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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 19586 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | ringgrp 19296 | . . . 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 2821 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
7 | 5, 6 | tngds 23251 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) = (dist‘𝑇)) |
8 | eqid 2821 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 1, 6 | abvmet 23179 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) ∈ (Met‘(Base‘𝑅))) |
10 | 7, 9 | eqeltrrd 2914 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (dist‘𝑇) ∈ (Met‘(Base‘𝑅))) |
11 | 1, 8 | abvf 19588 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹:(Base‘𝑅)⟶ℝ) |
12 | eqid 2821 | . . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
13 | 5, 8, 12 | tngngp2 23255 | . . . 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 711 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmGrp) |
16 | reex 10622 | . . . . . 6 ⊢ ℝ ∈ V | |
17 | 5, 8, 16 | tngnm 23254 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐹:(Base‘𝑅)⟶ℝ) → 𝐹 = (norm‘𝑇)) |
18 | 4, 11, 17 | syl2anc 586 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 = (norm‘𝑇)) |
19 | eqidd 2822 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑅)) | |
20 | 5, 8 | tngbas 23244 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑇)) |
21 | eqid 2821 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
22 | 5, 21 | tngplusg 23245 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (+g‘𝑅) = (+g‘𝑇)) |
23 | 22 | oveqdr 7178 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
24 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
25 | 5, 24 | tngmulr 23247 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (.r‘𝑅) = (.r‘𝑇)) |
26 | 25 | oveqdr 7178 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑇)𝑦)) |
27 | 19, 20, 23, 26 | abvpropd 19607 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → (AbsVal‘𝑅) = (AbsVal‘𝑇)) |
28 | 1, 27 | syl5eq 2868 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐴 = (AbsVal‘𝑇)) |
29 | 18, 28 | eleq12d 2907 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ 𝐴 ↔ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
30 | 29 | ibi 269 | . 2 ⊢ (𝐹 ∈ 𝐴 → (norm‘𝑇) ∈ (AbsVal‘𝑇)) |
31 | eqid 2821 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
32 | eqid 2821 | . . 3 ⊢ (AbsVal‘𝑇) = (AbsVal‘𝑇) | |
33 | 31, 32 | isnrg 23263 | . 2 ⊢ (𝑇 ∈ NrmRing ↔ (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
34 | 15, 30, 33 | sylanbrc 585 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∘ ccom 5554 ⟶wf 6346 ‘cfv 6350 (class class class)co 7150 ℝcr 10530 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 distcds 16568 Grpcgrp 18097 -gcsg 18099 Ringcrg 19291 AbsValcabv 19581 Metcmet 20525 normcnm 23180 NrmGrpcngp 23181 toNrmGrp ctng 23182 NrmRingcnrg 23183 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ico 12738 df-seq 13364 df-exp 13424 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-plusg 16572 df-mulr 16573 df-tset 16578 df-ds 16581 df-rest 16690 df-topn 16691 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mgp 19234 df-ur 19246 df-ring 19293 df-abv 19582 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 df-tng 23188 df-nrg 23189 |
This theorem is referenced by: (None) |
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