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Theorem List for Metamath Proof Explorer - 18401-18500   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremiiuni 18401 The base set of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 15-Jan-2014.)
 |-  ( 0 [,] 1
 )  =  U. II
 
Theoremdfii2 18402 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  II  =  ( (
 topGen `  ran  (,) )t  (
 0 [,] 1 ) )
 
Theoremdfii3 18403 Alternate definition of the unit interval. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 3-Sep-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  II  =  ( Jt  ( 0 [,] 1
 ) )
 
Theoremdfii4 18404 Alternate definition of the unit interval. (Contributed by Mario Carneiro, 3-Sep-2015.)
 |-  I  =  (flds  ( 0 [,] 1
 ) )   =>    |-  II  =  ( TopOpen `  I )
 
Theoremdfii5 18405 The unit interval expressed as an order topology. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  =  (ordTop `  (  <_  i^i  ( ( 0 [,] 1 )  X.  ( 0 [,] 1
 ) ) ) )
 
Theoremiicmp 18406 The unit interval is compact. (Contributed by Jeff Madsen, 2-Sep-2009.) (Revised by Mario Carneiro, 13-Jun-2014.)
 |-  II  e.  Comp
 
Theoremiicon 18407 The unit interval is connected. (Contributed by Mario Carneiro, 11-Feb-2015.)
 |-  II  e.  Con
 
Theoremcncfval 18408* The value of the continuous complex function operation is the set of continuous functions from  A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( A -cn-> B )  =  { f  e.  ( B  ^m  A )  |  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  <  z  ->  ( abs `  ( ( f `
  x )  -  ( f `  w ) ) )  < 
 y ) } )
 
Theoremelcncf 18409* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 11-Oct-2007.) (Revised by Mario Carneiro, 9-Nov-2013.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( x  -  w ) )  <  z  ->  ( abs `  ( ( F `
  x )  -  ( F `  w ) ) )  <  y
 ) ) ) )
 
Theoremelcncf2 18410* Version of elcncf 18409 with arguments commuted. (Contributed by Mario Carneiro, 28-Apr-2014.)
 |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  ( F  e.  ( A -cn-> B )  <->  ( F : A
 --> B  /\  A. x  e.  A  A. y  e.  RR+  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  x ) )  <  z  ->  ( abs `  ( ( F `
  w )  -  ( F `  x ) ) )  <  y
 ) ) ) )
 
Theoremcncfrss 18411 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  A  C_  CC )
 
Theoremcncfrss2 18412 Reverse closure of the continuous function predicate. (Contributed by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  B  C_  CC )
 
Theoremcncff 18413 A continuous complex function's domain and codomain. (Contributed by Paul Chapman, 17-Jan-2008.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( F  e.  ( A -cn-> B )  ->  F : A --> B )
 
Theoremcncfi 18414* Defining property of a continuous function. (Contributed by Mario Carneiro, 30-Apr-2014.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( ( F  e.  ( A -cn-> B )  /\  C  e.  A  /\  R  e.  RR+ )  ->  E. z  e.  RR+  A. w  e.  A  ( ( abs `  ( w  -  C ) )  <  z  ->  ( abs `  ( ( F `  w )  -  ( F `  C ) ) )  <  R ) )
 
Theoremelcncf1di 18415* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  ( ph  ->  F : A --> B )   &    |-  ( ph  ->  ( ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ ) )   &    |-  ( ph  ->  ( ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  ( ( abs `  ( x  -  w ) )  <  Z  ->  ( abs `  ( ( F `
  x )  -  ( F `  w ) ) )  <  y
 ) ) )   =>    |-  ( ph  ->  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A
 -cn-> B ) ) )
 
Theoremelcncf1ii 18416* Membership in the set of continuous complex functions from  A to  B. (Contributed by Paul Chapman, 26-Nov-2007.)
 |-  F : A --> B   &    |-  (
 ( x  e.  A  /\  y  e.  RR+ )  ->  Z  e.  RR+ )   &    |-  (
 ( ( x  e.  A  /\  w  e.  A )  /\  y  e.  RR+ )  ->  (
 ( abs `  ( x  -  w ) )  <  Z  ->  ( abs `  (
 ( F `  x )  -  ( F `  w ) ) )  <  y ) )   =>    |-  ( ( A  C_  CC  /\  B  C_  CC )  ->  F  e.  ( A -cn-> B ) )
 
Theoremrescncf 18417 A continuous complex function restricted to a subset is continuous. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 25-Aug-2014.)
 |-  ( C  C_  A  ->  ( F  e.  ( A -cn-> B )  ->  ( F  |`  C )  e.  ( C -cn-> B ) ) )
 
Theoremcncffvrn 18418 Change the codomain of a continuous complex function. (Contributed by Paul Chapman, 18-Oct-2007.) (Revised by Mario Carneiro, 1-May-2015.)
 |-  ( ( C  C_  CC  /\  F  e.  ( A -cn-> B ) ) 
 ->  ( F  e.  ( A -cn-> C )  <->  F : A --> C ) )
 
