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Mirrors > Home > MPE Home > Th. List > conjnsg | Structured version Visualization version GIF version |
Description: A normal subgroup is unchanged under conjugation. (Contributed by Mario Carneiro, 18-Jan-2015.) |
Ref | Expression |
---|---|
conjghm.x | ⊢ 𝑋 = (Base‘𝐺) |
conjghm.p | ⊢ + = (+g‘𝐺) |
conjghm.m | ⊢ − = (-g‘𝐺) |
conjsubg.f | ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) |
Ref | Expression |
---|---|
conjnsg | ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 = ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nsgsubg 18309 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) | |
2 | eqid 2821 | . . . . . 6 ⊢ {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} | |
3 | conjghm.x | . . . . . 6 ⊢ 𝑋 = (Base‘𝐺) | |
4 | conjghm.p | . . . . . 6 ⊢ + = (+g‘𝐺) | |
5 | 2, 3, 4 | isnsg4 18318 | . . . . 5 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋)) |
6 | 5 | simprbi 499 | . . . 4 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} = 𝑋) |
7 | 6 | eleq2d 2898 | . . 3 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → (𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)} ↔ 𝐴 ∈ 𝑋)) |
8 | 7 | biimpar 480 | . 2 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}) |
9 | conjghm.m | . . 3 ⊢ − = (-g‘𝐺) | |
10 | conjsubg.f | . . 3 ⊢ 𝐹 = (𝑥 ∈ 𝑆 ↦ ((𝐴 + 𝑥) − 𝐴)) | |
11 | 3, 4, 9, 10, 2 | conjnmz 18391 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝐴 ∈ {𝑦 ∈ 𝑋 ∣ ∀𝑧 ∈ 𝑋 ((𝑦 + 𝑧) ∈ 𝑆 ↔ (𝑧 + 𝑦) ∈ 𝑆)}) → 𝑆 = ran 𝐹) |
12 | 1, 8, 11 | syl2an2r 683 | 1 ⊢ ((𝑆 ∈ (NrmSGrp‘𝐺) ∧ 𝐴 ∈ 𝑋) → 𝑆 = ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 {crab 3142 ↦ cmpt 5145 ran crn 5555 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 -gcsg 18104 SubGrpcsubg 18272 NrmSGrpcnsg 18273 |
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 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-1st 7688 df-2nd 7689 df-0g 16714 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-grp 18105 df-minusg 18106 df-sbg 18107 df-subg 18275 df-nsg 18276 |
This theorem is referenced by: sylow3lem6 18756 |
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