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Mirrors > Home > MPE Home > Th. List > subgmulgcl | Structured version Visualization version GIF version |
Description: Closure of the group multiple (exponentiation) operation in a subgroup. (Contributed by Mario Carneiro, 13-Jan-2015.) |
Ref | Expression |
---|---|
subgmulgcl.t | ⊢ · = (.g‘𝐺) |
Ref | Expression |
---|---|
subgmulgcl | ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2825 | . 2 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | subgmulgcl.t | . 2 ⊢ · = (.g‘𝐺) | |
3 | eqid 2825 | . 2 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
4 | subgrcl 17957 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝐺 ∈ Grp) | |
5 | 1 | subgss 17953 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → 𝑆 ⊆ (Base‘𝐺)) |
6 | 3 | subgcl 17962 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆 ∧ 𝑦 ∈ 𝑆) → (𝑥(+g‘𝐺)𝑦) ∈ 𝑆) |
7 | eqid 2825 | . 2 ⊢ (0g‘𝐺) = (0g‘𝐺) | |
8 | 7 | subg0cl 17960 | . 2 ⊢ (𝑆 ∈ (SubGrp‘𝐺) → (0g‘𝐺) ∈ 𝑆) |
9 | eqid 2825 | . 2 ⊢ (invg‘𝐺) = (invg‘𝐺) | |
10 | 9 | subginvcl 17961 | . 2 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑥 ∈ 𝑆) → ((invg‘𝐺)‘𝑥) ∈ 𝑆) |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | mulgsubcl 17916 | 1 ⊢ ((𝑆 ∈ (SubGrp‘𝐺) ∧ 𝑁 ∈ ℤ ∧ 𝑋 ∈ 𝑆) → (𝑁 · 𝑋) ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ‘cfv 6127 (class class class)co 6910 ℤcz 11711 Basecbs 16229 +gcplusg 16312 0gc0g 16460 Grpcgrp 17783 invgcminusg 17784 .gcmg 17901 SubGrpcsubg 17946 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-2 11421 df-n0 11626 df-z 11712 df-uz 11976 df-fz 12627 df-seq 13103 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-mulg 17902 df-subg 17949 |
This theorem is referenced by: subgmulg 17966 cycsubgss 17979 cycsubg2cl 17990 odf1o1 18345 pgpfac1lem2 18835 pgpfac1lem3a 18836 pgpfac1lem3 18837 zsssubrg 20171 |
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