Theoremcncfss 18419 The set of continuous functions is expanded when the range is expanded. (Contributed by Mario Carneiro, 30-Aug-2014.)
 |-  ( ( B  C_  C  /\  C  C_  CC )  ->  ( A -cn-> B )  C_  ( A -cn-> C ) )
 
Theoremclimcncf 18420 Image of a limit under a continuous map. (Contributed by Mario Carneiro, 7-Apr-2015.)
 |-  Z  =  ( ZZ>= `  M )   &    |-  ( ph  ->  M  e.  ZZ )   &    |-  ( ph  ->  F  e.  ( A -cn-> B ) )   &    |-  ( ph  ->  G : Z
 --> A )   &    |-  ( ph  ->  G  ~~>  D )   &    |-  ( ph  ->  D  e.  A )   =>    |-  ( ph  ->  ( F  o.  G )  ~~>  ( F `  D ) )
 
Theoremabscncf 18421 Absolute value is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremrecncf 18422 Real part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Re  e.  ( CC
 -cn-> RR )
 
Theoremimcncf 18423 Imaginary part is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  Im  e.  ( CC
 -cn-> RR )
 
Theoremcjcncf 18424 Complex conjugate is continuous. (Contributed by Paul Chapman, 21-Oct-2007.) (Revised by Mario Carneiro, 28-Apr-2014.)
 |-  *  e.  ( CC
 -cn-> CC )
 
Theoremmulc1cncf 18425* 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 18426* 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 18427 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 18428 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 18429 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 18430 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 18431* 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 18432* 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 18433* Composition of continuous functions.  -cn-> analog of cnmpt11f 17374. (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 18434* Composition of continuous functions.  -cn-> analog of cnmpt12f 17376. (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 18435* 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 18436* 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 18437* 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 18438* 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 18439 Absolute value is continuous. Alternate proof of abscncf 18421. (Contributed by NM, 6-Jun-2008.) (Revised by Mario Carneiro, 10-Sep-2015.) (Proof modification is discouraged.)
 |- 
 abs  e.  ( CC -cn-> RR )
 
Theoremcncfcnvcn 18440 Rewrite cmphaushmeo 17507 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 18441* Lemma for iirevcn 18444 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 18442* 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 18443 Reverse the unit interval. (Contributed by Jeff Madsen, 2-Sep-2009.)
 |-  ( X  e.  (
 0 [,] 1 )  ->  ( 1  -  X )  e.  ( 0 [,] 1 ) )
 
Theoremiirevcn 18444 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 18445 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 18446 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 18447 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 18448 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 18449 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 18450 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 18451 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 18452* 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 18453 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 18454 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 18455* 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 18456* 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 18457* 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 18458* 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 18459 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 18460* 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 18461 The extended reals are homeomorphic to the interval  [
0 ,  1 ]. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  II  ~=  (ordTop `  <_  )
 
Theoremxrcmp 18462 The topology of the extended reals is compact. Since the topology of the extended reals extends the topology of the reals (by xrtgioo 18328), this means that  RR* is a compactification of  RR. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Comp
 
Theoremxrcon 18463 The topology of the extended reals is connected. (Contributed by Mario Carneiro, 9-Sep-2015.)
 |-  (ordTop `  <_  )  e. 
 Con
 
Theoremicccvx 18464 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 18465* 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 18466* 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 18467* The canonical bijection from  ( RR  X.  RR ) to  CC described in cnref1o 10365 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 18468* Lemma for cnheibor 18469. (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 18469* 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 18470 The topology on the complexes is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  J  =  ( TopOpen ` fld )   =>    |-  J  e. 𝑛Locally  Comp
 
Theoremrellycmp 18471 The topology on the reals is locally compact. (Contributed by Mario Carneiro, 2-Mar-2015.)
 |-  ( topGen `  ran  (,) )  e. 𝑛Locally  Comp
 
Theorembndth 18472* 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 18473* 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 18474* 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 18475* Lemma for lebnum 18478. 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 18476* Lemma for lebnum 18478. As a finite sum of point-to-set distance functions, which are continuous by metdscn 18376, 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 18477* Lemma for lebnum 18478. 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 18478* 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 18479* Generalize lebnum 18478 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 18480* Specialize the Lebesgue number lemma lebnum 18478 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 18481 Extend class notation with the class of homotopies between two continuous functions.
 class Htpy
 
Syntaxcphtpy 18482 Extend class notation with the class of path homotopies between two continuous functions.
 class  PHtpy
 
Syntaxcphtpc 18483 Extend class notation with the path homotopy relation.
 class  ~=ph
 
Definitiondf-htpy 18484* 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 18485* Define the class of path homotopies between two paths  F ,  G : II --> X; these are homotopies (in the sense of df-htpy 18484) 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 18486* 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 18487 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 18488 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 18489* 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 18490* 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 18491* 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 18492* 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 18493 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 18494* 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 18495* 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 18496 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 18497 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 18498 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 18499 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 18500* 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 ) )
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