Theorem pconpi1 27126

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 ~=ph𝑔 Q )

Proof of Theorem pconpi1

Dummy variables f h x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		pconpi1.x	. . 3 |- X = U. J
2	1	pconcn 27113	. 2 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )
3		eqid 2443	. . . . 5 |- ( J pi1 ( f ` 0 ) ) = ( J pi1 ( f ` 0 ) )
4		eqid 2443	. . . . 5 |- ( J pi1 ( f ` 1 ) ) = ( J pi1 ( f ` 1 ) )
5		eqid 2443	. . . . 5 |- ( Base ` ( J pi1 ( f ` 0 ) ) ) = ( Base ` ( J pi1 ( f ` 0 ) ) )
6		eqid 2443	. . . . 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 991	. . . . . . 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 27114	. . . . . . 7 |- ( J e. PCon -> J e. Top )
9	7, 8	syl 16	. . . . . 6 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> J e. Top )
10	1	toptopon 18538	. . . . . 6 |- ( J e. Top <-> J e. (TopOn ` X ) )
11	9, 10	sylib 196	. . . . 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 755	. . . . 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 6099	. . . . . . 7 |- ( x = y -> ( 1 - x ) = ( 1 - y ) )
14	13	fveq2d 5695	. . . . . 6 |- ( x = y -> ( f ` ( 1 - x ) ) = ( f ` ( 1 - y ) ) )
15	14	cbvmptv 4383	. . . . 5 |- ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( f ` ( 1 - y ) ) )
16	3, 4, 5, 6, 11, 12, 15	pi1xfrgim 20630	. . . 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 763	. . . . . . 7 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( f ` 0 ) = A )
18	17	oveq2d 6107	. . . . . 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 )
20	18, 19	syl6eqr 2493	. . . . 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 764	. . . . . . 7 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> ( f ` 1 ) = B )
22	21	oveq2d 6107	. . . . . 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 )
24	22, 23	syl6eqr 2493	. . . . 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 )
25	20, 24	oveq12d 6109	. . . 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 ) )
26	16, 25	eleqtrd 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 15798	. . 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 ~=ph𝑔 Q )
28	26, 27	syl 16	. 2 |- ( ( ( J e. PCon /\ A e. X /\ B e. X ) /\ ( f e. ( II Cn J ) /\ ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) ) ) -> P ~=ph𝑔 Q )
29	2, 28	rexlimddv 2845	1 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> P ~=ph𝑔 Q )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 965   = wceq 1369   e. wcel 1756   <.cop 3883   U.cuni 4091   class class class wbr 4292   e. cmpt 4350   ran crn 4841   ` cfv 5418  (class class class)co 6091   [cec 7099   0cc0 9282   1c1 9283   - cmin 9595   [,]cicc 11303   Basecbs 14174   GrpIso cgim 15785   ~=ph_𝑔 cgic 15786   Topctop 18498  TopOnctopon 18499   Cn ccn 18828   IIcii 20451   ~=ph cphtpc 20541   *pcpco 20572   pi1 cpi1 20575  PConcpcon 27108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360  ax-addf 9361  ax-mulf 9362

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-se 4680  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-isom 5427  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-of 6320  df-om 6477  df-1st 6577  df-2nd 6578  df-supp 6691  df-recs 6832  df-rdg 6866  df-1o 6920  df-2o 6921  df-oadd 6924  df-er 7101  df-ec 7103  df-qs 7107  df-map 7216  df-ixp 7264  df-en 7311  df-dom 7312  df-sdom 7313  df-fin 7314  df-fsupp 7621  df-fi 7661  df-sup 7691  df-oi 7724  df-card 8109  df-cda 8337  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-n0 10580  df-z 10647  df-dec 10756  df-uz 10862  df-q 10954  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-ioo 11304  df-icc 11307  df-fz 11438  df-fzo 11549  df-seq 11807  df-exp 11866  df-hash 12104  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-struct 14176  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-ress 14181  df-plusg 14251  df-mulr 14252  df-starv 14253  df-sca 14254  df-vsca 14255  df-ip 14256  df-tset 14257  df-ple 14258  df-ds 14260  df-unif 14261  df-hom 14262  df-cco 14263  df-rest 14361  df-topn 14362  df-0g 14380  df-gsum 14381  df-topgen 14382  df-pt 14383  df-prds 14386  df-xrs 14440  df-qtop 14445  df-imas 14446  df-divs 14447  df-xps 14448  df-mre 14524  df-mrc 14525  df-acs 14527  df-mnd 15415  df-submnd 15465  df-grp 15545  df-mulg 15548  df-ghm 15745  df-gim 15787  df-gic 15788  df-cntz 15835  df-cmn 16279  df-psmet 17809  df-xmet 17810  df-met 17811  df-bl 17812  df-mopn 17813  df-cnfld 17819  df-top 18503  df-bases 18505  df-topon 18506  df-topsp 18507  df-cld 18623  df-cn 18831  df-cnp 18832  df-tx 19135  df-hmeo 19328  df-xms 19895  df-ms 19896  df-tms 19897  df-ii 20453  df-htpy 20542  df-phtpy 20543  df-phtpc 20564  df-pco 20577  df-om1 20578  df-pi1 20580  df-pcon 27110

This theorem is referenced by:  sconpi1  27128