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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 2818 | . . . 4 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
3 | 2 | issubg 18217 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) ↔ (𝐺 ∈ Grp ∧ 𝑆 ⊆ (Base‘𝐺) ∧ (𝐺 ↾s 𝑆) ∈ Grp)) |
4 | 3 | simp3bi 1139 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
5 | 1, 4 | eqeltrid 2914 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐻 ∈ Grp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ 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 subg0cl 18225 subginvcl 18226 subgcl 18227 issubg2 18232 issubgrpd 18234 subsubg 18240 resghm 18312 resghm2b 18314 subgga 18368 gasubg 18370 odsubdvds 18625 pgp0 18650 subgpgp 18651 sylow2blem2 18675 slwhash 18678 fislw 18679 subglsm 18728 pj1ghm 18758 subgabl 18885 cntrabl 18892 cycsubgcyg 18950 subgdmdprd 19085 subgdprd 19086 ablfacrplem 19116 pgpfaclem1 19132 pgpfaclem3 19134 ablfaclem3 19138 issubrg2 19484 subdrgint 19511 islss3 19660 mplgrp 20158 zringcyg 20566 cnmsgngrp 20651 psgnghm 20652 scmatghm 21070 m2cpmrngiso 21294 subgtgp 22641 subgngp 23171 reefgim 24965 amgmlemALT 44832 |
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