmptc 18409* A constant function is a continuous function on  CC. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 7-Sep-2015.)
 |-  ( ( A  e.  T  /\  S  C_  CC  /\  T  C_  CC )  ->  ( x  e.  S  |->  A )  e.  ( S -cn-> T ) )
 
Theoremcncfmptid 18410* The identity function is a continuous function on  CC. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 17-May-2016.)
 |-  ( ( S  C_  T  /\  T  C_  CC )  ->  ( x  e.  S  |->  x )  e.  ( S -cn-> T ) )
 
Theoremcncfmpt1f 18411* Composition of continuous functions.  -cn-> analog of cnmpt11f 17352. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  ( ph  ->  F  e.  ( CC -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( F `
  A ) )  e.  ( X -cn-> CC ) )
 
Theoremcncfmpt2f 18412* Composition of continuous functions.  -cn-> analog of cnmpt12f 17354. (Contributed by Mario Carneiro, 3-Sep-2014.)
 |-  J  =  ( TopOpen ` fld )   &    |-  ( ph  ->  F  e.  (
 ( J  tX  J )  Cn  J ) )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> CC )
 )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X -cn-> CC ) )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( X -cn-> CC ) )
 
Theoremcncfmpt2ss 18413* Composition of continuous functions in a subset. (Contributed by Mario Carneiro, 17-May-2016.)
 |-  J  =  ( TopOpen ` fld )   &    |-  F  e.  ( ( J  tX  J )  Cn  J )   &    |-  ( ph  ->  ( x  e.  X  |->  A )  e.  ( X -cn-> S ) )   &    |-  ( ph  ->  ( x  e.  X  |->  B )  e.  ( X
 -cn-> S ) )   &    |-  S  C_ 
 CC   &    |-  ( ( A  e.  S  /\  B  e.  S )  ->  ( A F B )  e.  S )   =>    |-  ( ph  ->  ( x  e.  X  |->  ( A F B ) )  e.  ( X -cn-> S ) )
 
Theoremcdivcncf 18414* Division with a constant numerator is continuous. (Contributed by Mario Carneiro, 28-Dec-2016.)
 |-  F  =  ( x  e.  ( CC  \  { 0 } )  |->  ( A  /  x ) )   =>    |-  ( A  e.  CC  ->  F  e.  ( ( CC  \  { 0 } ) -cn-> CC )
 )
 
Theoremnegcncf 18415* The negative function is continuous. (Contributed by Mario Carneiro, 30-Dec-2016.)
 |-  F  =  ( x  e.  A  |->  -u x )   =>    |-  ( A  C_  CC  ->  F  e.  ( A
 -cn-> CC ) )
 
Theoremnegfcncf 18416* The negative of a continuous complex function is continuous. (Contributed by Paul Chapman, 21-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  G  =  ( x  e.  A  |->  -u ( F `  x ) )   =>    |-  ( F  e.  ( A -cn-> CC )  ->  G  e.  ( A -cn-> CC )
 )
 
TheoremabscncfALT 18417 Absolute value is continuous. Alternate proof of abscncf 18399. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremcncfcnvcn 18418 Rewrite cmphaushmeo 17485 for functions on the complexes. (Contributed by Mario Carneiro, 19-Feb-2015.)
 |-  J  =  ( TopOpen ` fld )   &    |-  K  =  ( Jt  X )   =>    |-  ( ( K  e.  Comp  /\  F  e.  ( X
 -cn-> Y ) )  ->  ( F : X -1-1-onto-> Y  <->  `' F  e.  ( Y -cn-> X ) ) )
 
Theoremcnmptre 18419* Lemma for iirevcn 18422 and related functions. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  R  =  ( TopOpen ` fld )   &    |-  J  =  ( ( topGen `  ran  (,) )t  A )   &    |-  K  =  ( ( topGen `  ran  (,) )t  B )   &    |-  ( ph  ->  A  C_ 
 RR )   &    |-  ( ph  ->  B 
 C_  RR )   &    |-  ( ( ph  /\  x  e.  A ) 
 ->  F  e.  B )   &    |-  ( ph  ->  ( x  e.  CC  |->  F )  e.  ( R  Cn  R ) )   =>    |-  ( ph  ->  ( x  e.  A  |->  F )  e.  ( J  Cn  K ) )
 
Theoremcnmpt2pc 18420* Piecewise definition of a continuous function on a real interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 5-Jun-2014.)
 |-  R  =  ( topGen `  ran  (,) )   &    |-  M  =  ( Rt  ( A [,] B ) )   &    |-  N  =  ( Rt  ( B [,] C ) )   &    |-  O  =  ( Rt  ( A [,] C ) )   &    |-  ( ph  ->  A  e.  RR )   &    |-  ( ph  ->  C  e.  RR )   &    |-  ( ph  ->  B  e.  ( A [,] C ) )   &    |-  ( ph  ->  J  e.  (TopOn `  X ) )   &    |-  ( ( ph  /\  ( x  =  B  /\  y  e.  X ) )  ->  D  =  E )   &    |-  ( ph  ->  ( x  e.  ( A [,] B ) ,  y  e.  X  |->  D )  e.  ( ( M  tX  J )  Cn  K ) )   &    |-  ( ph  ->  ( x  e.  ( B [,] C ) ,  y  e.  X  |->  E )  e.  ( ( N  tX  J )  Cn  K ) )   =>    |-  ( ph  ->  ( x  e.  ( A [,] C ) ,  y  e.  X  |->  if ( x  <_  B ,  D ,  E ) )  e.  ( ( O  tX  J )  Cn  K ) )
 
