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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 19585 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝑅 ∈ Ring) |
3 | ringgrp 19295 | . . . 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 2798 | . . . . 5 ⊢ (-g‘𝑅) = (-g‘𝑅) | |
7 | 5, 6 | tngds 23254 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) = (dist‘𝑇)) |
8 | eqid 2798 | . . . . 5 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
9 | 8, 1, 6 | abvmet 23182 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∘ (-g‘𝑅)) ∈ (Met‘(Base‘𝑅))) |
10 | 7, 9 | eqeltrrd 2891 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (dist‘𝑇) ∈ (Met‘(Base‘𝑅))) |
11 | 1, 8 | abvf 19587 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹:(Base‘𝑅)⟶ℝ) |
12 | eqid 2798 | . . . . 5 ⊢ (dist‘𝑇) = (dist‘𝑇) | |
13 | 5, 8, 12 | tngngp2 23258 | . . . 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 712 | . 2 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmGrp) |
16 | reex 10617 | . . . . . 6 ⊢ ℝ ∈ V | |
17 | 5, 8, 16 | tngnm 23257 | . . . . 5 ⊢ ((𝑅 ∈ Grp ∧ 𝐹:(Base‘𝑅)⟶ℝ) → 𝐹 = (norm‘𝑇)) |
18 | 4, 11, 17 | syl2anc 587 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐹 = (norm‘𝑇)) |
19 | eqidd 2799 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑅)) | |
20 | 5, 8 | tngbas 23247 | . . . . . 6 ⊢ (𝐹 ∈ 𝐴 → (Base‘𝑅) = (Base‘𝑇)) |
21 | eqid 2798 | . . . . . . . 8 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
22 | 5, 21 | tngplusg 23248 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (+g‘𝑅) = (+g‘𝑇)) |
23 | 22 | oveqdr 7163 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(+g‘𝑅)𝑦) = (𝑥(+g‘𝑇)𝑦)) |
24 | eqid 2798 | . . . . . . . 8 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
25 | 5, 24 | tngmulr 23250 | . . . . . . 7 ⊢ (𝐹 ∈ 𝐴 → (.r‘𝑅) = (.r‘𝑇)) |
26 | 25 | oveqdr 7163 | . . . . . 6 ⊢ ((𝐹 ∈ 𝐴 ∧ (𝑥 ∈ (Base‘𝑅) ∧ 𝑦 ∈ (Base‘𝑅))) → (𝑥(.r‘𝑅)𝑦) = (𝑥(.r‘𝑇)𝑦)) |
27 | 19, 20, 23, 26 | abvpropd 19606 | . . . . 5 ⊢ (𝐹 ∈ 𝐴 → (AbsVal‘𝑅) = (AbsVal‘𝑇)) |
28 | 1, 27 | syl5eq 2845 | . . . 4 ⊢ (𝐹 ∈ 𝐴 → 𝐴 = (AbsVal‘𝑇)) |
29 | 18, 28 | eleq12d 2884 | . . 3 ⊢ (𝐹 ∈ 𝐴 → (𝐹 ∈ 𝐴 ↔ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
30 | 29 | ibi 270 | . 2 ⊢ (𝐹 ∈ 𝐴 → (norm‘𝑇) ∈ (AbsVal‘𝑇)) |
31 | eqid 2798 | . . 3 ⊢ (norm‘𝑇) = (norm‘𝑇) | |
32 | eqid 2798 | . . 3 ⊢ (AbsVal‘𝑇) = (AbsVal‘𝑇) | |
33 | 31, 32 | isnrg 23266 | . 2 ⊢ (𝑇 ∈ NrmRing ↔ (𝑇 ∈ NrmGrp ∧ (norm‘𝑇) ∈ (AbsVal‘𝑇))) |
34 | 15, 30, 33 | sylanbrc 586 | 1 ⊢ (𝐹 ∈ 𝐴 → 𝑇 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 distcds 16566 Grpcgrp 18095 -gcsg 18097 Ringcrg 19290 AbsValcabv 19580 Metcmet 20077 normcnm 23183 NrmGrpcngp 23184 toNrmGrp ctng 23185 NrmRingcnrg 23186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-sup 8890 df-inf 8891 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ico 12732 df-seq 13365 df-exp 13426 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-plusg 16570 df-mulr 16571 df-tset 16576 df-ds 16579 df-rest 16688 df-topn 16689 df-0g 16707 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mgp 19233 df-ur 19245 df-ring 19292 df-abv 19581 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-xms 22927 df-ms 22928 df-nm 23189 df-ngp 23190 df-tng 23191 df-nrg 23192 |
This theorem is referenced by: (None) |
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