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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zhmnrg | Structured version Visualization version GIF version |
Description: The ℤ-module built from a normed ring is also a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
zlmlem2.1 | ⊢ 𝑊 = (ℤMod‘𝐺) |
Ref | Expression |
---|---|
zhmnrg | ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝐺) | |
2 | 1 | a1i 11 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (Base‘𝐺) = (Base‘𝐺)) |
3 | zlmlem2.1 | . . . . . . . . 9 ⊢ 𝑊 = (ℤMod‘𝐺) | |
4 | 3, 1 | zlmbas 21052 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝑊) |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (Base‘𝐺) = (Base‘𝑊)) |
6 | eqid 2733 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 3, 6 | zlmplusg 21054 | . . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝑊) |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (+g‘𝐺) = (+g‘𝑊)) |
9 | 8 | oveqdr 7432 | . . . . . . 7 ⊢ ((𝐺 ∈ NrmRing ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑊)𝑦)) |
10 | 2, 5, 9 | grppropd 18833 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ Grp ↔ 𝑊 ∈ Grp)) |
11 | eqid 2733 | . . . . . . . . 9 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
12 | 3, 11 | zlmds 32880 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (dist‘𝐺) = (dist‘𝑊)) |
13 | 12 | reseq1d 5978 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝑊) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
14 | eqid 2733 | . . . . . . . . 9 ⊢ (TopSet‘𝐺) = (TopSet‘𝐺) | |
15 | 3, 14 | zlmtset 32882 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (TopSet‘𝐺) = (TopSet‘𝑊)) |
16 | 5, 15 | topnpropd 17378 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (TopOpen‘𝐺) = (TopOpen‘𝑊)) |
17 | 2, 5, 13, 16 | mspropd 23962 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ MetSp ↔ 𝑊 ∈ MetSp)) |
18 | eqid 2733 | . . . . . . . . 9 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
19 | 3, 18 | zlmnm 32884 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) = (norm‘𝑊)) |
20 | 5, 8 | grpsubpropd 18924 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (-g‘𝐺) = (-g‘𝑊)) |
21 | 19, 20 | coeq12d 5862 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∘ (-g‘𝐺)) = ((norm‘𝑊) ∘ (-g‘𝑊))) |
22 | 21, 12 | sseq12d 4014 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺) ↔ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
23 | 10, 17, 22 | 3anbi123d 1437 | . . . . 5 ⊢ (𝐺 ∈ NrmRing → ((𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺)) ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ MetSp ∧ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊)))) |
24 | eqid 2733 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
25 | 18, 24, 11 | isngp 24087 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
26 | eqid 2733 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
27 | eqid 2733 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
28 | eqid 2733 | . . . . . 6 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
29 | 26, 27, 28 | isngp 24087 | . . . . 5 ⊢ (𝑊 ∈ NrmGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ MetSp ∧ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
30 | 23, 25, 29 | 3bitr4g 314 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmGrp ↔ 𝑊 ∈ NrmGrp)) |
31 | eqid 2733 | . . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
32 | 3, 31 | zlmmulr 21056 | . . . . . . 7 ⊢ (.r‘𝐺) = (.r‘𝑊) |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (.r‘𝐺) = (.r‘𝑊)) |
34 | 5, 8, 33 | abvpropd2 32107 | . . . . 5 ⊢ (𝐺 ∈ NrmRing → (AbsVal‘𝐺) = (AbsVal‘𝑊)) |
35 | 19, 34 | eleq12d 2828 | . . . 4 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∈ (AbsVal‘𝐺) ↔ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
36 | 30, 35 | anbi12d 632 | . . 3 ⊢ (𝐺 ∈ NrmRing → ((𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺)) ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊)))) |
37 | eqid 2733 | . . . 4 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
38 | 18, 37 | isnrg 24159 | . . 3 ⊢ (𝐺 ∈ NrmRing ↔ (𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺))) |
39 | eqid 2733 | . . . 4 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊) | |
40 | 26, 39 | isnrg 24159 | . . 3 ⊢ (𝑊 ∈ NrmRing ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
41 | 36, 38, 40 | 3bitr4g 314 | . 2 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmRing ↔ 𝑊 ∈ NrmRing)) |
42 | 41 | ibi 267 | 1 ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ⊆ wss 3947 × cxp 5673 ∘ ccom 5679 ‘cfv 6540 Basecbs 17140 +gcplusg 17193 .rcmulr 17194 TopSetcts 17199 distcds 17202 Grpcgrp 18815 -gcsg 18817 AbsValcabv 20412 ℤModczlm 21034 MetSpcms 23806 normcnm 24067 NrmGrpcngp 24068 NrmRingcnrg 24070 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7720 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-om 7851 df-1st 7970 df-2nd 7971 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-plusg 17206 df-mulr 17207 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ds 17215 df-rest 17364 df-topn 17365 df-0g 17383 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mgp 19980 df-ring 20049 df-abv 20413 df-zlm 21038 df-top 22378 df-topon 22395 df-topsp 22417 df-xms 23808 df-ms 23809 df-nm 24073 df-ngp 24074 df-nrg 24076 |
This theorem is referenced by: cnzh 32888 rezh 32889 qqhnm 32908 |
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