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 1129 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ ℤ) | |
2 | zringbas 20551 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
3 | nlmlmod 23214 | . . . . . . . . 9 ⊢ (𝑍 ∈ NrmMod → 𝑍 ∈ LMod) | |
4 | nmmulg.z | . . . . . . . . . 10 ⊢ 𝑍 = (ℤMod‘𝑅) | |
5 | 4 | zlmlmod 20598 | . . . . . . . . 9 ⊢ (𝑅 ∈ Abel ↔ 𝑍 ∈ LMod) |
6 | 3, 5 | sylibr 235 | . . . . . . . 8 ⊢ (𝑍 ∈ NrmMod → 𝑅 ∈ Abel) |
7 | 6 | 3ad2ant1 1125 | . . . . . . 7 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Abel) |
8 | 4 | zlmsca 20596 | . . . . . . 7 ⊢ (𝑅 ∈ Abel → ℤring = (Scalar‘𝑍)) |
9 | 7, 8 | syl 17 | . . . . . 6 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤring = (Scalar‘𝑍)) |
10 | 9 | fveq2d 6667 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (Base‘ℤring) = (Base‘(Scalar‘𝑍))) |
11 | 2, 10 | syl5eq 2865 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤ = (Base‘(Scalar‘𝑍))) |
12 | 1, 11 | eleqtrd 2912 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ (Base‘(Scalar‘𝑍))) |
13 | nmmulg.x | . . . . 5 ⊢ 𝐵 = (Base‘𝑅) | |
14 | 4, 13 | zlmbas 20593 | . . . 4 ⊢ 𝐵 = (Base‘𝑍) |
15 | eqid 2818 | . . . 4 ⊢ (norm‘𝑍) = (norm‘𝑍) | |
16 | nmmulg.t | . . . . 5 ⊢ · = (.g‘𝑅) | |
17 | 4, 16 | zlmvsca 20597 | . . . 4 ⊢ · = ( ·𝑠 ‘𝑍) |
18 | eqid 2818 | . . . 4 ⊢ (Scalar‘𝑍) = (Scalar‘𝑍) | |
19 | eqid 2818 | . . . 4 ⊢ (Base‘(Scalar‘𝑍)) = (Base‘(Scalar‘𝑍)) | |
20 | eqid 2818 | . . . 4 ⊢ (norm‘(Scalar‘𝑍)) = (norm‘(Scalar‘𝑍)) | |
21 | 14, 15, 17, 18, 19, 20 | nmvs 23212 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑍)) ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
22 | 12, 21 | syld3an2 1403 | . 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 31106 | . . . 4 ⊢ (𝑅 ∈ Abel → 𝑁 = (norm‘𝑍)) |
25 | 7, 24 | syl 17 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑁 = (norm‘𝑍)) |
26 | 25 | fveq1d 6665 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((norm‘𝑍)‘(𝑀 · 𝑋))) |
27 | zzsnm 31101 | . . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) | |
28 | 27 | 3ad2ant2 1126 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘ℤring)‘𝑀)) |
29 | 9 | fveq2d 6667 | . . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (norm‘ℤring) = (norm‘(Scalar‘𝑍))) |
30 | 29 | fveq1d 6665 | . . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘ℤring)‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
31 | 28, 30 | eqtrd 2853 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀)) |
32 | 25 | fveq1d 6665 | . . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ((norm‘𝑍)‘𝑋)) |
33 | 31, 32 | oveq12d 7163 | . 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((abs‘𝑀) · (𝑁‘𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋))) |
34 | 22, 26, 33 | 3eqtr4d 2863 | 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 1079 = wceq 1528 ∈ wcel 2105 ‘cfv 6348 (class class class)co 7145 · cmul 10530 ℤcz 11969 abscabs 14581 Basecbs 16471 Scalarcsca 16556 .gcmg 18162 Abelcabl 18836 LModclmod 19563 ℤringzring 20545 ℤModczlm 20576 normcnm 23113 NrmModcnlm 23117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 ax-pre-sup 10603 ax-addf 10604 ax-mulf 10605 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-nn 11627 df-2 11688 df-3 11689 df-4 11690 df-5 11691 df-6 11692 df-7 11693 df-8 11694 df-9 11695 df-n0 11886 df-z 11970 df-dec 12087 df-uz 12232 df-rp 12378 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-cj 14446 df-re 14447 df-im 14448 df-sqrt 14582 df-abs 14583 df-struct 16473 df-ndx 16474 df-slot 16475 df-base 16477 df-sets 16478 df-ress 16479 df-plusg 16566 df-mulr 16567 df-starv 16568 df-sca 16569 df-vsca 16570 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-mulg 18163 df-subg 18214 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-subrg 19462 df-lmod 19565 df-cnfld 20474 df-zring 20546 df-zlm 20580 df-nm 23119 df-nlm 23123 |
This theorem is referenced by: zrhnm 31109 |
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