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| Mirrors > Home > MPE Home > Th. List > nsgsubg | Structured version Visualization version GIF version | ||
| Description: A normal subgroup is a subgroup. (Contributed by Mario Carneiro, 18-Jan-2015.) |
| Ref | Expression |
|---|---|
| nsgsubg | ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | eqid 2737 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
| 3 | 1, 2 | isnsg 19173 | . 2 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) ↔ (𝑆 ∈ (SubGrp‘𝐺) ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)((𝑥(+g‘𝐺)𝑦) ∈ 𝑆 ↔ (𝑦(+g‘𝐺)𝑥) ∈ 𝑆))) |
| 4 | 3 | simplbi 497 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 SubGrpcsubg 19138 NrmSGrpcnsg 19139 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-subg 19141 df-nsg 19142 |
| This theorem is referenced by: nsgconj 19177 isnsg3 19178 trivnsgd 19190 eqgcpbl 19200 qusgrp 19204 quseccl 19205 qusadd 19206 qus0 19207 qusinv 19208 qussub 19209 ecqusaddcl 19211 ghmnsgima 19258 ghmnsgpreima 19259 conjnsg 19272 qusghm 19273 ghmqusnsglem1 19298 ghmqusnsglem2 19299 ghmqusnsg 19300 ghmquskerlem1 19301 ghmquskerlem2 19303 ghmquskerlem3 19304 ghmqusker 19305 sylow3lem4 19648 prmgrpsimpgd 20134 rhmqusnsg 21295 rngqiprngimf1lem 21304 rngqiprngimf1 21310 rngqiprngimfo 21311 rngqiprngfulem4 21324 rngqipring1 21326 clsnsg 24118 qustgpopn 24128 qustgphaus 24131 cyc3genpm 33172 qusker 33377 qus0g 33435 qusima 33436 qusrn 33437 nsgqus0 33438 nsgmgclem 33439 nsgmgc 33440 nsgqusf1olem1 33441 nsgqusf1olem2 33442 nsgqusf1olem3 33443 lmhmqusker 33445 rhmquskerlem 33453 qsnzr 33483 opprqusplusg 33517 opprqus0g 33518 qsdrngilem 33522 qsdrngi 33523 |
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