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| Mirrors > Home > MPE Home > Th. List > subggrp | Structured version Visualization version GIF version | ||
| Description: A subgroup is a group. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| Ref | Expression |
|---|---|
| subggrp.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
| Ref | Expression |
|---|---|
| subggrp | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | subggrp.h | . 2 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
| 2 | eqid 2737 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 3 | 2 | issubg 19144 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
| 4 | 3 | simp3bi 1148 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
| 5 | 1, 4 | eqeltrid 2845 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 Grpcgrp 18951 SubGrpcsubg 19138 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fv 6569 df-ov 7434 df-subg 19141 |
| This theorem is referenced by: subg0 19150 subginv 19151 subg0cl 19152 subginvcl 19153 subgcl 19154 issubg2 19159 issubgrpd 19161 subsubg 19167 resghm 19250 resghm2b 19252 subgga 19318 gasubg 19320 odsubdvds 19589 pgp0 19614 subgpgp 19615 sylow2blem2 19639 slwhash 19642 fislw 19643 subglsm 19691 pj1ghm 19721 subgabl 19854 cntrabl 19861 cycsubgcyg 19919 subgdmdprd 20054 subgdprd 20055 ablfacrplem 20085 pgpfaclem1 20101 pgpfaclem3 20103 ablfaclem3 20107 issubrg2 20592 subdrgint 20804 islss3 20957 zringcyg 21480 cnmsgngrp 21597 psgnghm 21598 mplgrp 22037 scmatghm 22539 subgtgp 24113 subgngp 24648 reefgim 26494 subgmulgcld 33048 ressply1sub 33595 amgmlemALT 49322 |
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