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Theorem pcophtb 21261
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 21227 . . . 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 21257 . . . . . . . . . 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 1009 . . . . . . . 8  |-  ( ph  ->  ( H `  0
)  =  ( G `
 1 ) )
93, 8eqtr4d 2511 . . . . . . 7  |-  ( ph  ->  ( F `  1
)  =  ( H `
 0 ) )
107simp1d 1008 . . . . . . . 8  |-  ( ph  ->  H  e.  ( II 
Cn  J ) )
1110, 4pco0 21246 . . . . . . 7  |-  ( ph  ->  ( ( H ( *p `  J ) G ) `  0
)  =  ( H `
 0 ) )
129, 11eqtr4d 2511 . . . . . 6  |-  ( ph  ->  ( F `  1
)  =  ( ( H ( *p `  J ) G ) `
 0 ) )
1312adantr 465 . . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F  e.  ( II  Cn  J
) )
162, 15erref 7328 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  F (  ~=ph  `  J ) F )
17 eqid 2467 . . . . . . . 8  |-  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 1 ) } )
185, 17pcorev 21259 . . . . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H
( *p `  J
) G ) ( 
~=ph  `  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) )
2113, 16, 20pcohtpy 21252 . . . 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 465 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( G `  1
) )
2317pcopt2 21255 . . . . 5  |-  ( ( F  e.  ( II 
Cn  J )  /\  ( F `  1 )  =  ( G ` 
1 ) )  -> 
( F ( *p
`  J ) ( ( 0 [,] 1
)  X.  { ( G `  1 ) } ) ) ( 
~=ph  `  J ) F )
2415, 22, 23syl2anc 661 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( ( 0 [,] 1 )  X. 
{ ( G ` 
1 ) } ) ) (  ~=ph  `  J
) F )
252, 21, 24ertrd 7324 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) F )
2610adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  H  e.  ( II  Cn  J
) )
274adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G  e.  ( II  Cn  J
) )
289adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  1 )  =  ( H `  0
) )
297simp3d 1010 . . . . . . 7  |-  ( ph  ->  ( H `  1
)  =  ( G `
 0 ) )
3029adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( H `  1 )  =  ( G `  0
) )
31 eqid 2467 . . . . . 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 21256 . . . . 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 21247 . . . . . . . 8  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( H `
 1 ) )
3433, 29eqtrd 2508 . . . . . . 7  |-  ( ph  ->  ( ( F ( *p `  J ) H ) `  1
)  =  ( G `
 0 ) )
3534adantr 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( ( F ( *p `  J ) H ) `
 1 )  =  ( G `  0
) )
36 simpr 461 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )
372, 27erref 7328 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  G (  ~=ph  `  J ) G )
3835, 36, 37pcohtpy 21252 . . . . 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 7326 . . . 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 465 . . . . . 6  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F `  0 )  =  ( G `  0
) )
4241eqcomd 2475 . . . . 5  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( G `  0 )  =  ( F `  0
) )
43 pcophtb.p . . . . . 6  |-  P  =  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )
4443pcopt 21254 . . . . 5  |-  ( ( G  e.  ( II 
Cn  J )  /\  ( G `  0 )  =  ( F ` 
0 ) )  -> 
( P ( *p
`  J ) G ) (  ~=ph  `  J
) G )
4527, 42, 44syl2anc 661 . . . 4  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( P
( *p `  J
) G ) ( 
~=ph  `  J ) G )
462, 39, 45ertrd 7324 . . 3  |-  ( (
ph  /\  ( F
( *p `  J
) H ) ( 
~=ph  `  J ) P )  ->  ( F
( *p `  J
) ( H ( *p `  J ) G ) ) ( 
~=ph  `  J ) G )
472, 25, 46ertr3d 7326 . 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 465 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ` 
