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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > zrhnm | Structured version Visualization version GIF version |
Description: The norm of the image by ℤRHom of an integer in a normed ring. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
nmmulg.x | ⊢ 𝐵 = (Base‘𝑅) |
nmmulg.n | ⊢ 𝑁 = (norm‘𝑅) |
nmmulg.z | ⊢ 𝑍 = (ℤMod‘𝑅) |
zrhnm.1 | ⊢ 𝐿 = (ℤRHom‘𝑅) |
Ref | Expression |
---|---|
zrhnm | ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (abs‘𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl3 1250 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ NzRing) | |
2 | nzrring 19629 | . . . . 5 ⊢ (𝑅 ∈ NzRing → 𝑅 ∈ Ring) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑅 ∈ Ring) |
4 | simpr 479 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℤ) | |
5 | zrhnm.1 | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑅) | |
6 | eqid 2825 | . . . . . 6 ⊢ (.g‘𝑅) = (.g‘𝑅) | |
7 | eqid 2825 | . . . . . 6 ⊢ (1r‘𝑅) = (1r‘𝑅) | |
8 | 5, 6, 7 | zrhmulg 20225 | . . . . 5 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝐿‘𝑀) = (𝑀(.g‘𝑅)(1r‘𝑅))) |
9 | 8 | fveq2d 6441 | . . . 4 ⊢ ((𝑅 ∈ Ring ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅)))) |
10 | 3, 4, 9 | syl2anc 579 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅)))) |
11 | simpl1 1246 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmMod) | |
12 | nmmulg.x | . . . . . 6 ⊢ 𝐵 = (Base‘𝑅) | |
13 | 12, 7 | ringidcl 18929 | . . . . 5 ⊢ (𝑅 ∈ Ring → (1r‘𝑅) ∈ 𝐵) |
14 | 3, 13 | syl 17 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (1r‘𝑅) ∈ 𝐵) |
15 | nmmulg.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑅) | |
16 | nmmulg.z | . . . . 5 ⊢ 𝑍 = (ℤMod‘𝑅) | |
17 | 12, 15, 16, 6 | nmmulg 30553 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ (1r‘𝑅) ∈ 𝐵) → (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅))) = ((abs‘𝑀) · (𝑁‘(1r‘𝑅)))) |
18 | 11, 4, 14, 17 | syl3anc 1494 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝑀(.g‘𝑅)(1r‘𝑅))) = ((abs‘𝑀) · (𝑁‘(1r‘𝑅)))) |
19 | 16, 15 | zlmnm 30551 | . . . . . . 7 ⊢ (𝑅 ∈ NzRing → 𝑁 = (norm‘𝑍)) |
20 | 1, 19 | syl 17 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑁 = (norm‘𝑍)) |
21 | 20 | fveq1d 6439 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = ((norm‘𝑍)‘(1r‘𝑅))) |
22 | simpl2 1248 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NrmRing) | |
23 | nrgring 22844 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmRing → 𝑍 ∈ Ring) | |
24 | 22, 23 | syl 17 | . . . . . . 7 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ Ring) |
25 | eqid 2825 | . . . . . . . . 9 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
26 | 7, 25 | nzrnz 19628 | . . . . . . . 8 ⊢ (𝑅 ∈ NzRing → (1r‘𝑅) ≠ (0g‘𝑅)) |
27 | 1, 26 | syl 17 | . . . . . . 7 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (1r‘𝑅) ≠ (0g‘𝑅)) |
28 | 16, 7 | zlm1 30548 | . . . . . . . 8 ⊢ (1r‘𝑅) = (1r‘𝑍) |
29 | 16, 25 | zlm0 30547 | . . . . . . . 8 ⊢ (0g‘𝑅) = (0g‘𝑍) |
30 | 28, 29 | isnzr 19627 | . . . . . . 7 ⊢ (𝑍 ∈ NzRing ↔ (𝑍 ∈ Ring ∧ (1r‘𝑅) ≠ (0g‘𝑅))) |
