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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 2821 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | eqid 2821 | . . 3 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
3 | 1, 2 | isnsg 18307 | . 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 500 | 1 ⊢ (𝑆 ∈ (NrmSGrp‘𝐺) → 𝑆 ∈ (SubGrp‘𝐺)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 +gcplusg 16565 SubGrpcsubg 18273 NrmSGrpcnsg 18274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fv 6363 df-ov 7159 df-subg 18276 df-nsg 18277 |
This theorem is referenced by: nsgconj 18311 isnsg3 18312 trivnsgd 18324 eqgcpbl 18334 qusgrp 18335 quseccl 18336 qusadd 18337 qus0 18338 qusinv 18339 qussub 18340 ghmnsgima 18382 ghmnsgpreima 18383 conjnsg 18394 qusghm 18395 sylow3lem4 18755 prmgrpsimpgd 19236 clsnsg 22718 qustgpopn 22728 qustgphaus 22731 cyc3genpm 30794 qusker 30918 |
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