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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 2760 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | issubg 17795 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
4 | 3 | simp3bi 1142 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
5 | 1, 4 | syl5eqel 2843 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
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: subg0 17801 subginv 17802 subg0cl 17803 subginvcl 17804 subgcl 17805 issubg2 17810 issubgrpd 17812 subsubg 17818 resghm 17877 resghm2b 17879 subgga 17933 gasubg 17935 odsubdvds 18186 pgp0 18211 subgpgp 18212 sylow2blem2 18236 slwhash 18239 fislw 18240 subglsm 18286 pj1ghm 18316 subgabl 18441 cycsubgcyg 18502 subgdmdprd 18633 subgdprd 18634 ablfacrplem 18664 pgpfaclem1 18680 pgpfaclem3 18682 ablfaclem3 18686 issubrg2 19002 islss3 19161 mplgrp 19652 zringcyg 20041 cnmsgngrp 20127 psgnghm 20128 scmatghm 20541 m2cpmrngiso 20765 subgtgp 22110 subgngp 22640 reefgim 24403 amgmlemALT 43062 |
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