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| Mirrors > Home > MPE Home > Th. List > subg0cl | Structured version Visualization version GIF version | ||
| Description: The group identity is an element of any subgroup. (Contributed by Mario Carneiro, 2-Dec-2014.) |
| Ref | Expression |
|---|---|
| subg0cl.i | ⊢ 0 = (0g‘𝐺) |
| Ref | Expression |
|---|---|
| subg0cl | ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | eqid 2737 | . . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
| 2 | 1 | subggrp 19147 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
| 3 | eqid 2737 | . . . 4 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
| 4 | eqid 2737 | . . . 4 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
| 5 | 3, 4 | grpidcl 18983 | . . 3 ⊢ ((𝐺 ↾s 𝑆) ∈ Grp → (0g‘(𝐺 ↾s 𝑆)) ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 6 | 2, 5 | syl 17 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘(𝐺 ↾s 𝑆)) ∈ (Base‘(𝐺 ↾s 𝑆))) |
| 7 | subg0cl.i | . . 3 ⊢ 0 = (0g‘𝐺) | |
| 8 | 1, 7 | subg0 19150 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘(𝐺 ↾s 𝑆))) |
| 9 | 1 | subgbas 19148 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
| 10 | 6, 8, 9 | 3eltr4d 2856 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 ↾s cress 17274 0gc0g 17484 Grpcgrp 18951 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-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| 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-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 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-riota 7388 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-er 8745 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-subg 19141 |
| This theorem is referenced by: subgmulgcl 19157 issubg3 19162 issubg4 19163 subgint 19168 trivsubgd 19171 eqger 19196 ghmpreima 19256 subgga 19318 gasubg 19320 sylow1lem5 19620 sylow2blem2 19639 sylow2blem3 19640 fislw 19643 sylow3lem3 19647 sylow3lem4 19648 lsm01 19689 lsm02 19690 lsmdisj 19699 lsmdisj2 19700 pj1lid 19719 pj1rid 19720 dmdprdd 20019 dprdfid 20037 dprdfeq0 20042 dprdsubg 20044 dprdres 20048 dprdz 20050 dprdsn 20056 dmdprdsplitlem 20057 dprddisj2 20059 dprd2da 20062 dmdprdsplit2lem 20065 ablfacrp 20086 ablfacrp2 20087 ablfac1c 20091 ablfac1eu 20093 pgpfac1lem3a 20096 pgpfac1lem3 20097 pgpfac1lem5 20099 pgpfaclem2 20102 pgpfaclem3 20103 prmgrpsimpgd 20134 primefld0cl 20807 abvres 20832 islss4 20960 dflidl2rng 21228 rnglidlrng 21257 rng2idl0 21277 rng2idlsubg0 21280 2idlcpblrng 21281 rng2idl1cntr 21315 subrgpsr 21998 mpllsslem 22020 0elcpmat 22728 opnsubg 24116 clssubg 24117 tgpconncompss 24122 plypf1 26251 dvply2g 26326 dvply2gOLD 26327 efsubm 26593 dchrptlem3 27310 gsumsubg 33049 elrgspnlem4 33249 subrdom 33288 nsgqus0 33438 nsgqusf1olem1 33441 ressply10g 33592 ressply1invg 33594 drgext0gsca 33642 fedgmullem2 33681 fldextrspunlsplem 33723 fldextrspunlsp 33724 algextdeglem4 33761 algextdeglem5 33762 rtelextdg2lem 33767 fsumcnsrcl 43178 cnsrplycl 43179 rngunsnply 43181 |
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