1 )  =  ( H `  0 ) )
50 simpr 461 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  F (  ~=ph  `  J ) G )
5110adantr 465 . . . . 5  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H  e.  ( II  Cn  J ) )
5248, 51erref 7328 . . . 4  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  H (  ~=ph  `  J ) H )
5349, 50, 52pcohtpy 21252 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) ( G ( *p `  J
) H ) )
54 eqid 2467 . . . . . . 7  |-  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } )
555, 54pcorev2 21260 . . . . . 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 4039 . . . . . . 7  |-  ( ph  ->  { ( F ` 
0 ) }  =  { ( G ` 
0 ) } )
5857xpeq2d 5023 . . . . . 6  |-  ( ph  ->  ( ( 0 [,] 1 )  X.  {
( F `  0
) } )  =  ( ( 0 [,] 1 )  X.  {
( G `  0
) } ) )
5943, 58syl5eq 2520 . . . . 5  |-  ( ph  ->  P  =  ( ( 0 [,] 1 )  X.  { ( G `
 0 ) } ) )
6056, 59breqtrrd 4473 . . . 4  |-  ( ph  ->  ( G ( *p
`  J ) H ) (  ~=ph  `  J
) P )
6160adantr 465 . . 3  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( G ( *p `  J ) H ) (  ~=ph  `  J ) P )
6248, 53, 61ertrd 7324 . 2  |-  ( (
ph  /\  F (  ~=ph  `  J ) G )  ->  ( F ( *p `  J ) H ) (  ~=ph  `  J ) P )
6347, 62impbida 830 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 369    /\ w3a 973    = wceq 1379    e. wcel 1767   ifcif 3939   {csn 4027   class class class wbr 4447    |-> cmpt 4505    X. cxp 4997   ` cfv 5586  (class class class)co 6282    Er wer 7305   0cc0 9488   1c1 9489    + caddc 9491    x. cmul 9493    <_ cle 9625    - cmin 9801    / cdiv 10202   2c2 10581   4c4 10583   [,]cicc 11528    Cn ccn 19488   IIcii 21111    ~=ph cphtpc 21201   *pcpco 21232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574  ax-inf2 8054  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566  ax-addf 9567  ax-mulf 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-se 4839  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-isom 5595  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-of 6522  df-om 6679  df-1st 6781  df-2nd 6782  df-supp 6899  df-recs 7039  df-rdg 7073  df-1o 7127  df-2o 7128  df-oadd 7131  df-er 7308  df-map 7419  df-ixp 7467  df-en 7514  df-dom 7515  df-sdom 7516  df-fin 7517  df-fsupp 7826  df-fi 7867  df-sup 7897  df-oi 7931  df-card 8316  df-cda 8544  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804  df-div 10203  df-nn 10533  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10973  df-uz 11079  df-q 11179  df-rp 11217  df-xneg 11314  df-xadd 11315  df-xmul 11316  df-ioo 11529  df-icc 11532  df-fz 11669  df-fzo 11789  df-seq 12071  df-exp 12130  df-hash 12368  df-cj 12889  df-re 12890  df-im 12891  df-sqrt 13025  df-abs 13026  df-struct 14485  df-ndx 14486  df-slot 14487  df-base 14488  df-sets 14489  df-ress 14490  df-plusg 14561  df-mulr 14562  df-starv 14563  df-sca 14564  df-vsca 14565  df-ip 14566  df-tset 14567  df-ple 14568  df-ds 14570  df-unif 14571  df-hom 14572  df-cco 14573  df-rest 14671  df-topn 14672  df-0g 14690  df-gsum 14691  df-topgen 14692  df-pt 14693  df-prds 14696  df-xrs 14750  df-qtop 14755  df-imas 14756  df-xps 14758  df-mre 14834  df-mrc 14835  df-acs 14837  df-mnd 15725  df-submnd 15775  df-mulg 15858  df-cntz 16147  df-cmn 16593  df-psmet 18179  df-xmet 18180  df-met 18181  df-bl 18182  df-mopn 18183  df-cnfld 18189  df-top 19163  df-bases 19165  df-topon 19166  df-topsp 19167  df-cld 19283  df-cn 19491  df-cnp 19492  df-tx 19795  df-hmeo 19988  df-xms 20555  df-ms 20556  df-tms 20557  df-ii 21113  df-htpy 21202  df-phtpy 21203  df-phtpc 21224  df-pco 21237
This theorem is referenced by:  sconpht2  28320
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