31 | 24, 27, 30 | sylanbrc 578 | . . . . . 6 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑍 ∈ NzRing) |
32 | eqid 2825 | . . . . . . 7 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
33 | 32, 28 | nm1 22848 | . . . . . 6 ⊢ ((𝑍 ∈ NrmRing ∧ 𝑍 ∈ NzRing) → ((norm‘𝑍)‘(1r‘𝑅)) = 1) |
34 | 22, 31, 33 | syl2anc 579 | . . . . 5 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((norm‘𝑍)‘(1r‘𝑅)) = 1) |
35 | 21, 34 | eqtrd 2861 | . . . 4 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(1r‘𝑅)) = 1) |
36 | 35 | oveq2d 6926 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((abs‘𝑀) · (𝑁‘(1r‘𝑅))) = ((abs‘𝑀) · 1)) |
37 | 10, 18, 36 | 3eqtrd 2865 | . 2 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = ((abs‘𝑀) · 1)) |
38 | 4 | zcnd 11818 | . . 3 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → 𝑀 ∈ ℂ) |
39 | abscl 14402 | . . . 4 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℝ) | |
40 | 39 | recnd 10392 | . . 3 ⊢ (𝑀 ∈ ℂ → (abs‘𝑀) ∈ ℂ) |
41 | mulid1 10361 | . . 3 ⊢ ((abs‘𝑀) ∈ ℂ → ((abs‘𝑀) · 1) = (abs‘𝑀)) | |
42 | 38, 40, 41 | 3syl 18 | . 2 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → ((abs‘𝑀) · 1) = (abs‘𝑀)) |
43 | 37, 42 | eqtrd 2861 | 1 ⊢ (((𝑍 ∈ NrmMod ∧ 𝑍 ∈ NrmRing ∧ 𝑅 ∈ NzRing) ∧ 𝑀 ∈ ℤ) → (𝑁‘(𝐿‘𝑀)) = (abs‘𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ≠ wne 2999 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 1c1 10260 · cmul 10264 ℤcz 11711 abscabs 14358 Basecbs 16229 0gc0g 16460 .gcmg 17901 1rcur 18862 Ringcrg 18908 NzRingcnzr 19625 ℤRHomczrh 20215 ℤModczlm 20216 normcnm 22758 NrmRingcnrg 22761 NrmModcnlm 22762 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-rep 4996 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-inf2 8822 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336 ax-pre-sup 10337 ax-addf 10338 ax-mulf 10339 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rmo 3125 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-int 4700 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-1o 7831 df-oadd 7835 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-fin 8232 df-sup 8623 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-div 11017 df-nn 11358 df-2 11421 df-3 11422 df-4 11423 df-5 11424 df-6 11425 df-7 11426 df-8 11427 df-9 11428 df-n0 11626 df-z 11712 df-dec 11829 df-uz 11976 df-rp 12120 df-ico 12476 df-fz 12627 df-fzo 12768 df-seq 13103 df-exp 13162 df-cj 14223 df-re 14224 df-im 14225 df-sqrt 14359 df-abs 14360 df-struct 16231 df-ndx 16232 df-slot 16233 df-base 16235 df-sets 16236 df-ress 16237 df-plusg 16325 df-mulr 16326 df-starv 16327 df-sca 16328 df-vsca 16329 df-tset 16331 df-ple 16332 df-ds 16334 df-unif 16335 df-0g 16462 df-mgm 17602 df-sgrp 17644 df-mnd 17655 df-mhm 17695 df-grp 17786 df-minusg 17787 df-mulg 17902 df-subg 17949 df-ghm 18016 df-cmn 18555 df-abl 18556 df-mgp 18851 df-ur 18863 df-ring 18910 df-cring 18911 df-rnghom 19078 df-subrg 19141 df-abv 19180 df-lmod 19228 df-nzr 19626 df-cnfld 20114 df-zring 20186 df-zrh 20219 df-zlm 20220 df-nm 22764 df-nrg 22767 df-nlm 22768 |
This theorem is referenced by: qqhnm 30575 |
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