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Mirrors > Home > MPE Home > Th. List > tngnrg | Structured version Visualization version GIF version |
Description: Given any absolute value over a ring, augmenting the ring with the absolute value produces a normed ring. (Contributed by Mario Carneiro, 4-Oct-2015.) |
Ref | Expression |
---|---|
tngnrg.t | ⊢ 𝑇 = (𝑅 toNrmGrp 𝐹) |
tngnrg.a | ⊢ 𝐴 = (AbsVal‘𝑅) |
Ref | Expression |
---|---|
tngnrg | ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tngnrg.a | . . . . 5 ⊢ 𝐴 = (AbsVal‘𝑅) | |
2 | 1 | abvrcl 19184 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | ringgrp 18913 | . . . 4 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
4 | 2, 3 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Grp) |
5 | tngnrg.t | . . . . 5 ⊢ 𝑇 = (𝑅 toNrmGrp 𝐹) | |
6 | eqid 2825 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
7 | 5, 6 | tngds 22829 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) = (dist‘𝑇)) |
8 | eqid 2825 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 1, 6 | abvmet 22757 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) ∈ (Met‘(Base‘𝑅))) |
10 | 7, 9 | eqeltrrd 2907 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (dist‘𝑇) ∈ (Met‘(Base‘𝑅))) |
11 | 1, 8 | abvf 19186 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹:(Base‘𝑅)⟶ℝ) |
12 | eqid 2825 | . . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
13 | 5, 8, 12 | tngngp2 22833 | . . . 4 ⊢ (𝐹:(Base‘𝑅)⟶ℝ → (𝑇 ∈ NrmGrp ↔ (𝑅 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝑅))))) |
14 | 11, 13 | syl 17 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝑇 ∈ NrmGrp ↔ (𝑅 ∈ Grp ∧ (dist‘𝑇) ∈ (Met‘(Base‘𝑅))))) |
15 | 4, 10, 14 | mpbir2and 704 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmGrp) |
16 | reex 10350 | . . . . . 6 ⊢ ℝ ∈ V | |
17 | 5, 8, 16 | tngnm 22832 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐹:(Base‘𝑅)⟶ℝ) → 𝐹 = (norm‘𝑇)) |
18 | 4, 11, 17 | syl2anc 579 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 = (norm‘𝑇)) |
19 | eqidd 2826 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑅)) | |
20 | 5, 8 | tngbas 22822 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑇)) |
21 | eqid 2825 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
22 | 5, 21 | tngplusg 22823 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (+g‘𝑅) = (+g‘𝑇)) |
23 | 22 | oveqdr 6938 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
24 | eqid 2825 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
25 | 5, 24 | tngmulr 22825 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (.r‘𝑅) = (.r‘𝑇)) |
26 | 25 | oveqdr 6938 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑇)𝑦)) |
27 | 19, 20, 23, 26 | abvpropd 19205 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → (AbsVal‘𝑅) = (AbsVal‘𝑇)) |
28 | 1, 27 | syl5eq 2873 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐴 = (AbsVal‘𝑇)) |
29 | 18, 28 | eleq12d 2900 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ 𝐴 ↔ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
30 | 29 | ibi 259 | . 2 ⊢ (𝐹 ∈ 𝐴 → (norm‘𝑇) ∈ (AbsVal‘𝑇)) |
31 | eqid 2825 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
32 | eqid 2825 | . . 3 ⊢ (AbsVal‘𝑇) = (AbsVal‘𝑇) | |
33 | 31, 32 | isnrg 22841 | . 2 ⊢ (𝑇 ∈ NrmRing ↔ (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
34 | 15, 30, 33 | sylanbrc 578 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1656 ∈ wcel 2164 ∘ ccom 5350 ⟶wf 6123 ‘cfv 6127 (class class class)co 6910 ℝcr 10258 Basecbs 16229 +gcplusg 16312 .rcmulr 16313 distcds 16321 Grpcgrp 17783 -gcsg 17785 Ringcrg 18908 AbsValcabv 19179 Metcmet 20099 normcnm 22758 NrmGrpcngp 22759 toNrmGrp ctng 22760 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-seq 13103 df-exp 13162 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 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-mgp 18851 df-ur 18863 df-ring 18910 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-tng 22766 df-nrg 22767 |
This theorem is referenced by: (None) |
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