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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 2728 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2728 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | isnsg 19110 | . 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 205 ∈ wcel 2099 ∀wral 3058 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 +gcplusg 17233 SubGrpcsubg 19075 NrmSGrpcnsg 19076 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fv 6556 df-ov 7423 df-subg 19078 df-nsg 19079 |
This theorem is referenced by: nsgconj 19114 isnsg3 19115 trivnsgd 19127 eqgcpbl 19137 qusgrp 19141 quseccl 19142 qusadd 19143 qus0 19144 qusinv 19145 qussub 19146 ecqusaddcl 19148 ghmnsgima 19194 ghmnsgpreima 19195 conjnsg 19208 qusghm 19209 ghmquskerlem1 19234 ghmquskerlem2 19236 ghmquskerlem3 19237 ghmqusker 19238 sylow3lem4 19585 prmgrpsimpgd 20071 rngqiprngimf1lem 21184 rngqiprngimf1 21190 rngqiprngimfo 21191 rngqiprngfulem4 21204 rngqipring1 21206 clsnsg 24027 qustgpopn 24037 qustgphaus 24040 cyc3genpm 32886 qusker 33074 qus0g 33130 qusima 33131 qusrn 33132 nsgqus0 33133 nsgmgclem 33134 nsgmgc 33135 nsgqusf1olem1 33136 nsgqusf1olem2 33137 nsgqusf1olem3 33138 lmhmqusker 33140 ghmqusnsglem1 33142 ghmqusnsglem2 33143 ghmqusnsg 33144 rhmquskerlem 33153 rhmqusnsg 33156 qsnzr 33184 opprqusplusg 33213 opprqus0g 33214 qsdrngilem 33218 qsdrngi 33219 |
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