Theorem pconpi1 28308

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 28295	. 2 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> E. f e. ( II Cn J ) ( ( f ` 0 ) = A /\ ( f ` 1 ) = B ) )
3		eqid 2460	. . . . 5 |- ( J pi1 ( f ` 0 ) ) = ( J pi1 ( f ` 0 ) )
4		eqid 2460	. . . . 5 |- ( J pi1 ( f ` 1 ) ) = ( J pi1 ( f ` 1 ) )
5		eqid 2460	. . . . 5 |- ( Base ` ( J pi1 ( f ` 0 ) ) ) = ( Base ` ( J pi1 ( f ` 0 ) ) )
6		eqid 2460	. . . . 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 994	. . . . . . 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 28296	. . . . . . 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 19194	. . . . . 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 6283	. . . . . . 7 |- ( x = y -> ( 1 - x ) = ( 1 - y ) )
14	13	fveq2d 5861	. . . . . 6 |- ( x = y -> ( f ` ( 1 - x ) ) = ( f ` ( 1 - y ) ) )
15	14	cbvmptv 4531	. . . . 5 |- ( x e. ( 0 [,] 1 ) |-> ( f ` ( 1 - x ) ) ) = ( y e. ( 0 [,] 1 ) |-> ( f ` ( 1 - y ) ) )
16	3, 4, 5, 6, 11, 12, 15	pi1xfrgim 21286	. . . 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 6291	. . . . . 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 2519	. . . . 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 6291	. . . . . 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 2519	. . . . 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 6293	. . . 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 2550	. . 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 16106	. . 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 2952	1 |- ( ( J e. PCon /\ A e. X /\ B e. X ) -> P ~=ph𝑔 Q )

Syntax hints:   -> wi 4   /\ wa 369   /\ w3a 968   = wceq 1374   e. wcel 1762   <.cop 4026   U.cuni 4238   class class class wbr 4440   |-> cmpt 4498   ran crn 4993   ` cfv 5579  (class class class)co 6275   [cec 7299   0cc0 9481   1c1 9482   - cmin 9794   [,]cicc 11521   Basecbs 14479   GrpIso cgim 16093   ~=ph_𝑔 cgic 16094   Topctop 19154  TopOnctopon 19155   Cn ccn 19484   IIcii 21107   ~=ph cphtpc 21197   *pcpco 21228   pi1 cpi1 21231  PConcpcon 28290

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-inf2 8047  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-iun 4320  df-iin 4321  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-se 4832  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-isom 5588  df-riota 6236  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-of 6515  df-om 6672  df-1st 6774  df-2nd 6775  df-supp 6892  df-recs 7032  df-rdg 7066  df-1o 7120  df-2o 7121  df-oadd 7124  df-er 7301  df-ec 7303  df-qs 7307  df-map 7412  df-ixp 7460  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-fsupp 7819  df-fi 7860  df-sup 7890  df-oi 7924  df-card 8309  df-cda 8537  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9796  df-neg 9797  df-div 10196  df-nn 10526  df-2 10583  df-3 10584  df-4 10585  df-5 10586  df-6 10587  df-7 10588  df-8 10589  df-9 10590  df-10 10591  df-n0 10785  df-z 10854  df-dec 10966  df-uz 11072  df-q 11172  df-rp 11210  df-xneg 11307  df-xadd 11308  df-xmul 11309  df-ioo 11522  df-icc 11525  df-fz 11662  df-fzo 11782  df-seq 12064  df-exp 12123  df-hash 12361  df-cj 12882  df-re 12883  df-im 12884  df-sqr 13018  df-abs 13019  df-struct 14481  df-ndx 14482  df-slot 14483  df-base 14484  df-sets 14485  df-ress 14486  df-plusg 14557  df-mulr 14558  df-starv 14559  df-sca 14560  df-vsca 14561  df-ip 14562  df-tset 14563  df-ple 14564  df-ds 14566  df-unif 14567  df-hom 14568  df-cco 14569  df-rest 14667  df-topn 14668  df-0g 14686  df-gsum 14687  df-topgen 14688  df-pt 14689  df-prds 14692  df-xrs 14746  df-qtop 14751  df-imas 14752  df-divs 14753  df-xps 14754  df-mre 14830  df-mrc 14831  df-acs 14833  df-mnd 15721  df-submnd 15771  df-grp 15851  df-mulg 15854  df-ghm 16053  df-gim 16095  df-gic 16096  df-cntz 16143  df-cmn 16589  df-psmet 18175  df-xmet 18176  df-met 18177  df-bl 18178  df-mopn 18179  df-cnfld 18185  df-top 19159  df-bases 19161  df-topon 19162  df-topsp 19163  df-cld 19279  df-cn 19487  df-cnp 19488  df-tx 19791  df-hmeo 19984  df-xms 20551  df-ms 20552  df-tms 20553  df-ii 21109  df-htpy 21198  df-phtpy 21199  df-phtpc 21220  df-pco 21233  df-om1 21234  df-pi1 21236  df-pcon 28292

This theorem is referenced by:  sconpi1  28310