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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 19197 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝑁) |
6 | subgrcl 19162 | . . . . 5 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
7 | 2, 3, 4 | nmzsubg 19196 | . . . . 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 19180 | . . . 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 713 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐻)) |
13 | 2 | ssrab3 4092 | . . . . . 6 ⊢ 𝑁 ⊆ 𝑋 |
14 | 13 | sseli 3991 | . . . . 5 ⊢ (𝑤 ∈ 𝑁 → 𝑤 ∈ 𝑋) |
15 | 2 | nmzbi 19195 | . . . . 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 19161 | . . . . 5 ⊢ (𝑁 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻)) |
19 | 8, 18 | syl 17 | . . . 4 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑁 = (Base‘𝐻)) |
20 | 19 | raleqdv 3324 | . . . 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 3344 | . . 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 233 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → ∀𝑧 ∈ (Base‘𝐻)∀𝑤 ∈ (Base‘𝐻)((𝑧 + 𝑤) ∈ 𝑆 ↔ (𝑤 + 𝑧) ∈ 𝑆)) |
23 | eqid 2735 | . . 3 ⊢ (Base‘𝐻) = (Base‘𝐻) | |
24 | 3 | fvexi 6921 | . . . . 5 ⊢ 𝑋 ∈ V |
25 | 24, 13 | ssexi 5328 | . . . 4 ⊢ 𝑁 ∈ V |
26 | 9, 4 | ressplusg 17336 | . . . 4 ⊢ (𝑁 ∈ V → + = (+g‘𝐻)) |
27 | 25, 26 | ax-mp 5 | . . 3 ⊢ + = (+g‘𝐻) |
28 | 23, 27 | isnsg 19186 | . 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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 Vcvv 3478 ⊆ wss 3963 ‘cfv 6563 (class class class)co 7431 Basecbs 17245 ↾s cress 17274 +gcplusg 17298 Grpcgrp 18964 SubGrpcsubg 19151 NrmSGrpcnsg 19152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 |
This theorem is referenced by: sylow3lem4 19663 sylow3lem6 19665 |
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