MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  pcophtb Unicode version

Theorem pcophtb 18527
Description: 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.)
Hypotheses
Ref Expression
pcophtb.h  |-  H  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) )
pcophtb.p  |-  P  =  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )
pcophtb.f  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
pcophtb.g  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
pcophtb.0  |-  ( ph  ->  ( F `  0
)  =  ( G `
 0 ) )
pcophtb.1  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
Assertion
Ref Expression
pcophtb  |-  ( ph  ->  ( ( F ( *p `  J ) H ) (  ~=ph  `  J ) P  <->  F (  ~=ph  `  J ) G ) )
Distinct variable groups:    x, G    x, J
Allowed substitution hints:    ph( x)    P( x)    F( x)    H( x)

Proof of Theorem pcophtb
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 phtpcer 18493 . . . 4  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
21a1i 10 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  (  ~=ph  `  J )  Er  (
II  Cn  J )
)
3 pcophtb.1 . . . . . . . 8  |-  ( ph  ->  ( F `  1
)  =  ( G `
 1 ) )
4 pcophtb.g . . . . . . . . . 10  |-  ( ph  ->  G  e.  ( II 
Cn  J ) )
5 pcophtb.h . . . . . . . . . . 11  |-  H  =  ( x  e.  ( 0 [,] 1 ) 
|->  ( G `  (
1  -  x ) ) )
65pcorevcl 18523 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  ( H  e.  ( II  Cn  J )  /\  ( H `  0 )  =  ( G ` 
1 )  /\  ( H `  1 )  =  ( G ` 
0 ) ) )
74, 6syl 15 . . . . . . . . 9  |-  ( ph  ->  ( H  e.  ( II  Cn  J )  /\  ( H ` 
0 )  =  ( G `  1 )  /\  ( H ` 
1 )  =  ( G `  0 ) ) )
87simp2d 968 . . . . . . . 8  |-  ( ph  ->  ( H `  0
)  =  ( G `
 1 ) )
93, 8eqtr4d 2318 . . . . . . 7  |-  ( ph  ->  ( F `  1
)  =  ( H `
 0 ) )
107simp1d 967 . . . . . . . 8  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
1110, 4pco0 18512 . . . . . . 7  |-  ( ph  ->  ( ( H ( *p `  J ) G ) `  0
)  =  ( H `
 0 ) )
129, 11eqtr4d 2318 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( ( H ( *p `  J ) G ) `
 0 ) )
1312adantr 451 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( ( H ( *p `  J ) G ) `  0
) )
14 pcophtb.f . . . . . . 7  |-  ( ph  ->  F  e.  ( II 
Cn  J ) )
1514adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F  e.  ( II  Cn  J
) )
162, 15erref 6680 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F (  ~=ph  `  J ) F )
17 eqid 2283 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )
185, 17pcorev 18525 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  ( H ( *p `  J ) G ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  {
( G `  1
) } ) )
194, 18syl 15 . . . . . 6  |-  ( ph  ->  ( H ( *p
`  J ) G ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) )
2019adantr 451 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H
( *p `  J
) G ) ( 
~=ph  `  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) )
2113, 16, 20pcohtpy 18518 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) ( F ( *p `  J ) ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } ) ) )
223adantr 451 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( G `  1
) )
2317pcopt2 18521 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  ( G ` 
1 ) )  -> 
( F ( *p
`  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) ) ( 
~=ph  `  J ) F )
2415, 22, 23syl2anc 642 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) ) (  ~=ph  `  J
) F )
252, 21, 24ertrd 6676 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) F )
2610adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  H  e.  ( II  Cn  J
) )
274adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G  e.  ( II  Cn  J
) )
289adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( H `  0
) )
297simp3d 969 . . . . . . 7  |-  ( ph  ->  ( H `  1
)  =  ( G `
 0 ) )
3029adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H `  1 )  =  ( G `  0
) )
31 eqid 2283 . . . . . 6  |-  ( y  e.  ( 0 [,] 1 )  |->  if ( y  <_  ( 1  /  2 ) ,  if ( y  <_ 
( 1  /  4
) ,  ( 2  x.  y ) ,  ( y  +  ( 1  /  4 ) ) ) ,  ( ( y  /  2
)  +  ( 1  /  2 ) ) ) )  =  ( y  e.  ( 0 [,] 1 )  |->  if ( y  <_  (
1  /  2 ) ,  if ( y  <_  ( 1  / 
4 ) ,  ( 2  x.  y ) ,  ( y  +  ( 1  /  4
) ) ) ,  ( ( y  / 
2 )  +  ( 1  /  2 ) ) ) )
3215, 26, 27, 28, 30, 31pcoass 18522 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( ( F ( *p `  J ) H ) ( *p `  J
) G ) ( 
~=ph  `  J ) ( F ( *p `  J ) ( H ( *p `  J
) G ) ) )
3314, 10pco1 18513 . . . . . . . 8  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( H `
 1 ) )
3433, 29eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( G `
