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Theorem pcophtb 21821
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 21787 . . . 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 21817 . . . . . . . . . 10  |-  ( G  e.  ( II  Cn  J )  ->  ( H  e.  ( II  Cn  J )  /\  ( H `  0 )  =  ( G ` 
1 )  /\  ( H `  1 )  =  ( G ` 
0 ) ) )
74, 6syl 17 . . . . . . . . 9  |-  ( ph  ->  ( H  e.  ( II  Cn  J )  /\  ( H ` 
0 )  =  ( G `  1 )  /\  ( H ` 
1 )  =  ( G `  0 ) ) )
87simp2d 1010 . . . . . . . 8  |-  ( ph  ->  ( H `  0
)  =  ( G `
 1 ) )
93, 8eqtr4d 2446 . . . . . . 7  |-  ( ph  ->  ( F `  1
)  =  ( H `
 0 ) )
107simp1d 1009 . . . . . . . 8  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
1110, 4pco0 21806 . . . . . . 7  |-  ( ph  ->  ( ( H ( *p `  J ) G ) `  0
)  =  ( H `
 0 ) )
129, 11eqtr4d 2446 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( ( H ( *p `  J ) G ) `
 0 ) )
1312adantr 463 . . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F  e.  ( II  Cn  J
) )
162, 15erref 7368 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F (  ~=ph  `  J ) F )
17 eqid 2402 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )
185, 17pcorev 21819 . . . . . . 7  |-  ( G  e.  ( II  Cn  J )  ->  ( H ( *p `  J ) G ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  {
( G `  1
) } ) )
194, 18syl 17 . . . . . 6  |-  ( ph  ->  ( H ( *p
`  J ) G ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) )
2019adantr 463 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H
( *p `  J
) G ) ( 
~=ph  `  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) )
2113, 16, 20pcohtpy 21812 . . . 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 463 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( G `  1
) )
2317pcopt2 21815 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  ( G ` 
1 ) )  -> 
( F ( *p
`  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) ) ( 
~=ph  `  J ) F )
2415, 22, 23syl2anc 659 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) ) (  ~=ph  `  J
) F )
252, 21, 24ertrd 7364 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) F )
2610adantr 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  H  e.  ( II  Cn  J
) )
274adantr 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G  e.  ( II  Cn  J
) )
289adantr 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( H `  0
) )
297simp3d 1011 . . . . . . 7  |-  ( ph  ->  ( H `  1
)  =  ( G `
 0 ) )
3029adantr 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H `  1 )  =  ( G `  0
) )
31 eqid 2402 . . . . . 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 21816 . . . . 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 21807 . . . . . . . 8  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( H `
 1 ) )
3433, 29eqtrd 2443 . . . . . . 7  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( G `
 0 ) )
3534adantr 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( ( F ( *p `  J ) H ) `
 1 )  =  ( G `  0
) )
36 simpr 459 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )
372, 27erref 7368 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G (  ~=ph  `  J ) G )
3835, 36, 37pcohtpy 21812 . . . . 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 7366 . . . 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 463 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  0 )  =  ( G `  0
) )
4241eqcomd 2410 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( G `  0 )  =  ( F `  0
) )
43 pcophtb.p . . . . . 6  |-  P  =  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )
4443pcopt 21814 . . . . 5  |-  ( ( G  e.  ( II 
Cn  J )  /\  ( G `  0 )  =  ( F ` 
0 ) )  -> 
( P ( *p
`  J ) G ) (  ~=ph  `  J
) G )
4527, 42, 44syl2anc 659 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( P
( *p `  J
) G ) ( 
~=ph  `  J ) G )
462, 39, 45ertrd 7364 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) G )
472, 25, 46ertr3d 7366 . 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 463 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ` 
1 )  =  ( H `  0 ) )
50 simpr 459 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  F (  ~=ph  `  J ) G )
5110adantr 463 . . . . 5  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H  e.  ( II  Cn  J ) )
