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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 2736 | . . . 4 ⊢ ((toInc‘𝑆)MGalConn(toInc‘𝑇)) = ((toInc‘𝑆)MGalConn(toInc‘𝑇)) | |
| 2 | nsgqusf1o.s | . . . . . 6 ⊢ 𝑆 = {ℎ ∈ (SubGrp‘𝐺) ∣ 𝑁 ⊆ ℎ} | |
| 3 | fvex 6918 | . . . . . 6 ⊢ (SubGrp‘𝐺) ∈ V | |
| 4 | 2, 3 | rabex2 5340 | . . . . 5 ⊢ 𝑆 ∈ V | 
| 5 | eqid 2736 | . . . . . 6 ⊢ (toInc‘𝑆) = (toInc‘𝑆) | |
| 6 | 5 | ipobas 18577 | . . . . 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 6919 | . . . . 5 ⊢ 𝑇 ∈ V | 
| 10 | eqid 2736 | . . . . . 6 ⊢ (toInc‘𝑇) = (toInc‘𝑇) | |
| 11 | 10 | ipobas 18577 | . . . . 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 18582 | . . . . 5 ⊢ (toInc‘𝑆) ∈ Poset | 
| 16 | 15 | a1i 11 | . . . 4 ⊢ (𝜑 → (toInc‘𝑆) ∈ Poset) | 
| 17 | 10 | ipopos 18582 | . . . . 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 33441 | . . . 4 ⊢ (𝜑 → 𝐸((toInc‘𝑆)MGalConn(toInc‘𝑇))𝐹) | 
| 26 | 1, 7, 12, 13, 14, 16, 18, 25 | mgcf1o 32994 | . . 3 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹) Isom ≤ , ≲ (ran 𝐹, ran 𝐸)) | 
| 27 | isof1o 7344 | . . 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 33444 | . . . . 5 ⊢ (𝜑 → ran 𝐹 = 𝑆) | 
| 30 | 29 | reseq2d 5996 | . . . 4 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹) = (𝐸 ↾ 𝑆)) | 
| 31 | nfv 1913 | . . . . . 6 ⊢ Ⅎℎ𝜑 | |
| 32 | vex 3483 | . . . . . . . . 9 ⊢ ℎ ∈ V | |
| 33 | 32 | mptex 7244 | . . . . . . . 8 ⊢ (𝑥 ∈ ℎ ↦ ({𝑥} ⊕ 𝑁)) ∈ V | 
| 34 | 33 | rnex 7933 | . . . . . . 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 6708 | . . . . 5 ⊢ (𝜑 → 𝐸 Fn 𝑆) | 
| 37 | fnresdm 6686 | . . . . 5 ⊢ (𝐸 Fn 𝑆 → (𝐸 ↾ 𝑆) = 𝐸) | |
| 38 | 36, 37 | syl 17 | . . . 4 ⊢ (𝜑 → (𝐸 ↾ 𝑆) = 𝐸) | 
| 39 | 30, 38 | eqtrd 2776 | . . 3 ⊢ (𝜑 → (𝐸 ↾ ran 𝐹) = 𝐸) | 
| 40 | 19, 2, 8, 13, 14, 20, 21, 22, 23, 24 | nsgqusf1olem2 33443 | . . 3 ⊢ (𝜑 → ran 𝐸 = 𝑇) | 
| 41 | 39, 29, 40 | f1oeq123d 6841 | . 2 ⊢ (𝜑 → ((𝐸 ↾ ran 𝐹):ran 𝐹–1-1-onto→ran 𝐸 ↔ 𝐸:𝑆–1-1-onto→𝑇)) | 
| 42 | 28, 41 | mpbid 232 | 1 ⊢ (𝜑 → 𝐸:𝑆–1-1-onto→𝑇) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 ∈ wcel 2107 {crab 3435 Vcvv 3479 ⊆ wss 3950 {csn 4625 ↦ cmpt 5224 ran crn 5685 ↾ cres 5686 Fn wfn 6555 –1-1-onto→wf1o 6559 ‘cfv 6560 Isom wiso 6561 (class class class)co 7432 Basecbs 17248 lecple 17305 /s cqus 17551 Posetcpo 18354 toInccipo 18573 SubGrpcsubg 19139 NrmSGrpcnsg 19140 ~QG cqg 19141 LSSumclsm 19653 MGalConncmgc 32970 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-tpos 8252 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-er 8746 df-ec 8748 df-qs 8752 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-sup 9483 df-inf 9484 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-fz 13549 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ocomp 17319 df-ds 17320 df-0g 17487 df-imas 17554 df-qus 17555 df-proset 18341 df-poset 18360 df-ipo 18574 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-submnd 18798 df-grp 18955 df-minusg 18956 df-subg 19142 df-nsg 19143 df-eqg 19144 df-ghm 19232 df-oppg 19365 df-lsm 19655 df-mnt 32971 df-mgc 32972 | 
| This theorem is referenced by: (None) | 
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