 0 ) )
3534adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( ( F ( *p `  J ) H ) `
 1 )  =  ( G `  0
) )
36 simpr 447 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )
372, 27erref 6680 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G (  ~=ph  `  J ) G )
3835, 36, 37pcohtpy 18518 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( ( F ( *p `  J ) H ) ( *p `  J
) G ) ( 
~=ph  `  J ) ( P ( *p `  J ) G ) )
392, 32, 38ertr3d 6678 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) ( P ( *p `  J ) G ) )
40 pcophtb.0 . . . . . . 7  |-  ( ph  ->  ( F `  0
)  =  ( G `
 0 ) )
4140adantr 451 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  0 )  =  ( G `  0
) )
4241eqcomd 2288 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( G `  0 )  =  ( F `  0
) )
43 pcophtb.p . . . . . 6  |-  P  =  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )
4443pcopt 18520 . . . . 5  |-  ( ( G  e.  ( II 
Cn  J )  /\  ( G `  0 )  =  ( F ` 
0 ) )  -> 
( P ( *p
`  J ) G ) (  ~=ph  `  J
) G )
4527, 42, 44syl2anc 642 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( P
( *p `  J
) G ) ( 
~=ph  `  J ) G )
462, 39, 45ertrd 6676 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) G )
472, 25, 46ertr3d 6678 . 2  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F (  ~=ph  `  J ) G )
481a1i 10 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
499adantr 451 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ` 
1 )  =  ( H `  0 ) )
50 simpr 447 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  F (  ~=ph  `  J ) G )
5110adantr 451 . . . . 5  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H  e.  ( II  Cn  J ) )
5248, 51erref 6680 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H (  ~=ph  `  J ) H )
5349, 50, 52pcohtpy 18518 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) ( G ( *p `  J
) H ) )
54 eqid 2283 . . . . . . 7  |-  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )
555, 54pcorev2 18526 . . . . . 6  |-  ( G  e.  ( II  Cn  J )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
564, 55syl 15 . . . . 5  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
0 ) } ) )
5740sneqd 3653 . . . . . . 7  |-  ( ph  ->  { ( F ` 
0 ) }  =  { ( G ` 
0 ) } )
5857xpeq2d 4713 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
5943, 58syl5eq 2327 . . . . 5  |-  ( ph  ->  P  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } ) )
6056, 59breqtrrd 4049 . . . 4  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) P )
6160adantr 451 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) P )
6248, 53, 61ertrd 6676 . 2  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) P )
6347, 62impbida 805 1  |-  ( ph  ->  ( ( F ( *p `  J ) H ) (  ~=ph  `  J ) P  <->  F (  ~=ph  `  J ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   ifcif 3565   {csn 3640   class class class wbr 4023    e. cmpt 4077    X. cxp 4687   ` cfv 5255  (class class class)co 5858    Er wer 6657   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742    <_ cle 8868    - cmin 9037    / cdiv 9423   2c2 9795   4c4 9797   [,]cicc 10659    Cn ccn 16954   IIcii 18379    ~=ph cphtpc 18467   *pcpco 18498
This theorem is referenced by:  sconpht2  23769
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-pre-sup 8815  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-isom 5264  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-2o 6480  df-oadd 6483  df-er 6660  df-map 6774  df-ixp 6818  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-fi 7165  df-sup 7194  df-oi 7225  df-card 7572  df-cda 7794  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-rp 10355  df-xneg 10452  df-xadd 10453  df-xmul 10454  df-ioo 10660  df-icc 10663  df-fz 10783  df-fzo 10871  df-seq 11047  df-exp 11105  df-hash 11338  df-cj 11584  df-re 11585  df-im 11586  df-sqr 11720  df-abs 11721  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-sca 13224  df-vsca 13225  df-tset 13227  df-ple 13228  df-ds 13230  df-hom 13232  df-cco 13233  df-rest 13327  df-topn 13328  df-topgen 13344  df-pt 13345  df-prds 13348  df-xrs 13403  df-0g 13404  df-gsum 13405  df-qtop 13410  df-imas 13411  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-mulg 14492  df-cntz 14793  df-cmn 15091  df-xmet 16373  df-met 16374  df-bl 16375  df-mopn 16376  df-cnfld 16378  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cld 16756  df-cn 16957  df-cnp 16958  df-tx 17257  df-hmeo 17446  df-xms 17885  df-ms 17886  df-tms 17887  df-ii 18381  df-htpy 18468  df-phtpy 18469  df-phtpc 18490  df-pco 18503
  Copyright terms: Public domain W3C validator