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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nsgqusf1o | Structured version Visualization version GIF version |
Description: The canonical projection homomorphism πΈ defines a bijective correspondence between the set π of subgroups of πΊ containing a normal subgroup π and the subgroups of the quotient group πΊ / π. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.) |
Ref | Expression |
---|---|
nsgqusf1o.b | β’ π΅ = (BaseβπΊ) |
nsgqusf1o.s | β’ π = {β β (SubGrpβπΊ) β£ π β β} |
nsgqusf1o.t | β’ π = (SubGrpβπ) |
nsgqusf1o.1 | β’ β€ = (leβ(toIncβπ)) |
nsgqusf1o.2 | β’ β² = (leβ(toIncβπ)) |
nsgqusf1o.q | β’ π = (πΊ /s (πΊ ~QG π)) |
nsgqusf1o.p | β’ β = (LSSumβπΊ) |
nsgqusf1o.e | β’ πΈ = (β β π β¦ ran (π₯ β β β¦ ({π₯} β π))) |
nsgqusf1o.f | β’ πΉ = (π β π β¦ {π β π΅ β£ ({π} β π) β π}) |
nsgqusf1o.n | β’ (π β π β (NrmSGrpβπΊ)) |
Ref | Expression |
---|---|
nsgqusf1o | β’ (π β πΈ:πβ1-1-ontoβπ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 β’ ((toIncβπ)MGalConn(toIncβπ)) = ((toIncβπ)MGalConn(toIncβπ)) | |
2 | nsgqusf1o.s | . . . . . 6 β’ π = {β β (SubGrpβπΊ) β£ π β β} | |
3 | fvex 6905 | . . . . . 6 β’ (SubGrpβπΊ) β V | |
4 | 2, 3 | rabex2 5335 | . . . . 5 β’ π β V |
5 | eqid 2733 | . . . . . 6 β’ (toIncβπ) = (toIncβπ) | |
6 | 5 | ipobas 18484 | . . . . 5 β’ (π β V β π = (Baseβ(toIncβπ))) |
7 | 4, 6 | ax-mp 5 | . . . 4 β’ π = (Baseβ(toIncβπ)) |
8 | nsgqusf1o.t | . . . . . 6 β’ π = (SubGrpβπ) | |
9 | 8 | fvexi 6906 | . . . . 5 β’ π β V |
10 | eqid 2733 | . . . . . 6 β’ (toIncβπ) = (toIncβπ) | |
11 | 10 | ipobas 18484 | . . . . 5 β’ (π β V β π = (Baseβ(toIncβπ))) |
12 | 9, 11 | ax-mp 5 | . . . 4 β’ π = (Baseβ(toIncβπ)) |
13 | nsgqusf1o.1 | . . . 4 β’ β€ = (leβ(toIncβπ)) | |
14 | nsgqusf1o.2 | . . . 4 β’ β² = (leβ(toIncβπ)) | |
15 | 5 | ipopos 18489 | . . . . 5 β’ (toIncβπ) β Poset |
16 | 15 | a1i 11 | . . . 4 β’ (π β (toIncβπ) β Poset) |
17 | 10 | ipopos 18489 | . . . . 5 β’ (toIncβπ) β Poset |
18 | 17 | a1i 11 | . . . 4 β’ (π β (toIncβπ) β Poset) |
19 | nsgqusf1o.b | . . . . 5 β’ π΅ = (BaseβπΊ) | |
20 | nsgqusf1o.q | . . . . 5 β’ π = (πΊ /s (πΊ ~QG π)) | |
21 | nsgqusf1o.p | . . . . 5 β’ β = (LSSumβπΊ) | |
22 | nsgqusf1o.e | . . . . 5 β’ πΈ = (β β π β¦ ran (π₯ β β β¦ ({π₯} β π))) | |
23 | nsgqusf1o.f | . . . . 5 β’ πΉ = (π β π β¦ {π β π΅ β£ ({π} β π) β π}) | |
24 | nsgqusf1o.n | . . . . 5 β’ (π β π β (NrmSGrpβπΊ)) | |
25 | 19, 2, 8, 1, 5, 10, 20, 21, 22, 23, 24 | nsgmgc 32523 | . . . 4 β’ (π β πΈ((toIncβπ)MGalConn(toIncβπ))πΉ) |
