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 2610 | . . . . . . . 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 19685 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝑊) |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (Base‘𝐺) = (Base‘𝑊)) |
6 | eqid 2610 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 3, 6 | zlmplusg 19686 | . . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝑊) |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (+g‘𝐺) = (+g‘𝑊)) |
9 | 8 | oveqdr 6573 | . . . . . . 7 ⊢ ((𝐺 ∈ NrmRing ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑊)𝑦)) |
10 | 2, 5, 9 | grppropd 17260 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ Grp ↔ 𝑊 ∈ Grp)) |
11 | eqid 2610 | . . . . . . . . 9 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
12 | 3, 11 | zlmds 29336 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (dist‘𝐺) = (dist‘𝑊)) |
13 | 12 | reseq1d 5316 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝑊) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
14 | eqid 2610 | . . . . . . . . 9 ⊢ (TopSet‘𝐺) = (TopSet‘𝐺) | |
15 | 3, 14 | zlmtset 29337 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (TopSet‘𝐺) = (TopSet‘𝑊)) |
16 | 5, 15 | topnpropd 15920 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (TopOpen‘𝐺) = (TopOpen‘𝑊)) |
17 | 2, 5, 13, 16 | mspropd 22089 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ MetSp ↔ 𝑊 ∈ MetSp)) |
18 | eqid 2610 | . . . . . . . . 9 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
19 | 3, 18 | zlmnm 29338 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) = (norm‘𝑊)) |
20 | 5, 8 | grpsubpropd 17343 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (-g‘𝐺) = (-g‘𝑊)) |
21 | 19, 20 | coeq12d 5208 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∘ (-g‘𝐺)) = ((norm‘𝑊) ∘ (-g‘𝑊))) |
22 | 21, 12 | sseq12d 3597 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺) ↔ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
23 | 10, 17, 22 | 3anbi123d 1391 | . . . . 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 2610 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
25 | 18, 24, 11 | isngp 22210 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
26 | eqid 2610 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
27 | eqid 2610 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
28 | eqid 2610 | . . . . . 6 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
29 | 26, 27, 28 | isngp 22210 | . . . . 5 ⊢ (𝑊 ∈ NrmGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ MetSp ∧ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
30 | 23, 25, 29 | 3bitr4g 302 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmGrp ↔ 𝑊 ∈ NrmGrp)) |
31 | eqid 2610 | . . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
32 | 3, 31 | zlmmulr 19687 | . . . . . . 7 ⊢ (.r‘𝐺) = (.r‘𝑊) |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (.r‘𝐺) = (.r‘𝑊)) |
34 | 5, 8, 33 | abvpropd2 28983 | . . . . 5 ⊢ (𝐺 ∈ NrmRing → (AbsVal‘𝐺) = (AbsVal‘𝑊)) |
35 | 19, 34 | eleq12d 2682 | . . . 4 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∈ (AbsVal‘𝐺) ↔ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
36 | 30, 35 | anbi12d 743 | . . 3 ⊢ (𝐺 ∈ NrmRing → ((𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺)) ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊)))) |
37 | eqid 2610 | . . . 4 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
38 | 18, 37 | isnrg 22274 | . . 3 ⊢ (𝐺 ∈ NrmRing ↔ (𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺))) |
39 | eqid 2610 | . . . 4 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊) | |
40 | 26, 39 | isnrg 22274 | . . 3 ⊢ (𝑊 ∈ NrmRing ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
41 | 36, 38, 40 | 3bitr4g 302 | . 2 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmRing ↔ 𝑊 ∈ NrmRing)) |
42 | 41 | ibi 255 | 1 ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 × cxp 5036 ∘ ccom 5042 ‘cfv 5804 Basecbs 15695 +gcplusg 15768 .rcmulr 15769 TopSetcts 15774 distcds 15777 Grpcgrp 17245 -gcsg 17247 AbsValcabv 18639 ℤModczlm 19668 MetSpcmt 21933 normcnm 22191 NrmGrpcngp 22192 NrmRingcnrg 22194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-plusg 15781 df-mulr 15782 df-sca 15784 df-vsca 15785 df-tset 15787 df-ds 15791 df-rest 15906 df-topn 15907 df-0g 15925 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mgp 18313 df-ring 18372 df-abv 18640 df-zlm 19672 df-top 20521 df-topon 20523 df-topsp 20524 df-xms 21935 df-ms 21936 df-nm 22197 df-ngp 22198 df-nrg 22200 |
This theorem is referenced by: cnzh 29342 rezh 29343 qqhnm 29362 |
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