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Theorem pconpi1 23768
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 23755 . 2  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) )
3 eqid 2283 . . . . . . 7  |-  ( J  pi 1  ( f `
 0 ) )  =  ( J  pi 1  ( f ` 
0 ) )
4 eqid 2283 . . . . . . 7  |-  ( J  pi 1  ( f `
 1 ) )  =  ( J  pi 1  ( f ` 
1 ) )
5 eqid 2283 . . . . . . 7  |-  ( Base `  ( J  pi 1 
( f `  0
) ) )  =  ( Base `  ( J  pi 1  ( f `
 0 ) ) )
6 eqid 2283 . . . . . . 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 23756 . . . . . . . . 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 16671 . . . . . . . 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 5866 . . . . . . . . 9  |-  ( x  =  y  ->  (
1  -  x )  =  ( 1  -  y ) )
1413fveq2d 5529 . . . . . . . 8  |-  ( x  =  y  ->  (
f `  ( 1  -  x ) )  =  ( f `  (
1  -  y ) ) )
1514cbvmptv 4111 . . . . . . 7  |-  ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( f `  ( 1  -  y
) ) )
163, 4, 5, 6, 11, 12, 15pi1xfrgim 18556 . . . . . 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 5874 . . . . . . . 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 2333 . . . . . . 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 5874 . . . . . . . 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 2333 . . . . . . 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 5876 . . . . . 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 2359 . . . . 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 14734 . . . . 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 2667 . 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 1623    e. wcel 1684   E.wrex 2544   <.cop 3643   U.cuni 3827   class class class wbr 4023    e. cmpt 4077   ran crn 4690   ` cfv 5255  (class class class)co 5858   [cec 6658   0cc0 8737   1c1 8738    - cmin 9037   [,]cicc 10659   Basecbs 13148   GrpIso cgim 14721    ~=ph𝑔 cgic 14722   Topctop 16631  TopOnctopon 16632    Cn ccn 16954   IIcii 18379    ~=ph cphtpc 18467   *pcpco 18498    pi 1 cpi1 18501  PConcpcon 23750
This theorem is referenced by:  sconpi1  23770
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-ec 6662  df-qs 6666  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-divs 13412  df-xps 13413  df-mre 13488  df-mrc 13489  df-acs 13491  df-mnd 14367  df-submnd 14416  df-grp 14489  df-mulg 14492  df-ghm 14681  df-gim 14723  df-gic 14724  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  df-om1 18504  df-pi1 18506  df-pcon 23752
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