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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 2825 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2825 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | isnsg 17981 | . 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 493 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∈ wcel 2164 ∀wral 3117 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 +gcplusg 16312 SubGrpcsubg 17946 NrmSGrpcnsg 17947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-iota 6090 df-fun 6129 df-fv 6135 df-ov 6913 df-subg 17949 df-nsg 17950 |
This theorem is referenced by: nsgconj 17985 isnsg3 17986 eqgcpbl 18006 qusgrp 18007 quseccl 18008 qusadd 18009 qus0 18010 qusinv 18011 qussub 18012 ghmnsgima 18042 ghmnsgpreima 18043 conjnsg 18054 qusghm 18055 sylow3lem4 18403 clsnsg 22290 qustgpopn 22300 qustgphaus 22303 |
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