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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 6856 | . . . . . 6 β’ (SubGrpβπΊ) β V | |
4 | 2, 3 | rabex2 5292 | . . . . 5 β’ π β V |
5 | eqid 2733 | . . . . . 6 β’ (toIncβπ) = (toIncβπ) | |
6 | 5 | ipobas 18425 | . . . . 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 6857 | . . . . 5 β’ π β V |
10 | eqid 2733 | . . . . . 6 β’ (toIncβπ) = (toIncβπ) | |
11 | 10 | ipobas 18425 | . . . . 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 18430 | . . . . 5 β’ (toIncβπ) β Poset |
16 | 15 | a1i 11 | . . . 4 β’ (π β (toIncβπ) β Poset) |
17 | 10 | ipopos 18430 | . . . . 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 32238 | . . . 4 β’ (π β πΈ((toIncβπ)MGalConn(toIncβπ))πΉ) |
26 | 1, 7, 12, 13, 14, 16, 18, 25 | mgcf1o 31912 | . . 3 β’ (π β (πΈ βΎ ran πΉ) Isom β€ , β² (ran πΉ, ran πΈ)) |
27 | isof1o 7269 | . . 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 32241 | . . . . 5 β’ (π β ran πΉ = π) |
30 | 29 | reseq2d 5938 | . . . 4 β’ (π β (πΈ βΎ ran πΉ) = (πΈ βΎ π)) |
31 | nfv 1918 | . . . . . 6 β’ β²βπ | |
32 | vex 3448 | . . . . . . . . 9 β’ β β V | |
33 | 32 | mptex 7174 | . . . . . . . 8 β’ (π₯ β β β¦ ({π₯} β π)) β V |
34 | 33 | rnex 7850 | . . . . . . 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 6643 | . . . . 5 β’ (π β πΈ Fn π) |
37 | fnresdm 6621 | . . . . 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 32240 | . . 3 β’ (π β ran πΈ = π) |
41 | 39, 29, 40 | f1oeq123d 6779 | . 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 3406 Vcvv 3444 β wss 3911 {csn 4587 β¦ cmpt 5189 ran crn 5635 βΎ cres 5636 Fn wfn 6492 β1-1-ontoβwf1o 6496 βcfv 6497 Isom wiso 6498 (class class class)co 7358 Basecbs 17088 lecple 17145 /s cqus 17392 Posetcpo 18201 toInccipo 18421 SubGrpcsubg 18927 NrmSGrpcnsg 18928 ~QG cqg 18929 LSSumclsm 19421 MGalConncmgc 31888 |
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 5243 ax-sep 5257 ax-nul 5264 ax-pow 5321 ax-pr 5385 ax-un 7673 ax-cnex 11112 ax-resscn 11113 ax-1cn 11114 ax-icn 11115 ax-addcl 11116 ax-addrcl 11117 ax-mulcl 11118 ax-mulrcl 11119 ax-mulcom 11120 ax-addass 11121 ax-mulass 11122 ax-distr 11123 ax-i2m1 11124 ax-1ne0 11125 ax-1rid 11126 ax-rnegex 11127 ax-rrecex 11128 ax-cnre 11129 ax-pre-lttri 11130 ax-pre-lttrn 11131 ax-pre-ltadd 11132 ax-pre-mulgt0 11133 |
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 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3741 df-csb 3857 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3930 df-nul 4284 df-if 4488 df-pw 4563 df-sn 4588 df-pr 4590 df-tp 4592 df-op 4594 df-uni 4867 df-iun 4957 df-br 5107 df-opab 5169 df-mpt 5190 df-tr 5224 df-id 5532 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5589 df-we 5591 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6254 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 df-riota 7314 df-ov 7361 df-oprab 7362 df-mpo 7363 df-om 7804 df-1st 7922 df-2nd 7923 df-tpos 8158 df-frecs 8213 df-wrecs 8244 df-recs 8318 df-rdg 8357 df-1o 8413 df-er 8651 df-ec 8653 df-qs 8657 df-map 8770 df-en 8887 df-dom 8888 df-sdom 8889 df-fin 8890 df-sup 9383 df-inf 9384 df-pnf 11196 df-mnf 11197 df-xr 11198 df-ltxr 11199 df-le 11200 df-sub 11392 df-neg 11393 df-nn 12159 df-2 12221 df-3 12222 df-4 12223 df-5 12224 df-6 12225 df-7 12226 df-8 12227 df-9 12228 df-n0 12419 df-z 12505 df-dec 12624 df-uz 12769 df-fz 13431 df-struct 17024 df-sets 17041 df-slot 17059 df-ndx 17071 df-base 17089 df-ress 17118 df-plusg 17151 df-mulr 17152 df-sca 17154 df-vsca 17155 df-ip 17156 df-tset 17157 df-ple 17158 df-ocomp 17159 df-ds 17160 df-0g 17328 df-imas 17395 df-qus 17396 df-proset 18189 df-poset 18207 df-ipo 18422 df-mgm 18502 df-sgrp 18551 df-mnd 18562 df-submnd 18607 df-grp 18756 df-minusg 18757 df-subg 18930 df-nsg 18931 df-eqg 18932 df-ghm 19011 df-oppg 19129 df-lsm 19423 df-mnt 31889 df-mgc 31890 |
This theorem is referenced by: (None) |
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