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Theorem pcophtb 18926
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 18892 . . . 4  |-  (  ~=ph  `  J )  Er  (
II  Cn  J )
21a1i 11 . . 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 18922 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  ( H  e.  ( II  Cn  J )  /\  ( H `  0 )  =  ( G ` 
1 )  /\  ( H `  1 )  =  ( G ` 
0 ) ) )
74, 6syl 16 . . . . . . . . 9  |-  ( ph  ->  ( H  e.  ( II  Cn  J )  /\  ( H ` 
0 )  =  ( G `  1 )  /\  ( H ` 
1 )  =  ( G `  0 ) ) )
87simp2d 970 . . . . . . . 8  |-  ( ph  ->  ( H `  0
)  =  ( G `
 1 ) )
93, 8eqtr4d 2423 . . . . . . 7  |-  ( ph  ->  ( F `  1
)  =  ( H `
 0 ) )
107simp1d 969 . . . . . . . 8  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
1110, 4pco0 18911 . . . . . . 7  |-  ( ph  ->  ( ( H ( *p `  J ) G ) `  0
)  =  ( H `
 0 ) )
129, 11eqtr4d 2423 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( ( H ( *p `  J ) G ) `
 0 ) )
1312adantr 452 . . . . 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 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F  e.  ( II  Cn  J
) )
162, 15erref 6862 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F (  ~=ph  `  J ) F )
17 eqid 2388 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )
185, 17pcorev 18924 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  ( H ( *p `  J ) G ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  {
( G `  1
) } ) )
194, 18syl 16 . . . . . 6  |-  ( ph  ->  ( H ( *p
`  J ) G ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) )
2019adantr 452 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H
( *p `  J
) G ) ( 
~=ph  `  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) )
2113, 16, 20pcohtpy 18917 . . . 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 452 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( G `  1
) )
2317pcopt2 18920 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  ( G ` 
1 ) )  -> 
( F ( *p
`  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) ) ( 
~=ph  `  J ) F )
2415, 22, 23syl2anc 643 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) ) (  ~=ph  `  J
) F )
252, 21, 24ertrd 6858 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) F )
2610adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  H  e.  ( II  Cn  J
) )
274adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G  e.  ( II  Cn  J
) )
289adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( H `  0
) )
297simp3d 971 . . . . . . 7  |-  ( ph  ->  ( H `  1
)  =  ( G `
 0 ) )
3029adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H `  1 )  =  ( G `  0
) )
31 eqid 2388 . . . . . 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 18921 . . . . 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 18912 . . . . . . . 8  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( H `
 1 ) )
3433, 29eqtrd 2420 . . . . . . 7  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( G `
 0 ) )
3534adantr 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( ( F ( *p `  J ) H ) `
 1 )  =  ( G `  0
) )
36 simpr 448 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )
372, 27erref 6862 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G (  ~=ph  `  J ) G )
3835, 36, 37pcohtpy 18917 . . . . 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 6860 . . . 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 452 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  0 )  =  ( G `  0
) )
4241eqcomd 2393 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( G `  0 )  =  ( F `  0
) )
43 pcophtb.p . . . . . 6  |-  P  =  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )
4443pcopt 18919 . . . . 5  |-  ( ( G  e.  ( II 
Cn  J )  /\  ( G `  0 )  =  ( F ` 
0 ) )  -> 
( P ( *p
`  J ) G ) (  ~=ph  `  J
) G )
4527, 42, 44syl2anc 643 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( P
( *p `  J
) G ) ( 
~=ph  `  J ) G )
462, 39, 45ertrd 6858 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) G )
472, 25, 46ertr3d 6860 . 2  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F (  ~=ph  `  J ) G )
481a1i 11 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  (  ~=ph  `  J
)  Er  ( II 
Cn  J ) )
499adantr 452 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ` 
1 )  =  ( H `  0 ) )
50 simpr 448 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  F (  ~=ph  `  J ) G )
5110adantr 452 . . . . 5  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H  e.  ( II  Cn  J ) )
