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Mirrors > Home > MPE Home > Th. List > subgbas | Structured version Visualization version GIF version |
Description: The base of the restricted group in a subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
Ref | Expression |
---|---|
subggrp.h | ⊢ 𝐻 = (𝐺 ↾s 𝑆) |
Ref | Expression |
---|---|
subgbas | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2734 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | subgss 19157 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
3 | subggrp.h | . . 3 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
4 | 3, 1 | ressbas2 17282 | . 2 ⊢ (𝑆 ⊆ (Base‘𝐺) → 𝑆 = (Base‘𝐻)) |
5 | 2, 4 | syl 17 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘𝐻)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ⊆ wss 3962 ‘cfv 6562 (class class class)co 7430 Basecbs 17244 ↾s cress 17273 SubGrpcsubg 19150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-1cn 11210 ax-addcl 11212 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-nn 12264 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-subg 19153 |
This theorem is referenced by: subg0 19162 subginv 19163 subg0cl 19164 subginvcl 19165 subgcl 19166 subgsub 19168 subgmulg 19170 issubg2 19171 subsubg 19179 nmznsg 19198 subgga 19330 gasubg 19332 odsubdvds 19603 pgp0 19628 subgpgp 19629 sylow2blem2 19653 sylow2blem3 19654 slwhash 19656 fislw 19657 sylow3lem4 19662 sylow3lem6 19664 subglsm 19705 pj1ghm 19735 subgabl 19868 cycsubgcyg 19933 subgdmdprd 20068 ablfacrplem 20099 ablfac1c 20105 pgpfaclem1 20115 pgpfaclem2 20116 pgpfaclem3 20117 ablfaclem3 20121 ablfac2 20123 subrngbas 20570 issubrng2 20574 subrgbas 20597 issubrg2 20608 pj1lmhm 21116 phssip 21693 scmatsgrp1 22543 subgtgp 24128 subgnm 24661 subgngp 24663 lssnlm 24737 cmscsscms 25420 cssbn 25422 reefgim 26508 efabl 26606 |
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