nmmulg - Mathbox for Thierry Arnoux

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Theorem nmmulg 33656

Description: The norm of a group product, provided the ℤ-module is normed. (Contributed by Thierry Arnoux, 8-Nov-2017.)

**Hypotheses**
Ref	Expression
nmmulg.x	⊢ 𝐵 = (Base‘𝑅)
nmmulg.n	⊢ 𝑁 = (norm‘𝑅)
nmmulg.z	⊢ 𝑍 = (ℤMod‘𝑅)
nmmulg.t	⊢ · = (._g‘𝑅)

**Assertion**
Ref	Expression
nmmulg	⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((abs‘𝑀) · (𝑁‘𝑋)))

**Proof of Theorem nmmulg**
Step	Hyp	Ref	Expression
1		simp2 1134	. . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ ℤ)
2		zringbas 21384	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
3		nlmlmod 24626	. . . . . . . . 9 ⊢ (𝑍 ∈ NrmMod → 𝑍 ∈ LMod)
4		nmmulg.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤMod‘𝑅)
5	4	zlmlmod 21457	. . . . . . . . 9 ⊢ (𝑅 ∈ Abel ↔ 𝑍 ∈ LMod)
6	3, 5	sylibr 233	. . . . . . . 8 ⊢ (𝑍 ∈ NrmMod → 𝑅 ∈ Abel)
7	6	3ad2ant1 1130	. . . . . . 7 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑅 ∈ Abel)
8	4	zlmsca 21455	. . . . . . 7 ⊢ (𝑅 ∈ Abel → ℤ_ring = (Scalar‘𝑍))
9	7, 8	syl 17	. . . . . 6 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤ_ring = (Scalar‘𝑍))
10	9	fveq2d 6898	. . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (Base‘ℤ_ring) = (Base‘(Scalar‘𝑍)))
11	2, 10	eqtrid 2777	. . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ℤ = (Base‘(Scalar‘𝑍)))
12	1, 11	eleqtrd 2827	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑀 ∈ (Base‘(Scalar‘𝑍)))
13		nmmulg.x	. . . . 5 ⊢ 𝐵 = (Base‘𝑅)
14	4, 13	zlmbas 21449	. . . 4 ⊢ 𝐵 = (Base‘𝑍)
15		eqid 2725	. . . 4 ⊢ (norm‘𝑍) = (norm‘𝑍)
16		nmmulg.t	. . . . 5 ⊢ · = (._g‘𝑅)
17	4, 16	zlmvsca 21456	. . . 4 ⊢ · = ( ·_𝑠 ‘𝑍)
18		eqid 2725	. . . 4 ⊢ (Scalar‘𝑍) = (Scalar‘𝑍)
19		eqid 2725	. . . 4 ⊢ (Base‘(Scalar‘𝑍)) = (Base‘(Scalar‘𝑍))
20		eqid 2725	. . . 4 ⊢ (norm‘(Scalar‘𝑍)) = (norm‘(Scalar‘𝑍))
21	14, 15, 17, 18, 19, 20	nmvs 24624	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ (Base‘(Scalar‘𝑍)) ∧ 𝑋 ∈ 𝐵) → ((norm‘𝑍)‘(𝑀 · 𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋)))
22	12, 21	syld3an2 1408	. 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 33654	. . . 4 ⊢ (𝑅 ∈ Abel → 𝑁 = (norm‘𝑍))
25	7, 24	syl 17	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑁 = (norm‘𝑍))
26	25	fveq1d 6896	. 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((norm‘𝑍)‘(𝑀 · 𝑋)))
27		zzsnm 33647	. . . . 5 ⊢ (𝑀 ∈ ℤ → (abs‘𝑀) = ((norm‘ℤ_ring)‘𝑀))
28	27	3ad2ant2 1131	. . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘ℤ_ring)‘𝑀))
29	9	fveq2d 6898	. . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (norm‘ℤ_ring) = (norm‘(Scalar‘𝑍)))
30	29	fveq1d 6896	. . . 4 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((norm‘ℤ_ring)‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀))
31	28, 30	eqtrd 2765	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (abs‘𝑀) = ((norm‘(Scalar‘𝑍))‘𝑀))
32	25	fveq1d 6896	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ((norm‘𝑍)‘𝑋))
33	31, 32	oveq12d 7435	. 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → ((abs‘𝑀) · (𝑁‘𝑋)) = (((norm‘(Scalar‘𝑍))‘𝑀) · ((norm‘𝑍)‘𝑋)))
34	22, 26, 33	3eqtr4d 2775	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 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6547 (class class class)co 7417 · cmul 11143 ℤcz 12588 abscabs 15214 Basecbs 17180 Scalarcsca 17236 ._gcmg 19028 Abelcabl 19741 LModclmod 20748 ℤ_ringczring 21377 ℤModczlm 21431 normcnm 24516 NrmModcnlm 24520

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5299 ax-nul 5306 ax-pow 5364 ax-pr 5428 ax-un 7739 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3775 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3965 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6499 df-fun 6549 df-fn 6550 df-f 6551 df-f1 6552 df-fo 6553 df-f1o 6554 df-fv 6555 df-riota 7373 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7870 df-1st 7992 df-2nd 7993 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-er 8723 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-rp 13007 df-fz 13517 df-fzo 13660 df-seq 14000 df-exp 14060 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-struct 17116 df-sets 17133 df-slot 17151 df-ndx 17163 df-base 17181 df-ress 17210 df-plusg 17246 df-mulr 17247 df-starv 17248 df-sca 17249 df-vsca 17250 df-ip 17251 df-tset 17252 df-ple 17253 df-ds 17255 df-unif 17256 df-0g 17423 df-mgm 18600 df-sgrp 18679 df-mnd 18695 df-grp 18898 df-minusg 18899 df-mulg 19029 df-subg 19083 df-cmn 19742 df-abl 19743 df-mgp 20080 df-rng 20098 df-ur 20127 df-ring 20180 df-cring 20181 df-subrng 20488 df-subrg 20513 df-lmod 20750 df-cnfld 21285 df-zring 21378 df-zlm 21435 df-nm 24522 df-nlm 24526

This theorem is referenced by: zrhnm 33657

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