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Mirrors > Home > MPE Home > Th. List > pi1xfrgim | Structured version Visualization version GIF version |
Description: The mapping πΊ between fundamental groups is an isomorphism. (Contributed by Mario Carneiro, 12-Feb-2015.) |
Ref | Expression |
---|---|
pi1xfr.p | β’ π = (π½ Ο1 (πΉβ0)) |
pi1xfr.q | β’ π = (π½ Ο1 (πΉβ1)) |
pi1xfr.b | β’ π΅ = (Baseβπ) |
pi1xfr.g | β’ πΊ = ran (π β βͺ π΅ β¦ β¨[π]( βphβπ½), [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½)β©) |
pi1xfr.j | β’ (π β π½ β (TopOnβπ)) |
pi1xfr.f | β’ (π β πΉ β (II Cn π½)) |
pi1xfr.i | β’ πΌ = (π₯ β (0[,]1) β¦ (πΉβ(1 β π₯))) |
Ref | Expression |
---|---|
pi1xfrgim | β’ (π β πΊ β (π GrpIso π)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pi1xfr.p | . . 3 β’ π = (π½ Ο1 (πΉβ0)) | |
2 | pi1xfr.q | . . 3 β’ π = (π½ Ο1 (πΉβ1)) | |
3 | pi1xfr.b | . . 3 β’ π΅ = (Baseβπ) | |
4 | pi1xfr.g | . . 3 β’ πΊ = ran (π β βͺ π΅ β¦ β¨[π]( βphβπ½), [(πΌ(*πβπ½)(π(*πβπ½)πΉ))]( βphβπ½)β©) | |
5 | pi1xfr.j | . . 3 β’ (π β π½ β (TopOnβπ)) | |
6 | pi1xfr.f | . . 3 β’ (π β πΉ β (II Cn π½)) | |
7 | pi1xfr.i | . . 3 β’ πΌ = (π₯ β (0[,]1) β¦ (πΉβ(1 β π₯))) | |
8 | 1, 2, 3, 4, 5, 6, 7 | pi1xfr 24803 | . 2 β’ (π β πΊ β (π GrpHom π)) |
9 | eqid 2731 | . . . 4 β’ ran (π¦ β βͺ (Baseβπ) β¦ β¨[π¦]( βphβπ½), [(πΉ(*πβπ½)(π¦(*πβπ½)πΌ))]( βphβπ½)β©) = ran (π¦ β βͺ (Baseβπ) β¦ β¨[π¦]( βphβπ½), [(πΉ(*πβπ½)(π¦(*πβπ½)πΌ))]( βphβπ½)β©) | |
10 | 1, 2, 3, 4, 5, 6, 7, 9 | pi1xfrcnv 24805 | . . 3 β’ (π β (β‘πΊ = ran (π¦ β βͺ (Baseβπ) β¦ β¨[π¦]( βphβπ½), [(πΉ(*πβπ½)(π¦(*πβπ½)πΌ))]( βphβπ½)β©) β§ β‘πΊ β (π GrpHom π))) |
11 | 10 | simprd 495 | . 2 β’ (π β β‘πΊ β (π GrpHom π)) |
12 | isgim2 19180 | . 2 β’ (πΊ β (π GrpIso π) β (πΊ β (π GrpHom π) β§ β‘πΊ β (π GrpHom π))) | |
13 | 8, 11, 12 | sylanbrc 582 | 1 β’ (π β πΊ β (π GrpIso π)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1540 β wcel 2105 β¨cop 4635 βͺ cuni 4909 β¦ cmpt 5232 β‘ccnv 5676 ran crn 5678 βcfv 6544 (class class class)co 7412 [cec 8704 0cc0 11113 1c1 11114 β cmin 11449 [,]cicc 13332 Basecbs 17149 GrpHom cghm 19128 GrpIso cgim 19172 TopOnctopon 22633 Cn ccn 22949 IIcii 24616 βphcphtpc 24716 *πcpco 24748 Ο1 cpi1 24751 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7728 ax-cnex 11169 ax-resscn 11170 ax-1cn 11171 ax-icn 11172 ax-addcl 11173 ax-addrcl 11174 ax-mulcl 11175 ax-mulrcl 11176 ax-mulcom 11177 ax-addass 11178 ax-mulass 11179 ax-distr 11180 ax-i2m1 11181 ax-1ne0 11182 ax-1rid 11183 ax-rnegex 11184 ax-rrecex 11185 ax-cnre 11186 ax-pre-lttri 11187 ax-pre-lttrn 11188 ax-pre-ltadd 11189 ax-pre-mulgt0 11190 ax-pre-sup 11191 ax-addf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7673 df-om 7859 df-1st 7978 df-2nd 7979 df-supp 8150 df-frecs 8269 df-wrecs 8300 df-recs 8374 df-rdg 8413 df-1o 8469 df-2o 8470 df-er 8706 df-ec 8708 df-qs 8712 df-map 8825 df-ixp 8895 df-en 8943 df-dom 8944 df-sdom 8945 df-fin 8946 df-fsupp 9365 df-fi 9409 df-sup 9440 df-inf 9441 df-oi 9508 df-card 9937 df-pnf 11255 df-mnf 11256 df-xr 11257 df-ltxr 11258 df-le 11259 df-sub 11451 df-neg 11452 df-div 11877 df-nn 12218 df-2 12280 df-3 12281 df-4 12282 df-5 12283 df-6 12284 df-7 12285 df-8 12286 df-9 12287 df-n0 12478 df-z 12564 df-dec 12683 df-uz 12828 df-q 12938 df-rp 12980 df-xneg 13097 df-xadd 13098 df-xmul 13099 df-ioo 13333 df-icc 13336 df-fz 13490 df-fzo 13633 df-seq 13972 df-exp 14033 df-hash 14296 df-cj 15051 df-re 15052 df-im 15053 df-sqrt 15187 df-abs 15188 df-struct 17085 df-sets 17102 df-slot 17120 df-ndx 17132 df-base 17150 df-ress 17179 df-plusg 17215 df-mulr 17216 df-starv 17217 df-sca 17218 df-vsca 17219 df-ip 17220 df-tset 17221 df-ple 17222 df-ds 17224 df-unif 17225 df-hom 17226 df-cco 17227 df-rest 17373 df-topn 17374 df-0g 17392 df-gsum 17393 df-topgen 17394 df-pt 17395 df-prds 17398 df-xrs 17453 df-qtop 17458 df-imas 17459 df-qus 17460 df-xps 17461 df-mre 17535 df-mrc 17536 df-acs 17538 df-mgm 18566 df-sgrp 18645 df-mnd 18661 df-submnd 18707 df-grp 18859 df-mulg 18988 df-ghm 19129 df-gim 19174 df-cntz 19223 df-cmn 19692 df-psmet 21137 df-xmet 21138 df-met 21139 df-bl 21140 df-mopn 21141 df-cnfld 21146 df-top 22617 df-topon 22634 df-topsp 22656 df-bases 22670 df-cld 22744 df-cn 22952 df-cnp 22953 df-tx 23287 df-hmeo 23480 df-xms 24047 df-ms 24048 df-tms 24049 df-ii 24618 df-htpy 24717 df-phtpy 24718 df-phtpc 24739 df-pco 24753 df-om1 24754 df-pi1 24756 |
This theorem is referenced by: pconnpi1 34523 |
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