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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 2736 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | subgss 18498 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
3 | subggrp.h | . . 3 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
4 | 3, 1 | ressbas2 16739 | . 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 1543 ∈ wcel 2112 ⊆ wss 3853 ‘cfv 6358 (class class class)co 7191 Basecbs 16666 ↾s cress 16667 SubGrpcsubg 18491 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-1cn 10752 ax-addcl 10754 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-nn 11796 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-subg 18494 |
This theorem is referenced by: subg0 18503 subginv 18504 subg0cl 18505 subginvcl 18506 subgcl 18507 subgsub 18509 subgmulg 18511 issubg2 18512 subsubg 18520 nmznsg 18538 subgga 18648 gasubg 18650 odsubdvds 18914 pgp0 18939 subgpgp 18940 sylow2blem2 18964 sylow2blem3 18965 slwhash 18967 fislw 18968 sylow3lem4 18973 sylow3lem6 18975 subglsm 19017 pj1ghm 19047 subgabl 19175 cycsubgcyg 19240 subgdmdprd 19375 ablfacrplem 19406 ablfac1c 19412 pgpfaclem1 19422 pgpfaclem2 19423 pgpfaclem3 19424 ablfaclem3 19428 ablfac2 19430 subrgbas 19763 issubrg2 19774 pj1lmhm 20091 phssip 20574 scmatsgrp1 21373 subgtgp 22956 subgnm 23485 subgngp 23487 lssnlm 23553 cmscsscms 24224 cssbn 24226 reefgim 25296 efabl 25393 |
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