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Mirrors > Home > MPE Home > Th. List > nmznsg | Structured version Visualization version GIF version |
Description: Any subgroup is a normal subgroup of its normalizer. (Contributed by Mario Carneiro, 19-Jan-2015.) |
Ref | Expression |
---|---|
elnmz.1 | ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} |
nmzsubg.2 | ⊢ 𝑋 = (Base‘𝐺) |
nmzsubg.3 | ⊢ + = (+g‘𝐺) |
nmznsg.4 | ⊢ 𝐻 = (𝐺 ↾s 𝑁) |
Ref | Expression |
---|---|
nmznsg | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 22 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | elnmz.1 | . . . 4 ⊢ 𝑁 = {𝑥 ∈ 𝑋 ∣ ∀𝑦 ∈ 𝑋 ((𝑥 + 𝑦) ∈ 𝑆 ↔ (𝑦 + 𝑥) ∈ 𝑆)} | |
3 | nmzsubg.2 | . . . 4 ⊢ 𝑋 = (Base‘𝐺) | |
4 | nmzsubg.3 | . . . 4 ⊢ + = (+g‘𝐺) | |
5 | 2, 3, 4 | ssnmz 19040 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁) |
6 | subgrcl 19005 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
7 | 2, 3, 4 | nmzsubg 19039 | . . . . 5 ⊢ (𝐺 ∈ Grp → 𝑁 ∈ (SubGrp‘𝐺)) |
8 | 6, 7 | syl 17 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 ∈ (SubGrp‘𝐺)) |
9 | nmznsg.4 | . . . . 5 ⊢ 𝐻 = (𝐺 ↾s 𝑁) | |
10 | 9 | subsubg 19023 | . . . 4 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑁))) |
11 | 8, 10 | syl 17 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝑆 ∈ (SubGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑆 ⊆ 𝑁))) |
12 | 1, 5, 11 | mpbir2and 711 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻)) |
13 | 2 | ssrab3 4079 | . . . . . 6 ⊢ 𝑁 ⊆ 𝑋 |
14 | 13 | sseli 3977 | . . . . 5 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋) |
15 | 2 | nmzbi 19038 | . . . . 5 ⊢ ((𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑋) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
16 | 14, 15 | sylan2 593 | . . . 4 ⊢ ((𝑧 ∈ 𝑁 ∧ 𝑤 ∈ 𝑁) → ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
17 | 16 | rgen2 3197 | . . 3 ⊢ ∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) |
18 | 9 | subgbas 19004 | . . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻)) |
19 | 8, 18 | syl 17 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻)) |
20 | 19 | raleqdv 3325 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))) |
21 | 19, 20 | raleqbidv 3342 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (∀𝑧 ∈ 𝑁 ∀𝑤 ∈ 𝑁 ((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆) ↔ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))) |
22 | 17, 21 | mpbii 232 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
23 | eqid 2732 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
24 | 3 | fvexi 6902 | . . . . 5 ⊢ 𝑋 ∈ V |
25 | 24, 13 | ssexi 5321 | . . . 4 ⊢ 𝑁 ∈ V |
26 | 9, 4 | ressplusg 17231 | . . . 4 ⊢ (𝑁 ∈ V → + = (+g‘𝐻)) |
27 | 25, 26 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝐻) |
28 | 23, 27 | isnsg 19029 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐻) ↔ (𝑆 ∈ (SubGrp‘𝐻) ∧ ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆))) |
29 | 12, 22, 28 | sylanbrc 583 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (NrmSGrp‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 {crab 3432 Vcvv 3474 ⊆ wss 3947 ‘cfv 6540 (class class class)co 7405 Basecbs 17140 ↾s cress 17169 +gcplusg 17193 Grpcgrp 18815 SubGrpcsubg 18994 NrmSGrpcnsg 18995 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-subg 18997 df-nsg 18998 |
This theorem is referenced by: sylow3lem4 19492 sylow3lem6 19494 |
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