Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 2622 | . . . . . . . 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 19866 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝑊) |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (Base‘𝐺) = (Base‘𝑊)) |
6 | eqid 2622 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 3, 6 | zlmplusg 19867 | . . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝑊) |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (+g‘𝐺) = (+g‘𝑊)) |
9 | 8 | oveqdr 6674 | . . . . . . 7 ⊢ ((𝐺 ∈ NrmRing ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑊)𝑦)) |
10 | 2, 5, 9 | grppropd 17437 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ Grp ↔ 𝑊 ∈ Grp)) |
11 | eqid 2622 | . . . . . . . . 9 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
12 | 3, 11 | zlmds 30008 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (dist‘𝐺) = (dist‘𝑊)) |
13 | 12 | reseq1d 5395 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝑊) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
14 | eqid 2622 | . . . . . . . . 9 ⊢ (TopSet‘𝐺) = (TopSet‘𝐺) | |
15 | 3, 14 | zlmtset 30009 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (TopSet‘𝐺) = (TopSet‘𝑊)) |
16 | 5, 15 | topnpropd 16097 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (TopOpen‘𝐺) = (TopOpen‘𝑊)) |
17 | 2, 5, 13, 16 | mspropd 22279 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ MetSp ↔ 𝑊 ∈ MetSp)) |
18 | eqid 2622 | . . . . . . . . 9 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
19 | 3, 18 | zlmnm 30010 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) = (norm‘𝑊)) |
20 | 5, 8 | grpsubpropd 17520 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (-g‘𝐺) = (-g‘𝑊)) |
21 | 19, 20 | coeq12d 5286 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∘ (-g‘𝐺)) = ((norm‘𝑊) ∘ (-g‘𝑊))) |
22 | 21, 12 | sseq12d 3634 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺) ↔ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
23 | 10, 17, 22 | 3anbi123d 1399 | . . . . 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 2622 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
25 | 18, 24, 11 | isngp 22400 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
26 | eqid 2622 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
27 | eqid 2622 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
28 | eqid 2622 | . . . . . 6 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
29 | 26, 27, 28 | isngp 22400 | . . . . 5 ⊢ (𝑊 ∈ NrmGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ MetSp ∧ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
30 | 23, 25, 29 | 3bitr4g 303 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmGrp ↔ 𝑊 ∈ NrmGrp)) |
31 | eqid 2622 | . . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
32 | 3, 31 | zlmmulr 19868 | . . . . . . 7 ⊢ (.r‘𝐺) = (.r‘𝑊) |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (.r‘𝐺) = (.r‘𝑊)) |
34 | 5, 8, 33 | abvpropd2 29652 | . . . . 5 ⊢ (𝐺 ∈ NrmRing → (AbsVal‘𝐺) = (AbsVal‘𝑊)) |
35 | 19, 34 | eleq12d 2695 | . . . 4 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∈ (AbsVal‘𝐺) ↔ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
36 | 30, 35 | anbi12d 747 | . . 3 ⊢ (𝐺 ∈ NrmRing → ((𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺)) ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊)))) |
37 | eqid 2622 | . . . 4 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
38 | 18, 37 | isnrg 22464 | . . 3 ⊢ (𝐺 ∈ NrmRing ↔ (𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺))) |
39 | eqid 2622 | . . . 4 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊) | |
40 | 26, 39 | isnrg 22464 | . . 3 ⊢ (𝑊 ∈ NrmRing ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
41 | 36, 38, 40 | 3bitr4g 303 | . 2 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmRing ↔ 𝑊 ∈ NrmRing)) |
42 | 41 | ibi 256 | 1 ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 × cxp 5112 ∘ ccom 5118 ‘cfv 5888 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 TopSetcts 15947 distcds 15950 Grpcgrp 17422 -gcsg 17424 AbsValcabv 18816 ℤModczlm 19849 MetSpcmt 22123 normcnm 22381 NrmGrpcngp 22382 NrmRingcnrg 22384 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-plusg 15954 df-mulr 15955 df-sca 15957 df-vsca 15958 df-tset 15960 df-ds 15964 df-rest 16083 df-topn 16084 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mgp 18490 df-ring 18549 df-abv 18817 df-zlm 19853 df-top 20699 df-topon 20716 df-topsp 20737 df-xms 22125 df-ms 22126 df-nm 22387 df-ngp 22388 df-nrg 22390 |
This theorem is referenced by: cnzh 30014 rezh 30015 qqhnm 30034 |
Copyright terms: Public domain | W3C validator |