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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 2738 | . . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
2 | 1 | subggrp 18400 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
3 | eqid 2738 | . . . 4 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
4 | eqid 2738 | . . . 4 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
5 | 3, 4 | grpidcl 18249 | . . 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 18403 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘(𝐺 ↾s 𝑆))) |
9 | 1 | subgbas 18401 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
10 | 6, 8, 9 | 3eltr4d 2848 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ‘cfv 6339 (class class class)co 7170 Basecbs 16586 ↾s cress 16587 0gc0g 16816 Grpcgrp 18219 SubGrpcsubg 18391 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2710 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7479 ax-cnex 10671 ax-resscn 10672 ax-1cn 10673 ax-icn 10674 ax-addcl 10675 ax-addrcl 10676 ax-mulcl 10677 ax-mulrcl 10678 ax-mulcom 10679 ax-addass 10680 ax-mulass 10681 ax-distr 10682 ax-i2m1 10683 ax-1ne0 10684 ax-1rid 10685 ax-rnegex 10686 ax-rrecex 10687 ax-cnre 10688 ax-pre-lttri 10689 ax-pre-lttrn 10690 ax-pre-ltadd 10691 ax-pre-mulgt0 10692 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-nel 3039 df-ral 3058 df-rex 3059 df-reu 3060 df-rmo 3061 df-rab 3062 df-v 3400 df-sbc 3681 df-csb 3791 df-dif 3846 df-un 3848 df-in 3850 df-ss 3860 df-pss 3862 df-nul 4212 df-if 4415 df-pw 4490 df-sn 4517 df-pr 4519 df-tp 4521 df-op 4523 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5429 df-eprel 5434 df-po 5442 df-so 5443 df-fr 5483 df-we 5485 df-xp 5531 df-rel 5532 df-cnv 5533 df-co 5534 df-dm 5535 df-rn 5536 df-res 5537 df-ima 5538 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6297 df-fun 6341 df-fn 6342 df-f 6343 df-f1 6344 df-fo 6345 df-f1o 6346 df-fv 6347 df-riota 7127 df-ov 7173 df-oprab 7174 df-mpo 7175 df-om 7600 df-wrecs 7976 df-recs 8037 df-rdg 8075 df-er 8320 df-en 8556 df-dom 8557 df-sdom 8558 df-pnf 10755 df-mnf 10756 df-xr 10757 df-ltxr 10758 df-le 10759 df-sub 10950 df-neg 10951 df-nn 11717 df-2 11779 df-ndx 16589 df-slot 16590 df-base 16592 df-sets 16593 df-ress 16594 df-plusg 16681 df-0g 16818 df-mgm 17968 df-sgrp 18017 df-mnd 18028 df-grp 18222 df-subg 18394 |
This theorem is referenced by: subgmulgcl 18410 issubg3 18415 issubg4 18416 subgint 18421 trivsubgd 18423 eqger 18448 ghmpreima 18498 subgga 18548 gasubg 18550 sylow1lem5 18845 sylow2blem2 18864 sylow2blem3 18865 fislw 18868 sylow3lem3 18872 sylow3lem4 18873 lsm01 18915 lsm02 18916 lsmdisj 18925 lsmdisj2 18926 pj1lid 18945 pj1rid 18946 dmdprdd 19240 dprdfid 19258 dprdfeq0 19263 dprdsubg 19265 dprdres 19269 dprdz 19271 dprdsn 19277 dmdprdsplitlem 19278 dprddisj2 19280 dprd2da 19283 dmdprdsplit2lem 19286 ablfacrp 19307 ablfacrp2 19308 ablfac1c 19312 ablfac1eu 19314 pgpfac1lem3a 19317 pgpfac1lem3 19318 pgpfac1lem5 19320 pgpfaclem2 19323 pgpfaclem3 19324 prmgrpsimpgd 19355 primefld0cl 19704 abvres 19729 islss4 19853 subrgpsr 20798 mpllsslem 20816 0elcpmat 21473 opnsubg 22859 clssubg 22860 tgpconncompss 22865 plypf1 24961 dvply2g 25033 efsubm 25295 dchrptlem3 26002 gsumsubg 30883 nsgqus0 31167 nsgqusf1olem1 31170 drgext0gsca 31251 fedgmullem2 31283 fsumcnsrcl 40563 cnsrplycl 40564 rngunsnply 40570 |
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