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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 2821 | . . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
2 | 1 | subggrp 18282 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
3 | eqid 2821 | . . . 4 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
4 | eqid 2821 | . . . 4 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
5 | 3, 4 | grpidcl 18131 | . . 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 18285 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘(𝐺 ↾s 𝑆))) |
9 | 1 | subgbas 18283 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
10 | 6, 8, 9 | 3eltr4d 2928 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 ↾s cress 16484 0gc0g 16713 Grpcgrp 18103 SubGrpcsubg 18273 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-ress 16491 df-plusg 16578 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-subg 18276 |
This theorem is referenced by: subgmulgcl 18292 issubg3 18297 issubg4 18298 subgint 18303 trivsubgd 18305 eqger 18330 ghmpreima 18380 subgga 18430 gasubg 18432 sylow1lem5 18727 sylow2blem2 18746 sylow2blem3 18747 fislw 18750 sylow3lem3 18754 sylow3lem4 18755 lsm01 18797 lsm02 18798 lsmdisj 18807 lsmdisj2 18808 pj1lid 18827 pj1rid 18828 dmdprdd 19121 dprdfid 19139 dprdfeq0 19144 dprdsubg 19146 dprdres 19150 dprdz 19152 dprdsn 19158 dmdprdsplitlem 19159 dprddisj2 19161 dprd2da 19164 dmdprdsplit2lem 19167 ablfacrp 19188 ablfacrp2 19189 ablfac1c 19193 ablfac1eu 19195 pgpfac1lem3a 19198 pgpfac1lem3 19199 pgpfac1lem5 19201 pgpfaclem2 19204 pgpfaclem3 19205 prmgrpsimpgd 19236 primefld0cl 19585 abvres 19610 islss4 19734 subrgpsr 20199 mpllsslem 20215 0elcpmat 21330 opnsubg 22716 clssubg 22717 tgpconncompss 22722 plypf1 24802 dvply2g 24874 efsubm 25135 dchrptlem3 25842 gsumsubg 30684 drgext0gsca 30994 fedgmullem2 31026 fsumcnsrcl 39786 cnsrplycl 39787 rngunsnply 39793 |
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