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Mirrors > Home > MPE Home > Th. List > grpcl | Structured version Visualization version GIF version |
Description: Closure of the operation of a group. (Contributed by NM, 14-Aug-2011.) |
Ref | Expression |
---|---|
grpcl.b | ⊢ 𝐵 = (Base‘𝐺) |
grpcl.p | ⊢ + = (+g‘𝐺) |
Ref | Expression |
---|---|
grpcl | ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | grpmnd 17252 | . 2 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd) | |
2 | grpcl.b | . . 3 ⊢ 𝐵 = (Base‘𝐺) | |
3 | grpcl.p | . . 3 ⊢ + = (+g‘𝐺) | |
4 | 2, 3 | mndcl 17124 | . 2 ⊢ ((𝐺 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1351 | 1 ⊢ ((𝐺 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 +gcplusg 15768 Mndcmnd 17117 Grpcgrp 17245 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
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-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 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-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 |
This theorem is referenced by: grprcan 17278 grprinv 17292 grplmulf1o 17312 grpinvadd 17316 grpsubf 17317 grpsubadd 17326 grpaddsubass 17328 grpnpcan 17330 grpsubsub4 17331 grppnpcan2 17332 dfgrp3 17337 grplactcnv 17341 imasgrp 17354 mulgcl 17382 mulgaddcomlem 17386 mulgdir 17396 subgcl 17427 grpissubg 17437 nsgacs 17453 nmzsubg 17458 nsgid 17463 eqger 17467 eqgcpbl 17471 qusgrp 17472 qusadd 17474 ghmrn 17496 idghm 17498 ghmpreima 17505 ghmnsgima 17507 ghmnsgpreima 17508 ghmf1o 17513 conjghm 17514 conjnmz 17517 qusghm 17520 gaid 17555 subgga 17556 gass 17557 gaorber 17564 gastacl 17565 gastacos 17566 cntzsubg 17592 galactghm 17646 lactghmga 17647 symgsssg 17710 symgfisg 17711 symggen 17713 sylow1lem2 17837 sylow2blem1 17858 sylow2blem2 17859 sylow2blem3 17860 sylow3lem1 17865 sylow3lem2 17866 subgdisj1 17927 ablsub4 18041 abladdsub4 18042 mulgdi 18055 mulgghm 18057 invghm 18062 ghmplusg 18072 odadd1 18074 odadd2 18075 odadd 18076 gex2abl 18077 gexexlem 18078 torsubg 18080 oddvdssubg 18081 frgpnabllem2 18100 ringacl 18401 ringpropd 18405 drngmcl 18583 abvtrivd 18663 idsrngd 18685 lmodacl 18697 lmodvacl 18700 lmodprop2d 18748 prdslmodd 18790 pwssplit2 18881 asclghm 19159 psraddcl 19204 mplind 19323 evlslem1 19336 evl1addd 19526 evpmodpmf1o 19761 scmataddcl 20141 mdetralt 20233 mdetunilem6 20242 opnsubg 21721 ghmcnp 21728 qustgpopn 21733 ngprcan 22224 ngpocelbl 22318 nmotri 22353 ncvspi 22764 cphipval2 22848 4cphipval2 22849 cphipval 22850 efsubm 24101 abvcxp 25104 ttgcontlem1 25565 abliso 29027 ogrpaddltbi 29050 ogrpaddltrbid 29052 ogrpinvlt 29055 archiabllem2a 29079 archiabllem2c 29080 archiabllem2b 29081 dvrdir 29121 matunitlindflem1 32575 gicabl 36687 isnumbasgrplem2 36693 mendlmod 36782 |
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