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 1062 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ ℤ) | |
2 | zringbas 19824 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | nlmlmod 22482 | . . . . . . . . 9 ⊢ (𝑍 ∈ NrmMod → 𝑍 ∈ LMod) | |
4 | nmmulg.z | . . . . . . . . . 10 ⊢ 𝑍 = (ℤMod‘𝑅) | |
5 | 4 | zlmlmod 19871 | . . . . . . . . 9 ⊢ (𝑅 ∈ Abel ↔ 𝑍 ∈ LMod) |
6 | 3, 5 | sylibr 224 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmMod → 𝑅 ∈ Abel) |
7 | 6 | 3ad2ant1 1082 | . . . . . . 7 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Abel) |
8 | 4 | zlmsca 19869 | . . . . . . 7 ⊢ (𝑅 ∈ Abel → ℤring = (Scalar‘𝑍)) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤring = (Scalar‘𝑍)) |
10 | 9 | fveq2d 6195 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (Base‘ℤring) = (Base‘(Scalar‘𝑍))) |
11 | 2, 10 | syl5eq 2668 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤ = (Base‘(Scalar‘𝑍))) |
12 | 1, 11 | eleqtrd 2703 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ (Base‘(Scalar‘𝑍))) |
13 | nmmulg.x | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
14 | 4, 13 | zlmbas 19866 | . . . 4 ⊢ 𝐵 = (Base‘𝑍) |
15 | eqid 2622 | . . . 4 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
16 | nmmulg.t | . . . . 5 ⊢ · = (.g‘𝑅) | |
17 | 4, 16 | zlmvsca 19870 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑍) |
18 | eqid 2622 | . . . 4 ⊢ (Scalar‘𝑍) = (Scalar‘𝑍) | |
19 | eqid 2622 | . . . 4 ⊢ (Base‘(Scalar‘𝑍)) = (Base‘(Scalar‘𝑍)) | |
20 | eqid 2622 | . . . 4 ⊢ (norm‘(Scalar‘𝑍)) = (norm‘(Scalar‘𝑍)) | |
21 | 14, 15, 17, 18, 19, 20 | nmvs 22480 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑍)) ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
22 | 12, 21 | syld3an2 1373 | . 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 30010 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑁 = (norm‘𝑍)) |
25 | 7, 24 | syl 17 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑁 = (norm‘𝑍)) |
26 | 25 | fveq1d 6193 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((norm‘𝑍)‘(𝑀 · 𝑋))) |
27 | zzsnm 30005 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) | |
28 | 27 | 3ad2ant2 1083 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) |
29 | 9 | fveq2d 6195 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (norm‘ℤring) = (norm‘(Scalar‘𝑍))) |
30 | 29 | fveq1d 6193 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘ℤring)‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
31 | 28, 30 | eqtrd 2656 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
32 | 25 | fveq1d 6193 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ((norm‘𝑍)‘𝑋)) |
33 | 31, 32 | oveq12d 6668 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((abs‘𝑀) · (𝑁‘𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
34 | 22, 26, 33 | 3eqtr4d 2666 | 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 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 · cmul 9941 ℤcz 11377 abscabs 13974 Basecbs 15857 Scalarcsca 15944 .gcmg 17540 Abelcabl 18194 LModclmod 18863 ℤringzring 19818 ℤModczlm 19849 normcnm 22381 NrmModcnlm 22385 |
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-inf2 8538 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 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
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-rmo 2920 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-int 4476 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-1o 7560 df-oadd 7564 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 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-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-minusg 17426 df-mulg 17541 df-subg 17591 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-lmod 18865 df-cnfld 19747 df-zring 19819 df-zlm 19853 df-nm 22387 df-nlm 22391 |
This theorem is referenced by: zrhnm 30013 |
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