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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 2818 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | issubg 18217 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
3 | 2 | simp1bi 1137 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3933 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 ↾s cress 16472 Grpcgrp 18041 SubGrpcsubg 18211 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-subg 18214 |
This theorem is referenced by: subg0 18223 subginv 18224 subgmulgcl 18230 subgsubm 18239 subsubg 18240 subgint 18241 isnsg 18245 nsgconj 18249 isnsg3 18250 ssnmz 18256 nmznsg 18258 eqger 18268 eqgid 18270 eqgen 18271 eqgcpbl 18272 qusgrp 18273 quseccl 18274 qusadd 18275 qus0 18276 qusinv 18277 qussub 18278 resghm2 18313 resghm2b 18314 conjsubg 18328 conjsubgen 18329 conjnmz 18330 conjnmzb 18331 qusghm 18333 subgga 18368 gastacos 18378 orbstafun 18379 cntrsubgnsg 18409 oppgsubg 18429 isslw 18662 sylow2blem1 18674 sylow2blem2 18675 sylow2blem3 18676 slwhash 18678 lsmval 18702 lsmelval 18703 lsmelvali 18704 lsmelvalm 18705 lsmsubg 18708 lsmless1 18714 lsmless2 18715 lsmless12 18716 lsmass 18724 lsm01 18726 lsm02 18727 subglsm 18728 lsmmod 18730 lsmcntz 18734 lsmcntzr 18735 lsmdisj2 18737 subgdisj1 18746 pj1f 18752 pj1id 18754 pj1lid 18756 pj1rid 18757 pj1ghm 18758 subgdmdprd 19085 subgdprd 19086 dprdsn 19087 pgpfaclem2 19133 cldsubg 22646 gsumsubg 30611 qusker 30845 |
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