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Theorem pconpi1 24052
Description: All fundamental groups in a path-connected space are isomorphic. (Contributed by Mario Carneiro, 12-Feb-2015.)
Hypotheses
Ref Expression
pconpi1.x  |-  X  = 
U. J
pconpi1.p  |-  P  =  ( J  pi 1  A )
pconpi1.q  |-  Q  =  ( J  pi 1  B )
pconpi1.s  |-  S  =  ( Base `  P
)
pconpi1.t  |-  T  =  ( Base `  Q
)
Assertion
Ref Expression
pconpi1  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  P  ~=ph𝑔  Q )

Proof of Theorem pconpi1
Dummy variables  f  h  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 pconpi1.x . . 3  |-  X  = 
U. J
21pconcn 24039 . 2  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) )
3 eqid 2316 . . . . . . 7  |-  ( J  pi 1  ( f `
 0 ) )  =  ( J  pi 1  ( f ` 
0 ) )
4 eqid 2316 . . . . . . 7  |-  ( J  pi 1  ( f `
 1 ) )  =  ( J  pi 1  ( f ` 
1 ) )
5 eqid 2316 . . . . . . 7  |-  ( Base `  ( J  pi 1 
( f `  0
) ) )  =  ( Base `  ( J  pi 1  ( f `
 0 ) ) )
6 eqid 2316 . . . . . . 7  |-  ran  (
h  e.  U. ( Base `  ( J  pi 1  ( f ` 
0 ) ) ) 
|->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )  =  ran  ( h  e.  U. ( Base `  ( J  pi 1  ( f ` 
0 ) ) ) 
|->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )
7 simpl1 958 . . . . . . . . 9  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  J  e. PCon )
8 pcontop 24040 . . . . . . . . 9  |-  ( J  e. PCon  ->  J  e.  Top )
97, 8syl 15 . . . . . . . 8  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  J  e.  Top )
101toptopon 16727 . . . . . . . 8  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
119, 10sylib 188 . . . . . . 7  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  J  e.  (TopOn `  X ) )
12 simprl 732 . . . . . . 7  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  f  e.  ( II  Cn  J ) )
13 oveq2 5908 . . . . . . . . 9  |-  ( x  =  y  ->  (
1  -  x )  =  ( 1  -  y ) )
1413fveq2d 5567 . . . . . . . 8  |-  ( x  =  y  ->  (
f `  ( 1  -  x ) )  =  ( f `  (
1  -  y ) ) )
1514cbvmptv 4148 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( f `  ( 1  -  y
) ) )
163, 4, 5, 6, 11, 12, 15pi1xfrgim 18609 . . . . . 6  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ran  ( h  e.  U. ( Base `  ( J  pi 1  ( f `
 0 ) ) )  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )  e.  (
( J  pi 1 
( f `  0
) ) GrpIso  ( J  pi 1  ( f `  1 ) ) ) )
17 simprrl 740 . . . . . . . . 9  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( f ` 
0 )  =  A )
1817oveq2d 5916 . . . . . . . 8  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi 1  ( f ` 
0 ) )  =  ( J  pi 1  A ) )
19 pconpi1.p . . . . . . . 8  |-  P  =  ( J  pi 1  A )
2018, 19syl6eqr 2366 . . . . . . 7  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi 1  ( f ` 
0 ) )  =  P )
21 simprrr 741 . . . . . . . . 9  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( f ` 
1 )  =  B )
2221oveq2d 5916 . . . . . . . 8  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi 1  ( f ` 
1 ) )  =  ( J  pi 1  B ) )
23 pconpi1.q . . . . . . . 8  |-  Q  =  ( J  pi 1  B )
2422, 23syl6eqr 2366 . . . . . . 7  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi 1  ( f ` 
1 ) )  =  Q )
2520, 24oveq12d 5918 . . . . . 6  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( ( J  pi 1  ( f `
 0 ) ) GrpIso 
( J  pi 1 
( f `  1
) ) )  =  ( P GrpIso  Q ) )
2616, 25eleqtrd 2392 . . . . 5  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ran  ( h  e.  U. ( Base `  ( J  pi 1  ( f `
 0 ) ) )  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )  e.  ( P GrpIso  Q ) )
27 brgici 14783 . . . . 5  |-  ( ran  ( h  e.  U. ( Base `  ( J  pi 1  ( f `  0 ) ) )  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )  e.  ( P GrpIso  Q )  ->  P  ~=ph𝑔  Q )
2826, 27syl 15 . . . 4  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  P  ~=ph𝑔 