5248, 51erref 6862 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H (  ~=ph  `  J ) H )
5349, 50, 52pcohtpy 18917 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) ( G ( *p `  J
) H ) )
54 eqid 2388 . . . . . . 7  |-  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )
555, 54pcorev2 18925 . . . . . 6  |-  ( G  e.  ( II  Cn  J )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
564, 55syl 16 . . . . 5  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
0 ) } ) )
5740sneqd 3771 . . . . . . 7  |-  ( ph  ->  { ( F ` 
0 ) }  =  { ( G ` 
0 ) } )
5857xpeq2d 4843 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
5943, 58syl5eq 2432 . . . . 5  |-  ( ph  ->  P  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } ) )
6056, 59breqtrrd 4180 . . . 4  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) P )
6160adantr 452 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) P )
6248, 53, 61ertrd 6858 . 2  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) P )
6347, 62impbida 806 1  |-  ( ph  ->  ( ( F ( *p `  J ) H ) (  ~=ph  `  J ) P  <->  F (  ~=ph  `  J ) G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   ifcif 3683   {csn 3758   class class class wbr 4154    e. cmpt 4208    X. cxp 4817   ` cfv 5395  (class class class)co 6021    Er wer 6839   0cc0 8924   1c1 8925    + caddc 8927    x. cmul 8929    <_ cle 9055    - cmin 9224    / cdiv 9610   2c2 9982   4c4 9984   [,]cicc 10852    Cn ccn 17211   IIcii 18777    ~=ph cphtpc 18866   *pcpco 18897
This theorem is referenced by:  sconpht2  24705
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-rep 4262  ax-sep 4272  ax-nul 4280  ax-pow 4319  ax-pr 4345  ax-un 4642  ax-inf2 7530  ax-cnex 8980  ax-resscn 8981  ax-1cn 8982  ax-icn 8983  ax-addcl 8984  ax-addrcl 8985  ax-mulcl 8986  ax-mulrcl 8987  ax-mulcom 8988  ax-addass 8989  ax-mulass 8990  ax-distr 8991  ax-i2m1 8992  ax-1ne0 8993  ax-1rid 8994  ax-rnegex 8995  ax-rrecex 8996  ax-cnre 8997  ax-pre-lttri 8998  ax-pre-lttrn 8999  ax-pre-ltadd 9000  ax-pre-mulgt0 9001  ax-pre-sup 9002  ax-addf 9003  ax-mulf 9004
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-nel 2554  df-ral 2655  df-rex 2656  df-reu 2657  df-rmo 2658  df-rab 2659  df-v 2902  df-sbc 3106  df-csb 3196  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-pss 3280  df-nul 3573  df-if 3684  df-pw 3745  df-sn 3764  df-pr 3765  df-tp 3766  df-op 3767  df-uni 3959  df-int 3994  df-iun 4038  df-iin 4039  df-br 4155  df-opab 4209  df-mpt 4210  df-tr 4245  df-eprel 4436  df-id 4440  df-po 4445  df-so 4446  df-fr 4483  df-se 4484  df-we 4485  df-ord 4526  df-on 4527  df-lim 4528  df-suc 4529  df-om 4787  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-isom 5404  df-ov 6024  df-oprab 6025  df-mpt2 6026  df-of 6245  df-1st 6289  df-2nd 6290  df-riota 6486  df-recs 6570  df-rdg 6605  df-1o 6661  df-2o 6662  df-oadd 6665  df-er 6842  df-map 6957  df-ixp 7001  df-en 7047  df-dom 7048  df-sdom 7049  df-fin 7050  df-fi 7352  df-sup 7382  df-oi 7413  df-card 7760  df-cda 7982  df-pnf 9056  df-mnf 9057  df-xr 9058  df-ltxr 9059  df-le 9060  df-sub 9226  df-neg 9227  df-div 9611  df-nn 9934  df-2 9991  df-3 9992  df-4 9993  df-5 9994  df-6 9995  df-7 9996  df-8 9997  df-9 9998  df-10 9999  df-n0 10155  df-z 10216  df-dec 10316  df-uz 10422  df-q 10508  df-rp 10546  df-xneg 10643  df-xadd 10644  df-xmul 10645  df-ioo 10853  df-icc 10856  df-fz 10977  df-fzo 11067  df-seq 11252  df-exp 11311  df-hash 11547  df-cj 11832  df-re 11833  df-im 11834  df-sqr 11968  df-abs 11969  df-struct 13399  df-ndx 13400  df-slot 13401  df-base 13402  df-sets 13403  df-ress 13404  df-plusg 13470  df-mulr 13471  df-starv 13472  df-sca 13473  df-vsca 13474  df-tset 13476  df-ple 13477  df-ds 13479  df-unif 13480  df-hom 13481  df-cco 13482  df-rest 13578  df-topn 13579  df-topgen 13595  df-pt 13596  df-prds 13599  df-xrs 13654  df-0g 13655  df-gsum 13656  df-qtop 13661  df-imas 13662  df-xps 13664  df-mre 13739  df-mrc 13740  df-acs 13742  df-mnd 14618  df-submnd 14667  df-mulg 14743  df-cntz 15044  df-cmn 15342  df-xmet 16620  df-met 16621  df-bl 16622  df-mopn 16623  df-cnfld 16628  df-top 16887  df-bases 16889  df-topon 16890  df-topsp 16891  df-cld 17007  df-cn 17214  df-cnp 17215  df-tx 17516  df-hmeo 17709  df-xms 18260  df-ms 18261  df-tms 18262  df-ii 18779  df-htpy 18867  df-phtpy 18868  df-phtpc 18889  df-pco 18902
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