26 | 1, 7, 12, 13, 14, 16, 18, 25 | mgcf1o 32173 | . . 3 β’ (π β (πΈ βΎ ran πΉ) Isom β€ , β² (ran πΉ, ran πΈ)) |
27 | isof1o 7320 | . . 3 β’ ((πΈ βΎ ran πΉ) Isom β€ , β² (ran πΉ, ran πΈ) β (πΈ βΎ ran πΉ):ran πΉβ1-1-ontoβran πΈ) | |
28 | 26, 27 | syl 17 | . 2 β’ (π β (πΈ βΎ ran πΉ):ran πΉβ1-1-ontoβran πΈ) |
29 | 19, 2, 8, 13, 14, 20, 21, 22, 23, 24 | nsgqusf1olem3 32526 | . . . . 5 β’ (π β ran πΉ = π) |
30 | 29 | reseq2d 5982 | . . . 4 β’ (π β (πΈ βΎ ran πΉ) = (πΈ βΎ π)) |
31 | nfv 1918 | . . . . . 6 β’ β²βπ | |
32 | vex 3479 | . . . . . . . . 9 β’ β β V | |
33 | 32 | mptex 7225 | . . . . . . . 8 β’ (π₯ β β β¦ ({π₯} β π)) β V |
34 | 33 | rnex 7903 | . . . . . . 7 β’ ran (π₯ β β β¦ ({π₯} β π)) β V |
35 | 34 | a1i 11 | . . . . . 6 β’ ((π β§ β β π) β ran (π₯ β β β¦ ({π₯} β π)) β V) |
36 | 31, 35, 22 | fnmptd 6692 | . . . . 5 β’ (π β πΈ Fn π) |
37 | fnresdm 6670 | . . . . 5 β’ (πΈ Fn π β (πΈ βΎ π) = πΈ) | |
38 | 36, 37 | syl 17 | . . . 4 β’ (π β (πΈ βΎ π) = πΈ) |
39 | 30, 38 | eqtrd 2773 | . . 3 β’ (π β (πΈ βΎ ran πΉ) = πΈ) |
40 | 19, 2, 8, 13, 14, 20, 21, 22, 23, 24 | nsgqusf1olem2 32525 | . . 3 β’ (π β ran πΈ = π) |
41 | 39, 29, 40 | f1oeq123d 6828 | . 2 β’ (π β ((πΈ βΎ ran πΉ):ran πΉβ1-1-ontoβran πΈ β πΈ:πβ1-1-ontoβπ)) |
42 | 28, 41 | mpbid 231 | 1 β’ (π β πΈ:πβ1-1-ontoβπ) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 397 = wceq 1542 β wcel 2107 {crab 3433 Vcvv 3475 β wss 3949 {csn 4629 β¦ cmpt 5232 ran crn 5678 βΎ cres 5679 Fn wfn 6539 β1-1-ontoβwf1o 6543 βcfv 6544 Isom wiso 6545 (class class class)co 7409 Basecbs 17144 lecple 17204 /s cqus 17451 Posetcpo 18260 toInccipo 18480 SubGrpcsubg 19000 NrmSGrpcnsg 19001 ~QG cqg 19002 LSSumclsm 19502 MGalConncmgc 32149 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 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-iun 5000 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-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 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-z 12559 df-dec 12678 df-uz 12823 df-fz 13485 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ocomp 17218 df-ds 17219 df-0g 17387 df-imas 17454 df-qus 17455 df-proset 18248 df-poset 18266 df-ipo 18481 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-submnd 18672 df-grp 18822 df-minusg 18823 df-subg 19003 df-nsg 19004 df-eqg 19005 df-ghm 19090 df-oppg 19210 df-lsm 19504 df-mnt 32150 df-mgc 32151 |
This theorem is referenced by: (None) |
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