Q )
2928expr 598 . . 3  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  f  e.  (
II  Cn  J )
)  ->  ( (
( f `  0
)  =  A  /\  ( f `  1
)  =  B )  ->  P  ~=ph𝑔 
Q ) )
3029rexlimdva 2701 . 2  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  ( E. f  e.  (
II  Cn  J )
( ( f ` 
0 )  =  A  /\  ( f ` 
1 )  =  B )  ->  P  ~=ph𝑔  Q ) )
312, 30mpd 14 1  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  P  ~=ph𝑔  Q )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1633    e. wcel 1701   E.wrex 2578   <.cop 3677   U.cuni 3864   class class class wbr 4060    e. cmpt 4114   ran crn 4727   ` cfv 5292  (class class class)co 5900   [cec 6700   0cc0 8782   1c1 8783    - cmin 9082   [,]cicc 10706   Basecbs 13195   GrpIso cgim 14770    ~=ph𝑔 cgic 14771   Topctop 16687  TopOnctopon 16688    Cn ccn 17010   IIcii 18431    ~=ph cphtpc 18520   *pcpco 18551    pi 1 cpi1 18554  PConcpcon 24034
This theorem is referenced by:  sconpi1  24054
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549  ax-inf2 7387  ax-cnex 8838  ax-resscn 8839  ax-1cn 8840  ax-icn 8841  ax-addcl 8842  ax-addrcl 8843  ax-mulcl 8844  ax-mulrcl 8845  ax-mulcom 8846  ax-addass 8847  ax-mulass 8848  ax-distr 8849  ax-i2m1 8850  ax-1ne0 8851  ax-1rid 8852  ax-rnegex 8853  ax-rrecex 8854  ax-cnre 8855  ax-pre-lttri 8856  ax-pre-lttrn 8857  ax-pre-ltadd 8858  ax-pre-mulgt0 8859  ax-pre-sup 8860  ax-addf 8861  ax-mulf 8862
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 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rmo 2585  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-pss 3202  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-tp 3682  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-tr 4151  df-eprel 4342  df-id 4346  df-po 4351  df-so 4352  df-fr 4389  df-se 4390  df-we 4391  df-ord 4432  df-on 4433  df-lim 4434  df-suc 4435  df-om 4694  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-isom 5301  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-of 6120  df-1st 6164  df-2nd 6165  df-riota 6346  df-recs 6430  df-rdg 6465  df-1o 6521  df-2o 6522  df-oadd 6525  df-er 6702  df-ec 6704  df-qs 6708  df-map 6817  df-ixp 6861  df-en 6907  df-dom 6908  df-sdom 6909  df-fin 6910  df-fi 7210  df-sup 7239  df-oi 7270  df-card 7617  df-cda 7839  df-pnf 8914  df-mnf 8915  df-xr 8916  df-ltxr 8917  df-le 8918  df-sub 9084  df-neg 9085  df-div 9469  df-nn 9792  df-2 9849  df-3 9850  df-4 9851  df-5 9852  df-6 9853  df-7 9854  df-8 9855  df-9 9856  df-10 9857  df-n0 10013  df-z 10072  df-dec 10172  df-uz 10278  df-q 10364  df-rp 10402  df-xneg 10499  df-xadd 10500  df-xmul 10501  df-ioo 10707  df-icc 10710  df-fz 10830  df-fzo 10918  df-seq 11094  df-exp 11152  df-hash 11385  df-cj 11631  df-re 11632  df-im 11633  df-sqr 11767  df-abs 11768  df-struct 13197  df-ndx 13198  df-slot 13199  df-base 13200  df-sets 13201  df-ress 13202  df-plusg 13268  df-mulr 13269  df-starv 13270  df-sca 13271  df-vsca 13272  df-tset 13274  df-ple 13275  df-ds 13277  df-unif 13278  df-hom 13279  df-cco 13280  df-rest 13376  df-topn 13377  df-topgen 13393  df-pt 13394  df-prds 13397  df-xrs 13452  df-0g 13453  df-gsum 13454  df-qtop 13459  df-imas 13460  df-divs 13461  df-xps 13462  df-mre 13537  df-mrc 13538  df-acs 13540  df-mnd 14416  df-submnd 14465  df-grp 14538  df-mulg 14541  df-ghm 14730  df-gim 14772  df-gic 14773  df-cntz 14842  df-cmn 15140  df-xmet 16425  df-met 16426  df-bl 16427  df-mopn 16428  df-cnfld 16433  df-top 16692  df-bases 16694  df-topon 16695  df-topsp 16696  df-cld 16812  df-cn 17013  df-cnp 17014  df-tx 17313  df-hmeo 17502  df-xms 17937  df-ms 17938  df-tms 17939  df-ii 18433  df-htpy 18521  df-phtpy 18522  df-phtpc 18543  df-pco 18556  df-om1 18557  df-pi1 18559  df-pcon 24036
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