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 2821 | . . . . . . . 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 20665 | . . . . . . . 8 ⊢ (Base‘𝐺) = (Base‘𝑊) |
5 | 4 | a1i 11 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (Base‘𝐺) = (Base‘𝑊)) |
6 | eqid 2821 | . . . . . . . . . 10 ⊢ (+g‘𝐺) = (+g‘𝐺) | |
7 | 3, 6 | zlmplusg 20666 | . . . . . . . . 9 ⊢ (+g‘𝐺) = (+g‘𝑊) |
8 | 7 | a1i 11 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (+g‘𝐺) = (+g‘𝑊)) |
9 | 8 | oveqdr 7184 | . . . . . . 7 ⊢ ((𝐺 ∈ NrmRing ∧ (𝑥 ∈ (Base‘𝐺) ∧ 𝑦 ∈ (Base‘𝐺))) → (𝑥(+g‘𝐺)𝑦) = (𝑥(+g‘𝑊)𝑦)) |
10 | 2, 5, 9 | grppropd 18118 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ Grp ↔ 𝑊 ∈ Grp)) |
11 | eqid 2821 | . . . . . . . . 9 ⊢ (dist‘𝐺) = (dist‘𝐺) | |
12 | 3, 11 | zlmds 31205 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (dist‘𝐺) = (dist‘𝑊)) |
13 | 12 | reseq1d 5852 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((dist‘𝐺) ↾ ((Base‘𝐺) × (Base‘𝐺))) = ((dist‘𝑊) ↾ ((Base‘𝐺) × (Base‘𝐺)))) |
14 | eqid 2821 | . . . . . . . . 9 ⊢ (TopSet‘𝐺) = (TopSet‘𝐺) | |
15 | 3, 14 | zlmtset 31206 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (TopSet‘𝐺) = (TopSet‘𝑊)) |
16 | 5, 15 | topnpropd 16710 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → (TopOpen‘𝐺) = (TopOpen‘𝑊)) |
17 | 2, 5, 13, 16 | mspropd 23084 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ MetSp ↔ 𝑊 ∈ MetSp)) |
18 | eqid 2821 | . . . . . . . . 9 ⊢ (norm‘𝐺) = (norm‘𝐺) | |
19 | 3, 18 | zlmnm 31207 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (norm‘𝐺) = (norm‘𝑊)) |
20 | 5, 8 | grpsubpropd 18204 | . . . . . . . 8 ⊢ (𝐺 ∈ NrmRing → (-g‘𝐺) = (-g‘𝑊)) |
21 | 19, 20 | coeq12d 5735 | . . . . . . 7 ⊢ (𝐺 ∈ NrmRing → ((norm‘𝐺) ∘ (-g‘𝐺)) = ((norm‘𝑊) ∘ (-g‘𝑊))) |
22 | 21, 12 | sseq12d 4000 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺) ↔ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
23 | 10, 17, 22 | 3anbi123d 1432 | . . . . 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 2821 | . . . . . 6 ⊢ (-g‘𝐺) = (-g‘𝐺) | |
25 | 18, 24, 11 | isngp 23205 | . . . . 5 ⊢ (𝐺 ∈ NrmGrp ↔ (𝐺 ∈ Grp ∧ 𝐺 ∈ MetSp ∧ ((norm‘𝐺) ∘ (-g‘𝐺)) ⊆ (dist‘𝐺))) |
26 | eqid 2821 | . . . . . 6 ⊢ (norm‘𝑊) = (norm‘𝑊) | |
27 | eqid 2821 | . . . . . 6 ⊢ (-g‘𝑊) = (-g‘𝑊) | |
28 | eqid 2821 | . . . . . 6 ⊢ (dist‘𝑊) = (dist‘𝑊) | |
29 | 26, 27, 28 | isngp 23205 | . . . . 5 ⊢ (𝑊 ∈ NrmGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ MetSp ∧ ((norm‘𝑊) ∘ (-g‘𝑊)) ⊆ (dist‘𝑊))) |
30 | 23, 25, 29 | 3bitr4g 316 | . . . 4 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmGrp ↔ 𝑊 ∈ NrmGrp)) |
31 | eqid 2821 | . . . . . . . 8 ⊢ (.r‘𝐺) = (.r‘𝐺) | |
32 | 3, 31 | zlmmulr 20667 | . . . . . . 7 ⊢ (.r‘𝐺) = (.r‘𝑊) |
33 | 32 | a1i 11 | . . . . . 6 ⊢ (𝐺 ∈ NrmRing → (.r‘𝐺) = (.r‘𝑊)) |
34 | 5, 8, 33 | abvpropd2 30639 | . . . . 5 ⊢ (𝐺 ∈ NrmRing → (AbsVal‘𝐺) = (AbsVal‘𝑊)) |
35 | 19, 34 | eleq12d 2907 | . . . 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 2821 | . . . 4 ⊢ (AbsVal‘𝐺) = (AbsVal‘𝐺) | |
38 | 18, 37 | isnrg 23269 | . . 3 ⊢ (𝐺 ∈ NrmRing ↔ (𝐺 ∈ NrmGrp ∧ (norm‘𝐺) ∈ (AbsVal‘𝐺))) |
39 | eqid 2821 | . . . 4 ⊢ (AbsVal‘𝑊) = (AbsVal‘𝑊) | |
40 | 26, 39 | isnrg 23269 | . . 3 ⊢ (𝑊 ∈ NrmRing ↔ (𝑊 ∈ NrmGrp ∧ (norm‘𝑊) ∈ (AbsVal‘𝑊))) |
41 | 36, 38, 40 | 3bitr4g 316 | . 2 ⊢ (𝐺 ∈ NrmRing → (𝐺 ∈ NrmRing ↔ 𝑊 ∈ NrmRing)) |
42 | 41 | ibi 269 | 1 ⊢ (𝐺 ∈ NrmRing → 𝑊 ∈ NrmRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ⊆ wss 3936 × cxp 5553 ∘ ccom 5559 ‘cfv 6355 Basecbs 16483 +gcplusg 16565 .rcmulr 16566 TopSetcts 16571 distcds 16574 Grpcgrp 18103 -gcsg 18105 AbsValcabv 19587 ℤModczlm 20648 MetSpcms 22928 normcnm 23186 NrmGrpcngp 23187 NrmRingcnrg 23189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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-rab 3147 df-v 3496 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 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-ndx 16486 df-slot 16487 df-base 16489 df-sets 16490 df-plusg 16578 df-mulr 16579 df-sca 16581 df-vsca 16582 df-tset 16584 df-ds 16587 df-rest 16696 df-topn 16697 df-0g 16715 df-mgm 17852 df-sgrp 17901 df-mnd 17912 df-grp 18106 df-minusg 18107 df-sbg 18108 df-mgp 19240 df-ring 19299 df-abv 19588 df-zlm 20652 df-top 21502 df-topon 21519 df-topsp 21541 df-xms 22930 df-ms 22931 df-nm 23192 df-ngp 23193 df-nrg 23195 |
This theorem is referenced by: cnzh 31211 rezh 31212 qqhnm 31231 |
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