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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 2824 | . . . 4 ⊢ (𝐺 ↾s 𝑆) = (𝐺 ↾s 𝑆) | |
2 | 1 | subggrp 18285 | . . 3 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (𝐺 ↾s 𝑆) ∈ Grp) |
3 | eqid 2824 | . . . 4 ⊢ (Base‘(𝐺 ↾s 𝑆)) = (Base‘(𝐺 ↾s 𝑆)) | |
4 | eqid 2824 | . . . 4 ⊢ (0g‘(𝐺 ↾s 𝑆)) = (0g‘(𝐺 ↾s 𝑆)) | |
5 | 3, 4 | grpidcl 18134 | . . 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 18288 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 = (0g‘(𝐺 ↾s 𝑆))) |
9 | 1 | subgbas 18286 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 = (Base‘(𝐺 ↾s 𝑆))) |
10 | 6, 8, 9 | 3eltr4d 2931 | 1 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 0 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2113 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 ↾s cress 16487 0gc0g 16716 Grpcgrp 18106 SubGrpcsubg 18276 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-er 8292 df-en 8513 df-dom 8514 df-sdom 8515 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-nn 11642 df-2 11703 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-0g 16718 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-grp 18109 df-subg 18279 |
This theorem is referenced by: subgmulgcl 18295 issubg3 18300 issubg4 18301 subgint 18306 trivsubgd 18308 eqger 18333 ghmpreima 18383 subgga 18433 gasubg 18435 sylow1lem5 18730 sylow2blem2 18749 sylow2blem3 18750 fislw 18753 sylow3lem3 18757 sylow3lem4 18758 lsm01 18800 lsm02 18801 lsmdisj 18810 lsmdisj2 18811 pj1lid 18830 pj1rid 18831 dmdprdd 19124 dprdfid 19142 dprdfeq0 19147 dprdsubg 19149 dprdres 19153 dprdz 19155 dprdsn 19161 dmdprdsplitlem 19162 dprddisj2 19164 dprd2da 19167 dmdprdsplit2lem 19170 ablfacrp 19191 ablfacrp2 19192 ablfac1c 19196 ablfac1eu 19198 pgpfac1lem3a 19201 pgpfac1lem3 19202 pgpfac1lem5 19204 pgpfaclem2 19207 pgpfaclem3 19208 prmgrpsimpgd 19239 primefld0cl 19588 abvres 19613 islss4 19737 subrgpsr 20202 mpllsslem 20218 0elcpmat 21333 opnsubg 22719 clssubg 22720 tgpconncompss 22725 plypf1 24805 dvply2g 24877 efsubm 25138 dchrptlem3 25845 gsumsubg 30688 drgext0gsca 30998 fedgmullem2 31030 fsumcnsrcl 39772 cnsrplycl 39773 rngunsnply 39779 |
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