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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 24529 | . . 3 ⊢ (𝐺 ∈ NrmRing → 𝐺 ∈ NrmGrp) | |
2 | subrgsubg 20476 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝐺) → 𝐴 ∈ (SubGrp‘𝐺)) | |
3 | subrgnrg.h | . . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
4 | 3 | subgngp 24494 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ NrmGrp) |
5 | 1, 2, 4 | syl2an 595 | . 2 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmGrp) |
6 | 2 | adantl 481 | . . . 4 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐴 ∈ (SubGrp‘𝐺)) |
7 | eqid 2726 | . . . . 5 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
8 | eqid 2726 | . . . . 5 ⊢ (norm‘𝐻) = (norm‘𝐻) | |
9 | 3, 7, 8 | subgnm 24492 | . . . 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 2726 | . . . . 5 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
12 | 7, 11 | nrgabv 24528 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) ∈ (AbsVal‘𝐺)) |
13 | eqid 2726 | . . . . 5 ⊢ (AbsVal‘𝐻) = (AbsVal‘𝐻) | |
14 | 11, 3, 13 | abvres 20679 | . . . 4 ⊢ (((norm‘𝐺) ∈ (AbsVal‘𝐺) ∧ 𝐴 ∈ (SubRing‘𝐺)) → ((norm‘𝐺) ↾ 𝐴) ∈ (AbsVal‘𝐻)) |
15 | 12, 14 | sylan 579 | . . 3 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → ((norm‘𝐺) ↾ 𝐴) ∈ (AbsVal‘𝐻)) |
16 | 10, 15 | eqeltrd 2827 | . 2 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → (norm‘𝐻) ∈ (AbsVal‘𝐻)) |
17 | 8, 13 | isnrg 24527 | . 2 ⊢ (𝐻 ∈ NrmRing ↔ (𝐻 ∈ NrmGrp ∧ (norm‘𝐻) ∈ (AbsVal‘𝐻))) |
18 | 5, 16, 17 | sylanbrc 582 | 1 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ↾ cres 5671 ‘cfv 6536 (class class class)co 7404 ↾s cress 17179 SubGrpcsubg 19044 SubRingcsubrg 20466 AbsValcabv 20656 normcnm 24435 NrmGrpcngp 24436 NrmRingcnrg 24438 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7721 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-er 8702 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-sup 9436 df-inf 9437 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-z 12560 df-dec 12679 df-uz 12824 df-q 12934 df-rp 12978 df-xneg 13095 df-xadd 13096 df-xmul 13097 df-ico 13333 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-tset 17222 df-ds 17225 df-rest 17374 df-topn 17375 df-0g 17393 df-topgen 17395 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-grp 18863 df-minusg 18864 df-sbg 18865 df-subg 19047 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-subrng 20443 df-subrg 20468 df-abv 20657 df-psmet 21227 df-xmet 21228 df-met 21229 df-bl 21230 df-mopn 21231 df-top 22746 df-topon 22763 df-topsp 22785 df-bases 22799 df-xms 24176 df-ms 24177 df-nm 24441 df-ngp 24442 df-nrg 24444 |
This theorem is referenced by: sranlm 24551 zringnrg 24654 isncvsngp 25027 tcphcph 25115 rezh 33480 rerrext 33518 |
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