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Mirrors > Home > MPE Home > Th. List > subgss | Structured version Visualization version GIF version |
Description: A subgroup is a subset. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
issubg.b | ⊢ 𝐵 = (Base‘𝐺) |
Ref | Expression |
---|---|
subgss | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issubg.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
2 | 1 | issubg 17795 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ 𝐵 ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
3 | 2 | simp2bi 1141 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ⊆ wss 3715 ‘cfv 6049 (class class class)co 6813 Basecbs 16059 ↾s cress 16060 Grpcgrp 17623 SubGrpcsubg 17789 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-rab 3059 df-v 3342 df-sbc 3577 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-op 4328 df-uni 4589 df-br 4805 df-opab 4865 df-mpt 4882 df-id 5174 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-iota 6012 df-fun 6051 df-fv 6057 df-ov 6816 df-subg 17792 |
This theorem is referenced by: subgbas 17799 subg0 17801 subginv 17802 subgsubcl 17806 subgsub 17807 subgmulgcl 17808 subgmulg 17809 issubg2 17810 issubg4 17814 subsubg 17818 subgint 17819 nsgconj 17828 nsgacs 17831 ssnmz 17837 eqger 17845 eqgid 17847 eqgen 17848 eqgcpbl 17849 lagsubg2 17856 lagsubg 17857 resghm 17877 ghmnsgima 17885 conjsubg 17893 conjsubgen 17894 conjnmz 17895 conjnmzb 17896 gicsubgen 17921 subgga 17933 gasubg 17935 gastacos 17943 orbstafun 17944 cntrsubgnsg 17973 oddvds2 18183 subgpgp 18212 odcau 18219 pgpssslw 18229 sylow2blem1 18235 sylow2blem2 18236 sylow2blem3 18237 slwhash 18239 fislw 18240 sylow2 18241 sylow3lem1 18242 sylow3lem2 18243 sylow3lem3 18244 sylow3lem4 18245 sylow3lem5 18246 sylow3lem6 18247 lsmval 18263 lsmelval 18264 lsmelvali 18265 lsmelvalm 18266 lsmsubg 18269 lsmub1 18271 lsmub2 18272 lsmless1 18274 lsmless2 18275 lsmless12 18276 lsmass 18283 subglsm 18286 lsmmod 18288 cntzrecd 18291 lsmcntz 18292 lsmcntzr 18293 lsmdisj2 18295 subgdisj1 18304 pj1f 18310 pj1id 18312 pj1lid 18314 pj1rid 18315 pj1ghm 18316 subgabl 18441 ablcntzd 18460 lsmcom 18461 dprdff 18611 dprdfadd 18619 dprdres 18627 dprdss 18628 subgdmdprd 18633 dprdcntz2 18637 dmdprdsplit2lem 18644 ablfacrp 18665 ablfac1eu 18672 pgpfac1lem1 18673 pgpfac1lem2 18674 pgpfac1lem3a 18675 pgpfac1lem3 18676 pgpfac1lem4 18677 pgpfac1lem5 18678 pgpfaclem1 18680 pgpfaclem2 18681 pgpfaclem3 18682 ablfaclem3 18686 ablfac2 18688 issubrg2 19002 issubrg3 19010 islss4 19164 mpllsslem 19637 phssip 20205 subgtgp 22110 subgntr 22111 opnsubg 22112 clssubg 22113 clsnsg 22114 cldsubg 22115 qustgpopn 22124 qustgphaus 22127 tgptsmscls 22154 subgnm 22638 subgngp 22640 lssnlm 22706 efgh 24486 efabl 24495 efsubm 24496 idomsubgmo 38278 |
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