Theoremiirev 18421 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] 1 )  ->  ( 1  -  X )  e.  ( 0 [,] 1 ) )
 
Theoremiirevcn 18422 The reversion function is a continuous map of the unit interval. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  ( x  e.  (
 0 [,] 1 )  |->  ( 1  -  x ) )  e.  ( II 
 Cn  II )
 
Theoremiihalf1 18423 Map the first half of  II into  II. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] ( 1  / 
 2 ) )  ->  ( 2  x.  X )  e.  ( 0 [,] 1 ) )
 
Theoremiihalf1cn 18424 The first half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  J  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] ( 1  / 
 2 ) ) )   =>    |-  ( x  e.  (
 0 [,] ( 1  / 
 2 ) )  |->  ( 2  x.  x ) )  e.  ( J  Cn  II )
 
Theoremiihalf2 18425 Map the second half of  II into  II. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 ( 1  /  2
 ) [,] 1 )  ->  ( ( 2  x.  X )  -  1
 )  e.  ( 0 [,] 1 ) )
 
Theoremiihalf2cn 18426 The second half function is a continuous map. (Contributed by Mario Carneiro, 6-Jun-2014.)
 |-  J  =  ( (
 topGen `  ran  (,) )t  (
 ( 1  /  2
 ) [,] 1 ) )   =>    |-  ( x  e.  (
 ( 1  /  2
 ) [,] 1 )  |->  ( ( 2  x.  x )  -  1 ) )  e.  ( J  Cn  II )
 
Theoremelii1 18427 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
 |-  ( X  e.  (
 0 [,] ( 1  / 
 2 ) )  <->  ( X  e.  ( 0 [,] 1
 )  /\  X  <_  ( 1  /  2 ) ) )
 
Theoremelii2 18428 Divide the unit interval into two pieces. (Contributed by Mario Carneiro, 7-Jun-2014.)
 |-  ( ( X  e.  ( 0 [,] 1
 )  /\  -.  X  <_  ( 1  /  2 ) )  ->  X  e.  ( ( 1  / 
 2 ) [,] 1
 ) )
 
Theoremiimulcl 18429 The unit interval is closed under multiplication. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( ( A  e.  ( 0 [,] 1
 )  /\  B  e.  ( 0 [,] 1
 ) )  ->  ( A  x.  B )  e.  ( 0 [,] 1
 ) )
 
Theoremiimulcn 18430* Multiplication is a continuous function on the unit interval. (Contributed by Mario Carneiro, 8-Jun-2014.)
 |-  ( x  e.  (
 0 [,] 1 ) ,  y  e.  ( 0 [,] 1 )  |->  ( x  x.  y ) )  e.  ( ( II  tX  II )  Cn  II )
 
Theoremicoopnst 18431 A half-open interval starting at  A 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 )  ->  ( A [,) C )  e.  J ) )
 
Theoremiocopnst 18432 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 18433* 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 18434* 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 18435* 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 18436* 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 18437 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 18438* 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 18439 The extended reals are homeomorphic to the interval  [
0 ,  1 ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  ~=  (ordTop `  <_  )
 
Theoremxrcmp 18440 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18306), this means that  RR* is a compactification of  RR. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Comp
 
Theoremxrcon 18441 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Con
 
Theoremicccvx 18442 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 18443* 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 18444* 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 18445* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 10344 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 18446* Lemma for cnheibor 18447. (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 18447* 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 18448 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. 𝑛Locally  Comp
 
Theoremrellycmp 18449 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e. 𝑛Locally  Comp
 
Theorembndth 18450* 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 18451* 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 18452* 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 18453* Lemma for lebnum 18456. 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 18454* Lemma for lebnum 18456. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18354, 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 18455* Lemma for lebnum 18456. 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 18456* 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 18457* Generalize lebnum 18456 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 18458* Specialize the Lebesgue number lemma lebnum 18456 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 18459 Extend class notation with the class of homotopies between two continuous functions.
 class Htpy
 
Syntaxcphtpy 18460 Extend class notation with the class of path homotopies between two continuous functions.
 class  PHtpy
 
Syntaxcphtpc 18461 Extend class notation with the path homotopy relation.
 class  ~=ph
 
Definitiondf-htpy 18462* 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 18463* Define the class of path homotopies between two paths  F ,  G : II --> X; these are homotopies (in the sense of df-htpy 18462) 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 18464* 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 18465 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 18466 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 18467* 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 18468* 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 18469* 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 18470* 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 18471 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 18472* 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 18473* 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 18474 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 18475 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 18476 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 18477 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 18478* 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 18479* 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 18480* 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 18481* 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 18482 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 18483* 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 18484* 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 18485 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 18486 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 18487 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 18488 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 18489* Lemma for reparpht 18490. (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 18490 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 18491 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 18492 Extend class notation with the concatenation operation for paths in a topological space.
 class  *p
 
Syntaxcomi 18493 Extend class notation with the loop space.
 class  Om 1
 
Syntaxcomn 18494 Extend class notation with the higher loop spaces.
 class  Om N
 
Syntaxcpi1 18495 Extend class notation with the fundamental group.
 class  pi 1
 
Syntaxcpin 18496 Extend class notation with the higher homotopy groups.
 class  pi N
 
Definitiondf-pco 18497* 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 18498* 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 18499* 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 18500* 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 )
 ) )
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