5248, 51erref 7368 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H (  ~=ph  `  J ) H )
5349, 50, 52pcohtpy 21812 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) ( G ( *p `  J
) H ) )
54 eqid 2402 . . . . . . 7  |-  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )
555, 54pcorev2 21820 . . . . . 6  |-  ( G  e.  ( II  Cn  J )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
564, 55syl 17 . . . . 5  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
0 ) } ) )
5740sneqd 3984 . . . . . . 7  |-  ( ph  ->  { ( F ` 
0 ) }  =  { ( G ` 
0 ) } )
5857xpeq2d 4847 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
5943, 58syl5eq 2455 . . . . 5  |-  ( ph  ->  P  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } ) )
6056, 59breqtrrd 4421 . . . 4  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) P )
6160adantr 463 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) P )
6248, 53, 61ertrd 7364 . 2  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) P )
6347, 62impbida 833 1  |-  ( ph  ->  ( ( F ( *p `  J ) H ) (  ~=ph  `  J ) P  <->  F (  ~=ph  `  J ) G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 367    /\ w3a 974    = wceq 1405    e. wcel 1842   ifcif 3885   {csn 3972   class class class wbr 4395    |-> cmpt 4453    X. cxp 4821   ` cfv 5569  (class class class)co 6278    Er wer 7345   0cc0 9522   1c1 9523    + caddc 9525    x. cmul 9527    <_ cle 9659    - cmin 9841    / cdiv 10247   2c2 10626   4c4 10628   [,]cicc 11585    Cn ccn 20018   IIcii 21671    ~=ph cphtpc 21761   *pcpco 21792
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574  ax-inf2 8091  ax-cnex 9578  ax-resscn 9579  ax-1cn 9580  ax-icn 9581  ax-addcl 9582  ax-addrcl 9583  ax-mulcl 9584  ax-mulrcl 9585  ax-mulcom 9586  ax-addass 9587  ax-mulass 9588  ax-distr 9589  ax-i2m1 9590  ax-1ne0 9591  ax-1rid 9592  ax-rnegex 9593  ax-rrecex 9594  ax-cnre 9595  ax-pre-lttri 9596  ax-pre-lttrn 9597  ax-pre-ltadd 9598  ax-pre-mulgt0 9599  ax-pre-sup 9600  ax-addf 9601  ax-mulf 9602
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rmo 2762  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-iun 4273  df-iin 4274  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-se 4783  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-pred 5367  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-isom 5578  df-riota 6240  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-of 6521  df-om 6684  df-1st 6784  df-2nd 6785  df-supp 6903  df-wrecs 7013  df-recs 7075  df-rdg 7113  df-1o 7167  df-2o 7168  df-oadd 7171  df-er 7348  df-map 7459  df-ixp 7508  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-fsupp 7864  df-fi 7905  df-sup 7935  df-oi 7969  df-card 8352  df-cda 8580  df-pnf 9660  df-mnf 9661  df-xr 9662  df-ltxr 9663  df-le 9664  df-sub 9843  df-neg 9844  df-div 10248  df-nn 10577  df-2 10635  df-3 10636  df-4 10637  df-5 10638  df-6 10639  df-7 10640  df-8 10641  df-9 10642  df-10 10643  df-n0 10837  df-z 10906  df-dec 11020  df-uz 11128  df-q 11228  df-rp 11266  df-xneg 11371  df-xadd 11372  df-xmul 11373  df-ioo 11586  df-icc 11589  df-fz 11727  df-fzo 11855  df-seq 12152  df-exp 12211  df-hash 12453  df-cj 13081  df-re 13082  df-im 13083  df-sqrt 13217  df-abs 13218  df-struct 14843  df-ndx 14844  df-slot 14845  df-base 14846  df-sets 14847  df-ress 14848  df-plusg 14922  df-mulr 14923  df-starv 14924  df-sca 14925  df-vsca 14926  df-ip 14927  df-tset 14928  df-ple 14929  df-ds 14931  df-unif 14932  df-hom 14933  df-cco 14934  df-rest 15037  df-topn 15038  df-0g 15056  df-gsum 15057  df-topgen 15058  df-pt 15059  df-prds 15062  df-xrs 15116  df-qtop 15121  df-imas 15122  df-xps 15124  df-mre 15200  df-mrc 15201  df-acs 15203  df-mgm 16196  df-sgrp 16235  df-mnd 16245  df-submnd 16291  df-mulg 16384  df-cntz 16679  df-cmn 17124  df-psmet 18731  df-xmet 18732  df-met 18733  df-bl 18734  df-mopn 18735  df-cnfld 18741  df-top 19691  df-bases 19693  df-topon 19694  df-topsp 19695  df-cld 19812  df-cn 20021  df-cnp 20022  df-tx 20355  df-hmeo 20548  df-xms 21115  df-ms 21116  df-tms 21117  df-ii 21673  df-htpy 21762  df-phtpy 21763  df-phtpc 21784  df-pco 21797
This theorem is referenced by:  sconpht2  29535
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