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Mirrors > Home > MPE Home > Th. List > subrgnrg | Structured version Visualization version GIF version |
Description: A normed ring restricted to a subring is a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
subrgnrg.h | ⊢ 𝐻 = (𝐺 ↾s 𝐴) |
Ref | Expression |
---|---|
subrgnrg | ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nrgngp 22843 | . . 3 ⊢ (𝐺 ∈ NrmRing → 𝐺 ∈ NrmGrp) | |
2 | subrgsubg 19149 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝐺) → 𝐴 ∈ (SubGrp‘𝐺)) | |
3 | subrgnrg.h | . . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
4 | 3 | subgngp 22816 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ NrmGrp) |
5 | 1, 2, 4 | syl2an 589 | . 2 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmGrp) |
6 | 2 | adantl 475 | . . . 4 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐴 ∈ (SubGrp‘𝐺)) |
7 | eqid 2825 | . . . . 5 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
8 | eqid 2825 | . . . . 5 ⊢ (norm‘𝐻) = (norm‘𝐻) | |
9 | 3, 7, 8 | subgnm 22814 | . . . 4 ⊢ (𝐴 ∈ (SubGrp‘𝐺) → (norm‘𝐻) = ((norm‘𝐺) ↾ 𝐴)) |
10 | 6, 9 | syl 17 | . . 3 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → (norm‘𝐻) = ((norm‘𝐺) ↾ 𝐴)) |
11 | eqid 2825 | . . . . 5 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
12 | 7, 11 | nrgabv 22842 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) ∈ (AbsVal‘𝐺)) |
13 | eqid 2825 | . . . . 5 ⊢ (AbsVal‘𝐻) = (AbsVal‘𝐻) | |
14 | 11, 3, 13 | abvres 19202 | . . . 4 ⊢ (((norm‘𝐺) ∈ (AbsVal‘𝐺) ∧ 𝐴 ∈ (SubRing‘𝐺)) → ((norm‘𝐺) ↾ 𝐴) ∈ (AbsVal‘𝐻)) |
15 | 12, 14 | sylan 575 | . . 3 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → ((norm‘𝐺) ↾ 𝐴) ∈ (AbsVal‘𝐻)) |
16 | 10, 15 | eqeltrd 2906 | . 2 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → (norm‘𝐻) ∈ (AbsVal‘𝐻)) |
17 | 8, 13 | isnrg 22841 | . 2 ⊢ (𝐻 ∈ NrmRing ↔ (𝐻 ∈ NrmGrp ∧ (norm‘𝐻) ∈ (AbsVal‘𝐻))) |
18 | 5, 16, 17 | sylanbrc 578 | 1 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ↾ cres 5348 ‘cfv 6127 (class class class)co 6910 ↾s cress 16230 SubGrpcsubg 17946 SubRingcsubrg 19139 AbsValcabv 19179 normcnm 22758 NrmGrpcngp 22759 NrmRingcnrg 22761 |
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-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 ax-pre-sup 10337 |
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-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-sup 8623 df-inf 8624 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-q 12079 df-rp 12120 df-xneg 12239 df-xadd 12240 df-xmul 12241 df-ico 12476 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-tset 16331 df-ds 16334 df-rest 16443 df-topn 16444 df-0g 16462 df-topgen 16464 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-grp 17786 df-minusg 17787 df-sbg 17788 df-subg 17949 df-mgp 18851 df-ring 18910 df-subrg 19141 df-abv 19180 df-psmet 20105 df-xmet 20106 df-met 20107 df-bl 20108 df-mopn 20109 df-top 21076 df-topon 21093 df-topsp 21115 df-bases 21128 df-xms 22502 df-ms 22503 df-nm 22764 df-ngp 22765 df-nrg 22767 |
This theorem is referenced by: sranlm 22865 zringnrg 22968 isncvsngp 23325 tcphcph 23412 rezh 30556 rerrext 30594 |
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