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Theorem pconpi1 29748
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  pi1  A )
pconpi1.q  |-  Q  =  ( J  pi1  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  ~=g𝑔  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 29735 . 2  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  E. f  e.  ( II  Cn  J
) ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) )
3 eqid 2429 . . . . 5  |-  ( J  pi1  ( f `
 0 ) )  =  ( J  pi1  ( f ` 
0 ) )
4 eqid 2429 . . . . 5  |-  ( J  pi1  ( f `
 1 ) )  =  ( J  pi1  ( f ` 
1 ) )
5 eqid 2429 . . . . 5  |-  ( Base `  ( J  pi1 
( f `  0
) ) )  =  ( Base `  ( J  pi1  ( f `
 0 ) ) )
6 eqid 2429 . . . . 5  |-  ran  (
h  e.  U. ( Base `  ( J  pi1  ( 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  pi1  ( f ` 
0 ) ) ) 
|->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )
7 simpl1 1008 . . . . . . 7  |-  ( ( ( 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 29736 . . . . . . 7  |-  ( J  e. PCon  ->  J  e.  Top )
97, 8syl 17 . . . . . 6  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  J  e.  Top )
101toptopon 19879 . . . . . 6  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
119, 10sylib 199 . . . . 5  |-  ( ( ( 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 762 . . . . 5  |-  ( ( ( 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 6313 . . . . . . 7  |-  ( x  =  y  ->  (
1  -  x )  =  ( 1  -  y ) )
1413fveq2d 5885 . . . . . 6  |-  ( x  =  y  ->  (
f `  ( 1  -  x ) )  =  ( f `  (
1  -  y ) ) )
1514cbvmptv 4518 . . . . 5  |-  ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) )  =  ( y  e.  ( 0 [,] 1
)  |->  ( f `  ( 1  -  y
) ) )
163, 4, 5, 6, 11, 12, 15pi1xfrgim 21982 . . . 4  |-  ( ( ( 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  pi1  ( f `
 0 ) ) )  |->  <. [ h ]
(  ~=ph  `  J ) ,  [ ( ( x  e.  ( 0 [,] 1 )  |->  ( f `
 ( 1  -  x ) ) ) ( *p `  J
) ( h ( *p `  J ) f ) ) ] (  ~=ph  `  J )
>. )  e.  (
( J  pi1 
( f `  0
) ) GrpIso  ( J  pi1  ( f `  1 ) ) ) )
17 simprrl 772 . . . . . . 7  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( f ` 
0 )  =  A )
1817oveq2d 6321 . . . . . 6  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi1  ( f ` 
0 ) )  =  ( J  pi1  A ) )
19 pconpi1.p . . . . . 6  |-  P  =  ( J  pi1  A )
2018, 19syl6eqr 2488 . . . . 5  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi1  ( f ` 
0 ) )  =  P )
21 simprrr 773 . . . . . . 7  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( f ` 
1 )  =  B )
2221oveq2d 6321 . . . . . 6  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi1  ( f ` 
1 ) )  =  ( J  pi1  B ) )
23 pconpi1.q . . . . . 6  |-  Q  =  ( J  pi1  B )
2422, 23syl6eqr 2488 . . . . 5  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( J  pi1  ( f ` 
1 ) )  =  Q )
2520, 24oveq12d 6323 . . . 4  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  ( ( J  pi1  ( f `
 0 ) ) GrpIso 
( J  pi1 
( f `  1
) ) )  =  ( P GrpIso  Q ) )
2616, 25eleqtrd 2519 . . 3  |-  ( ( ( 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  pi1  ( 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 16885 . . 3  |-  ( ran  ( h  e.  U. ( Base `  ( J  pi1  ( 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  ~=g𝑔  Q )
2826, 27syl 17 . 2  |-  ( ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  /\  ( f  e.  ( II  Cn  J )  /\  ( ( f `
 0 )  =  A  /\  ( f `
 1 )  =  B ) ) )  ->  P  ~=g𝑔  Q )
292, 28rexlimddv 2928 1  |-  ( ( J  e. PCon  /\  A  e.  X  /\  B  e.  X )  ->  P  ~=g𝑔  Q )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1870   <.cop 4008   U.cuni 4222   class class class wbr 4426    |-> cmpt 4484   ran crn 4855   ` cfv 5601  (class class class)co 6305   [cec 7369   0cc0 9538   1c1 9539    - cmin 9859   [,]cicc 11638   Basecbs 15084   GrpIso cgim 16872    ~=g𝑔 cgic 16873   Topctop 19848  TopOnctopon 19849    Cn ccn 20171   IIcii 21803    ~=ph cphtpc 21893   *pcpco 21924    pi1 cpi1 21927  PConcpcon 29730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-ec 7373  df-qs 7377  df-map 7482  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ioo 11639  df-icc 11642  df-fz 11783  df-fzo 11914  df-seq 12211  df-exp 12270  df-hash 12513  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-qus 15366  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-grp 16624  df-mulg 16627  df-ghm 16832  df-gim 16874  df-gic 16875  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-cn 20174  df-cnp 20175  df-tx 20508  df-hmeo 20701  df-xms 21266  df-ms 21267  df-tms 21268  df-ii 21805  df-htpy 21894  df-phtpy 21895  df-phtpc 21916  df-pco 21929  df-om1 21930  df-pi1 21932  df-pcon 29732
This theorem is referenced by:  sconpi1  29750
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