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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 2737 | . . 3 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
| 2 | 1 | subgss 19145 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
| 3 | subggrp.h | . . 3 ⊢ 𝐻 = (𝐺 ↾s 𝑆) | |
| 4 | 3, 1 | ressbas2 17283 | . 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 1540 ∈ wcel 2108 ⊆ wss 3951 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 SubGrpcsubg 19138 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-1cn 11213 ax-addcl 11215 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-nn 12267 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-subg 19141 |
| This theorem is referenced by: subg0 19150 subginv 19151 subg0cl 19152 subginvcl 19153 subgcl 19154 subgsub 19156 subgmulg 19158 issubg2 19159 subsubg 19167 nmznsg 19186 subgga 19318 gasubg 19320 odsubdvds 19589 pgp0 19614 subgpgp 19615 sylow2blem2 19639 sylow2blem3 19640 slwhash 19642 fislw 19643 sylow3lem4 19648 sylow3lem6 19650 subglsm 19691 pj1ghm 19721 subgabl 19854 cycsubgcyg 19919 subgdmdprd 20054 ablfacrplem 20085 ablfac1c 20091 pgpfaclem1 20101 pgpfaclem2 20102 pgpfaclem3 20103 ablfaclem3 20107 ablfac2 20109 subrngbas 20554 issubrng2 20558 subrgbas 20581 issubrg2 20592 pj1lmhm 21099 phssip 21676 scmatsgrp1 22528 subgtgp 24113 subgnm 24646 subgngp 24648 lssnlm 24722 cmscsscms 25407 cssbn 25409 reefgim 26494 efabl 26592 |
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