![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > subgrcl | Structured version Visualization version GIF version |
Description: Reverse closure for the subgroup predicate. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
subgrcl | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | issubg 17952 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
3 | 2 | simp1bi 1179 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2164 ⊆ wss 3798 ‘cfv 6127 (class class class)co 6910 Basecbs 16229 ↾s cress 16230 Grpcgrp 17783 SubGrpcsubg 17946 |
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 |
This theorem is referenced by: subg0 17958 subginv 17959 subgmulgcl 17965 subgsubm 17974 subsubg 17975 subgint 17976 isnsg 17981 nsgconj 17985 isnsg3 17986 ssnmz 17994 nmznsg 17996 eqger 18002 eqgid 18004 eqgen 18005 eqgcpbl 18006 qusgrp 18007 quseccl 18008 qusadd 18009 qus0 18010 qusinv 18011 qussub 18012 resghm2 18035 resghm2b 18036 conjsubg 18050 conjsubgen 18051 conjnmz 18052 conjnmzb 18053 qusghm 18055 subgga 18090 gastacos 18100 orbstafun 18101 cntrsubgnsg 18130 oppgsubg 18150 isslw 18381 sylow2blem1 18393 sylow2blem2 18394 sylow2blem3 18395 slwhash 18397 lsmval 18421 lsmelval 18422 lsmelvali 18423 lsmelvalm 18424 lsmsubg 18427 lsmless1 18432 lsmless2 18433 lsmless12 18434 lsmass 18441 lsm01 18442 lsm02 18443 subglsm 18444 lsmmod 18446 lsmcntz 18450 lsmcntzr 18451 lsmdisj2 18453 subgdisj1 18462 pj1f 18468 pj1id 18470 pj1lid 18472 pj1rid 18473 pj1ghm 18474 subgdmdprd 18794 subgdprd 18795 dprdsn 18796 pgpfaclem2 18842 cldsubg 22291 |
Copyright terms: Public domain | W3C validator |