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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > nmmulg | Structured version Visualization version GIF version |
Description: The norm of a group product, provided the ℤ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.) |
Ref | Expression |
---|---|
nmmulg.x | ⊢ 𝐵 = (Base‘𝑅) |
nmmulg.n | ⊢ 𝑁 = (norm‘𝑅) |
nmmulg.z | ⊢ 𝑍 = (ℤMod‘𝑅) |
nmmulg.t | ⊢ · = (.g‘𝑅) |
Ref | Expression |
---|---|
nmmulg | ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 1128 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ ℤ) | |
2 | zringbas 20220 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | nlmlmod 22890 | . . . . . . . . 9 ⊢ (𝑍 ∈ NrmMod → 𝑍 ∈ LMod) | |
4 | nmmulg.z | . . . . . . . . . 10 ⊢ 𝑍 = (ℤMod‘𝑅) | |
5 | 4 | zlmlmod 20267 | . . . . . . . . 9 ⊢ (𝑅 ∈ Abel ↔ 𝑍 ∈ LMod) |
6 | 3, 5 | sylibr 226 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmMod → 𝑅 ∈ Abel) |
7 | 6 | 3ad2ant1 1124 | . . . . . . 7 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Abel) |
8 | 4 | zlmsca 20265 | . . . . . . 7 ⊢ (𝑅 ∈ Abel → ℤring = (Scalar‘𝑍)) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤring = (Scalar‘𝑍)) |
10 | 9 | fveq2d 6450 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (Base‘ℤring) = (Base‘(Scalar‘𝑍))) |
11 | 2, 10 | syl5eq 2826 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤ = (Base‘(Scalar‘𝑍))) |
12 | 1, 11 | eleqtrd 2861 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ (Base‘(Scalar‘𝑍))) |
13 | nmmulg.x | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
14 | 4, 13 | zlmbas 20262 | . . . 4 ⊢ 𝐵 = (Base‘𝑍) |
15 | eqid 2778 | . . . 4 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
16 | nmmulg.t | . . . . 5 ⊢ · = (.g‘𝑅) | |
17 | 4, 16 | zlmvsca 20266 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑍) |
18 | eqid 2778 | . . . 4 ⊢ (Scalar‘𝑍) = (Scalar‘𝑍) | |
19 | eqid 2778 | . . . 4 ⊢ (Base‘(Scalar‘𝑍)) = (Base‘(Scalar‘𝑍)) | |
20 | eqid 2778 | . . . 4 ⊢ (norm‘(Scalar‘𝑍)) = (norm‘(Scalar‘𝑍)) | |
21 | 14, 15, 17, 18, 19, 20 | nmvs 22888 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑍)) ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
22 | 12, 21 | syld3an2 1480 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
23 | nmmulg.n | . . . . 5 ⊢ 𝑁 = (norm‘𝑅) | |
24 | 4, 23 | zlmnm 30608 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑁 = (norm‘𝑍)) |
25 | 7, 24 | syl 17 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑁 = (norm‘𝑍)) |
26 | 25 | fveq1d 6448 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((norm‘𝑍)‘(𝑀 · 𝑋))) |
27 | zzsnm 30603 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) | |
28 | 27 | 3ad2ant2 1125 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) |
29 | 9 | fveq2d 6450 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (norm‘ℤring) = (norm‘(Scalar‘𝑍))) |
30 | 29 | fveq1d 6448 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘ℤring)‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
31 | 28, 30 | eqtrd 2814 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
32 | 25 | fveq1d 6448 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ((norm‘𝑍)‘𝑋)) |
33 | 31, 32 | oveq12d 6940 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((abs‘𝑀) · (𝑁‘𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
34 | 22, 26, 33 | 3eqtr4d 2824 | 1 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ‘cfv 6135 (class class class)co 6922 · cmul 10277 ℤcz 11728 abscabs 14381 Basecbs 16255 Scalarcsca 16341 .gcmg 17927 Abelcabl 18580 LModclmod 19255 ℤringzring 20214 ℤModczlm 20245 normcnm 22789 NrmModcnlm 22793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-inf2 8835 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 ax-pre-sup 10350 ax-addf 10351 ax-mulf 10352 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-nel 3076 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-oadd 7847 df-er 8026 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-div 11033 df-nn 11375 df-2 11438 df-3 11439 df-4 11440 df-5 11441 df-6 11442 df-7 11443 df-8 11444 df-9 11445 df-n0 11643 df-z 11729 df-dec 11846 df-uz 11993 df-rp 12138 df-fz 12644 df-fzo 12785 df-seq 13120 df-exp 13179 df-cj 14246 df-re 14247 df-im 14248 df-sqrt 14382 df-abs 14383 df-struct 16257 df-ndx 16258 df-slot 16259 df-base 16261 df-sets 16262 df-ress 16263 df-plusg 16351 df-mulr 16352 df-starv 16353 df-sca 16354 df-vsca 16355 df-tset 16357 df-ple 16358 df-ds 16360 df-unif 16361 df-0g 16488 df-mgm 17628 df-sgrp 17670 df-mnd 17681 df-grp 17812 df-minusg 17813 df-mulg 17928 df-subg 17975 df-cmn 18581 df-abl 18582 df-mgp 18877 df-ur 18889 df-ring 18936 df-cring 18937 df-subrg 19170 df-lmod 19257 df-cnfld 20143 df-zring 20215 df-zlm 20249 df-nm 22795 df-nlm 22799 |
This theorem is referenced by: zrhnm 30611 |
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