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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 24599 | . . 3 ⊢ (𝐺 ∈ NrmRing → 𝐺 ∈ NrmGrp) | |
2 | subrgsubg 20523 | . . 3 ⊢ (𝐴 ∈ (SubRing‘𝐺) → 𝐴 ∈ (SubGrp‘𝐺)) | |
3 | subrgnrg.h | . . . 4 ⊢ 𝐻 = (𝐺 ↾s 𝐴) | |
4 | 3 | subgngp 24564 | . . 3 ⊢ ((𝐺 ∈ NrmGrp ∧ 𝐴 ∈ (SubGrp‘𝐺)) → 𝐻 ∈ NrmGrp) |
5 | 1, 2, 4 | syl2an 594 | . 2 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmGrp) |
6 | 2 | adantl 480 | . . . 4 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐴 ∈ (SubGrp‘𝐺)) |
7 | eqid 2728 | . . . . 5 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
8 | eqid 2728 | . . . . 5 ⊢ (norm‘𝐻) = (norm‘𝐻) | |
9 | 3, 7, 8 | subgnm 24562 | . . . 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 2728 | . . . . 5 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
12 | 7, 11 | nrgabv 24598 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) ∈ (AbsVal‘𝐺)) |
13 | eqid 2728 | . . . . 5 ⊢ (AbsVal‘𝐻) = (AbsVal‘𝐻) | |
14 | 11, 3, 13 | abvres 20726 | . . . 4 ⊢ (((norm‘𝐺) ∈ (AbsVal‘𝐺) ∧ 𝐴 ∈ (SubRing‘𝐺)) → ((norm‘𝐺) ↾ 𝐴) ∈ (AbsVal‘𝐻)) |
15 | 12, 14 | sylan 578 | . . 3 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → ((norm‘𝐺) ↾ 𝐴) ∈ (AbsVal‘𝐻)) |
16 | 10, 15 | eqeltrd 2829 | . 2 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → (norm‘𝐻) ∈ (AbsVal‘𝐻)) |
17 | 8, 13 | isnrg 24597 | . 2 ⊢ (𝐻 ∈ NrmRing ↔ (𝐻 ∈ NrmGrp ∧ (norm‘𝐻) ∈ (AbsVal‘𝐻))) |
18 | 5, 16, 17 | sylanbrc 581 | 1 ⊢ ((𝐺 ∈ NrmRing ∧ 𝐴 ∈ (SubRing‘𝐺)) → 𝐻 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ↾ cres 5684 ‘cfv 6553 (class class class)co 7426 ↾s cress 17216 SubGrpcsubg 19082 SubRingcsubrg 20513 AbsValcabv 20703 normcnm 24505 NrmGrpcngp 24506 NrmRingcnrg 24508 |
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 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-sup 9473 df-inf 9474 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-q 12971 df-rp 13015 df-xneg 13132 df-xadd 13133 df-xmul 13134 df-ico 13370 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-tset 17259 df-ds 17262 df-rest 17411 df-topn 17412 df-0g 17430 df-topgen 17432 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-grp 18900 df-minusg 18901 df-sbg 18902 df-subg 19085 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-subrng 20490 df-subrg 20515 df-abv 20704 df-psmet 21278 df-xmet 21279 df-met 21280 df-bl 21281 df-mopn 21282 df-top 22816 df-topon 22833 df-topsp 22855 df-bases 22869 df-xms 24246 df-ms 24247 df-nm 24511 df-ngp 24512 df-nrg 24514 |
This theorem is referenced by: sranlm 24621 zringnrg 24724 isncvsngp 25097 tcphcph 25185 rezh 33605 rerrext 33643 |
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