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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 𝐸 defines a bijective correspondence between the set 𝑆 of subgroups of 𝐺 containing a normal subgroup 𝑁 and the subgroups of the quotient group 𝐺 / 𝑁. This theorem is sometimes called the correspondence theorem, or the fourth isomorphism theorem. (Contributed by Thierry Arnoux, 4-Aug-2024.) |
Ref | Expression |
---|---|
nsgqusf1o.b | ⊢ 𝐵 = (Base‘𝐺) |
nsgqusf1o.s | ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} |
nsgqusf1o.t | ⊢ 𝑇 = (SubGrp‘𝑄) |
nsgqusf1o.1 | ⊢ ≤ = (le‘(toInc‘𝑆)) |
nsgqusf1o.2 | ⊢ ≲ = (le‘(toInc‘𝑇)) |
nsgqusf1o.q | ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) |
nsgqusf1o.p | ⊢ ⊕ = (LSSum‘𝐺) |
nsgqusf1o.e | ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) |
nsgqusf1o.f | ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) |
nsgqusf1o.n | ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) |
Ref | Expression |
---|---|
nsgqusf1o | ⊢ (𝜑 → 𝐸:𝑆–1-1-onto→𝑇) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ ((toInc‘𝑆)MGalConn(toInc‘𝑇)) = ((toInc‘𝑆)MGalConn(toInc‘𝑇)) | |
2 | nsgqusf1o.s | . . . . . 6 ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} | |
3 | fvex 6902 | . . . . . 6 ⊢ (SubGrp‘𝐺) ∈ V | |
4 | 2, 3 | rabex2 5334 | . . . . 5 ⊢ 𝑆 ∈ V |
5 | eqid 2733 | . . . . . 6 ⊢ (toInc‘𝑆) = (toInc‘𝑆) | |
6 | 5 | ipobas 18481 | . . . . 5 ⊢ (𝑆 ∈ V → 𝑆 = (Base‘(toInc‘𝑆))) |
7 | 4, 6 | ax-mp 5 | . . . 4 ⊢ 𝑆 = (Base‘(toInc‘𝑆)) |
8 | nsgqusf1o.t | . . . . . 6 ⊢ 𝑇 = (SubGrp‘𝑄) | |
9 | 8 | fvexi 6903 | . . . . 5 ⊢ 𝑇 ∈ V |
10 | eqid 2733 | . . . . . 6 ⊢ (toInc‘𝑇) = (toInc‘𝑇) | |
11 | 10 | ipobas 18481 | . . . . 5 ⊢ (𝑇 ∈ V → 𝑇 = (Base‘(toInc‘𝑇))) |
12 | 9, 11 | ax-mp 5 | . . . 4 ⊢ 𝑇 = (Base‘(toInc‘𝑇)) |
13 | nsgqusf1o.1 | . . . 4 ⊢ ≤ = (le‘(toInc‘𝑆)) | |
14 | nsgqusf1o.2 | . . . 4 ⊢ ≲ = (le‘(toInc‘𝑇)) | |
15 | 5 | ipopos 18486 | . . . . 5 ⊢ (toInc‘𝑆) ∈ Poset |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (toInc‘𝑆) ∈ Poset) |
17 | 10 | ipopos 18486 | . . . . 5 ⊢ (toInc‘𝑇) ∈ Poset |
18 | 17 | a1i 11 | . . . 4 ⊢ (𝜑 → (toInc‘𝑇) ∈ Poset) |
19 | nsgqusf1o.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐺) | |
20 | nsgqusf1o.q | . . . . 5 ⊢ 𝑄 = (𝐺 /s (𝐺 ~QG 𝑁)) | |
21 | nsgqusf1o.p | . . . . 5 ⊢ ⊕ = (LSSum‘𝐺) | |
22 | nsgqusf1o.e | . . . . 5 ⊢ 𝐸 = (ℎ ∈ 𝑆 ↦ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁))) | |
23 | nsgqusf1o.f | . . . . 5 ⊢ 𝐹 = (𝑓 ∈ 𝑇 ↦ {𝑎 ∈ 𝐵 ∣ ({𝑎} ⊕ 𝑁) ∈ 𝑓}) | |
24 | nsgqusf1o.n | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (NrmSGrp‘𝐺)) | |
25 | 19, 2, 8, 1, 5, 10, 20, 21, 22, 23, 24 | nsgmgc 32512 | . . . 4 ⊢ (𝜑 → 𝐸((toInc‘𝑆)MGalConn(toInc‘𝑇))𝐹) |
26 | 1, 7, 12, 13, 14, 16, 18, 25 | mgcf1o 32161 | . . 3 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹) Isom ≤ , ≲ (ran 𝐹, ran 𝐸)) |
27 | isof1o 7317 | . . 3 ⊢ ((𝐸 ↾ ran 𝐹) Isom ≤ , ≲ (ran 𝐹, ran 𝐸) → (𝐸 ↾ ran 𝐹):ran 𝐹–1-1-onto→ran 𝐸) | |
28 | 26, 27 | syl 17 | . 2 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹):ran 𝐹–1-1-onto→ran 𝐸) |
29 | 19, 2, 8, 13, 14, 20, 21, 22, 23, 24 | nsgqusf1olem3 32515 | . . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝑆) |
30 | 29 | reseq2d 5980 | . . . 4 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹) = (𝐸 ↾ 𝑆)) |
31 | nfv 1918 | . . . . . 6 ⊢ Ⅎℎ𝜑 | |
32 | vex 3479 | . . . . . . . . 9 ⊢ ℎ ∈ V | |
33 | 32 | mptex 7222 | . . . . . . . 8 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V |
34 | 33 | rnex 7900 | . . . . . . 7 ⊢ ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V |
35 | 34 | a1i 11 | . . . . . 6 ⊢ ((𝜑 ∧ ℎ ∈ 𝑆) → ran (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V) |
36 | 31, 35, 22 | fnmptd 6689 | . . . . 5 ⊢ (𝜑 → 𝐸 Fn 𝑆) |
37 | fnresdm 6667 | . . . . 5 ⊢ (𝐸 Fn 𝑆 → (𝐸 ↾ 𝑆) = 𝐸) | |
38 | 36, 37 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐸 ↾ 𝑆) = 𝐸) |
39 | 30, 38 | eqtrd 2773 | . . 3 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹) = 𝐸) |
40 | 19, 2, 8, 13, 14, 20, 21, 22, 23, 24 | nsgqusf1olem2 32514 | . . 3 ⊢ (𝜑 → ran 𝐸 = 𝑇) |
41 | 39, 29, 40 | f1oeq123d 6825 | . 2 ⊢ (𝜑 → ((𝐸 ↾ ran 𝐹):ran 𝐹–1-1-onto→ran 𝐸 ↔ 𝐸:𝑆–1-1-onto→𝑇)) |
42 | 28, 41 | mpbid 231 | 1 ⊢ (𝜑 → 𝐸:𝑆–1-1-onto→𝑇) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3433 Vcvv 3475 ⊆ wss 3948 {csn 4628 ↦ cmpt 5231 ran crn 5677 ↾ cres 5678 Fn wfn 6536 –1-1-onto→wf1o 6540 ‘cfv 6541 Isom wiso 6542 (class class class)co 7406 Basecbs 17141 lecple 17201 /s cqus 17448 Posetcpo 18257 toInccipo 18477 SubGrpcsubg 18995 NrmSGrpcnsg 18996 ~QG cqg 18997 LSSumclsm 19497 MGalConncmgc 32137 |
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 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722 ax-cnex 11163 ax-resscn 11164 ax-1cn 11165 ax-icn 11166 ax-addcl 11167 ax-addrcl 11168 ax-mulcl 11169 ax-mulrcl 11170 ax-mulcom 11171 ax-addass 11172 ax-mulass 11173 ax-distr 11174 ax-i2m1 11175 ax-1ne0 11176 ax-1rid 11177 ax-rnegex 11178 ax-rrecex 11179 ax-cnre 11180 ax-pre-lttri 11181 ax-pre-lttrn 11182 ax-pre-ltadd 11183 ax-pre-mulgt0 11184 |
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 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6298 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-isom 6550 df-riota 7362 df-ov 7409 df-oprab 7410 df-mpo 7411 df-om 7853 df-1st 7972 df-2nd 7973 df-tpos 8208 df-frecs 8263 df-wrecs 8294 df-recs 8368 df-rdg 8407 df-1o 8463 df-er 8700 df-ec 8702 df-qs 8706 df-map 8819 df-en 8937 df-dom 8938 df-sdom 8939 df-fin 8940 df-sup 9434 df-inf 9435 df-pnf 11247 df-mnf 11248 df-xr 11249 df-ltxr 11250 df-le 11251 df-sub 11443 df-neg 11444 df-nn 12210 df-2 12272 df-3 12273 df-4 12274 df-5 12275 df-6 12276 df-7 12277 df-8 12278 df-9 12279 df-n0 12470 df-z 12556 df-dec 12675 df-uz 12820 df-fz 13482 df-struct 17077 df-sets 17094 df-slot 17112 df-ndx 17124 df-base 17142 df-ress 17171 df-plusg 17207 df-mulr 17208 df-sca 17210 df-vsca 17211 df-ip 17212 df-tset 17213 df-ple 17214 df-ocomp 17215 df-ds 17216 df-0g 17384 df-imas 17451 df-qus 17452 df-proset 18245 df-poset 18263 df-ipo 18478 df-mgm 18558 df-sgrp 18607 df-mnd 18623 df-submnd 18669 df-grp 18819 df-minusg 18820 df-subg 18998 df-nsg 18999 df-eqg 19000 df-ghm 19085 df-oppg 19205 df-lsm 19499 df-mnt 32138 df-mgc 32139 |
This theorem is referenced by: (None) |
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