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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 2610 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | issubg 17417 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
3 | 2 | simp1bi 1069 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ⊆ wss 3540 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 ↾s cress 15696 Grpcgrp 17245 SubGrpcsubg 17411 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fv 5812 df-ov 6552 df-subg 17414 |
This theorem is referenced by: subg0 17423 subginv 17424 subgmulgcl 17430 subgsubm 17439 subsubg 17440 subgint 17441 isnsg 17446 nsgconj 17450 isnsg3 17451 ssnmz 17459 nmznsg 17461 eqger 17467 eqgid 17469 eqgen 17470 eqgcpbl 17471 qusgrp 17472 quseccl 17473 qusadd 17474 qus0 17475 qusinv 17476 qussub 17477 resghm2 17500 resghm2b 17501 conjsubg 17515 conjsubgen 17516 conjnmz 17517 conjnmzb 17518 qusghm 17520 subgga 17556 gastacos 17566 orbstafun 17567 cntrsubgnsg 17596 oppgsubg 17616 isslw 17846 sylow2blem1 17858 sylow2blem2 17859 sylow2blem3 17860 slwhash 17862 lsmval 17886 lsmelval 17887 lsmelvali 17888 lsmelvalm 17889 lsmsubg 17892 lsmless1 17897 lsmless2 17898 lsmless12 17899 lsmass 17906 lsm01 17907 lsm02 17908 subglsm 17909 lsmmod 17911 lsmcntz 17915 lsmcntzr 17916 lsmdisj2 17918 subgdisj1 17927 pj1f 17933 pj1id 17935 pj1lid 17937 pj1rid 17938 pj1ghm 17939 subgdmdprd 18256 subgdprd 18257 dprdsn 18258 pgpfaclem2 18304 cldsubg 21724 |
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