nmmulg - Mathbox for Thierry Arnoux

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Theorem nmmulg 33700

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 21396	. . . . 5 ⊢ ℤ = (Base‘ℤ_ring)
3		nlmlmod 24639	. . . . . . . . 9 ⊢ (𝑍 ∈ NrmMod → 𝑍 ∈ LMod)
4		nmmulg.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤMod‘𝑅)
5	4	zlmlmod 21469	. . . . . . . . 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 21467	. . . . . . 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 6900	. . . . 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 21461	. . . 4 ⊢ 𝐵 = (Base‘𝑍)
15		eqid 2725	. . . 4 ⊢ (norm‘𝑍) = (norm‘𝑍)
16		nmmulg.t	. . . . 5 ⊢ · = (._g‘𝑅)
17	4, 16	zlmvsca 21468	. . . 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 24637	. . 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 33698	. . . 4 ⊢ (𝑅 ∈ Abel → 𝑁 = (norm‘𝑍))
25	7, 24	syl 17	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → 𝑁 = (norm‘𝑍))
26	25	fveq1d 6898	. 2 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘(𝑀 · 𝑋)) = ((norm‘𝑍)‘(𝑀 · 𝑋)))
27		zzsnm 33691	. . . . 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 6900	. . . . 5 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (norm‘ℤ_ring) = (norm‘(Scalar‘𝑍)))
30	29	fveq1d 6898	. . . 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 6898	. . 3 ⊢ ((𝑍 ∈ NrmMod ∧ 𝑀 ∈ ℤ ∧ 𝑋 ∈ 𝐵) → (𝑁‘𝑋) = ((norm‘𝑍)‘𝑋))
33	31, 32	oveq12d 7437	. 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 6549 (class class class)co 7419 · cmul 11145 ℤcz 12591 abscabs 15217 Basecbs 17183 Scalarcsca 17239 ._gcmg 19031 Abelcabl 19748 LModclmod 20755 ℤ_ringczring 21389 ℤModczlm 21443 normcnm 24529 NrmModcnlm 24533

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 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-pre-sup 11218 ax-addf 11219 ax-mulf 11220

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 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-rp 13010 df-fz 13520 df-fzo 13663 df-seq 14003 df-exp 14063 df-cj 15082 df-re 15083 df-im 15084 df-sqrt 15218 df-abs 15219 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-0g 17426 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-grp 18901 df-minusg 18902 df-mulg 19032 df-subg 19086 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-subrng 20495 df-subrg 20520 df-lmod 20757 df-cnfld 21297 df-zring 21390 df-zlm 21447 df-nm 24535 df-nlm 24539

This theorem is referenced by: zrhnm 33701

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