rezh - Mathbox for Thierry Arnoux

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Theorem rezh 30015

Description: The ℤ-module of ℝ is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.)

**Assertion**
Ref	Expression
rezh	⊢ (ℤMod‘ℝ_fld) ∈ NrmMod

Proof of Theorem rezh Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cnnrg 22584	. . . . 5 ⊢ ℂ_fld ∈ NrmRing
2		resubdrg 19954	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
3	2	simpli 474	. . . . 5 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
4		df-refld 19951	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
5	4	subrgnrg 22477	. . . . 5 ⊢ ((ℂ_fld ∈ NrmRing ∧ ℝ ∈ (SubRing‘ℂ_fld)) → ℝ_fld ∈ NrmRing)
6	1, 3, 5	mp2an 708	. . . 4 ⊢ ℝ_fld ∈ NrmRing
7		eqid 2622	. . . . 5 ⊢ (ℤMod‘ℝ_fld) = (ℤMod‘ℝ_fld)
8	7	zhmnrg 30011	. . . 4 ⊢ (ℝ_fld ∈ NrmRing → (ℤMod‘ℝ_fld) ∈ NrmRing)
9		nrgngp 22466	. . . 4 ⊢ ((ℤMod‘ℝ_fld) ∈ NrmRing → (ℤMod‘ℝ_fld) ∈ NrmGrp)
10	6, 8, 9	mp2b 10	. . 3 ⊢ (ℤMod‘ℝ_fld) ∈ NrmGrp
11		nrgring 22467	. . . . 5 ⊢ (ℝ_fld ∈ NrmRing → ℝ_fld ∈ Ring)
12		ringabl 18580	. . . . 5 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Abel)
13	6, 11, 12	mp2b 10	. . . 4 ⊢ ℝ_fld ∈ Abel
14	7	zlmlmod 19871	. . . 4 ⊢ (ℝ_fld ∈ Abel ↔ (ℤMod‘ℝ_fld) ∈ LMod)
15	13, 14	mpbi 220	. . 3 ⊢ (ℤMod‘ℝ_fld) ∈ LMod
16		zringnrg 22591	. . 3 ⊢ ℤ_ring ∈ NrmRing
17	10, 15, 16	3pm3.2i 1239	. 2 ⊢ ((ℤMod‘ℝ_fld) ∈ NrmGrp ∧ (ℤMod‘ℝ_fld) ∈ LMod ∧ ℤ_ring ∈ NrmRing)
18		simpl 473	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℤ)
19	18	zcnd 11483	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℂ)
20		simpr 477	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
21	20	recnd 10068	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
22	19, 21	absmuld 14193	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥)))
23		subrgsubg 18786	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
24	3, 23	ax-mp 5	. . . . . . . 8 ⊢ ℝ ∈ (SubGrp‘ℂ_fld)
25		eqid 2622	. . . . . . . . 9 ⊢ (._g‘ℂ_fld) = (._g‘ℂ_fld)
26		eqid 2622	. . . . . . . . . . 11 ⊢ (._g‘ℝ_fld) = (._g‘ℝ_fld)
27	7, 26	zlmvsca 19870	. . . . . . . . . 10 ⊢ (._g‘ℝ_fld) = ( ·_𝑠 ‘(ℤMod‘ℝ_fld))
28	27	eqcomi 2631	. . . . . . . . 9 ⊢ ( ·_𝑠 ‘(ℤMod‘ℝ_fld)) = (._g‘ℝ_fld)
29	25, 4, 28	subgmulg 17608	. . . . . . . 8 ⊢ ((ℝ ∈ (SubGrp‘ℂ_fld) ∧ 𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥))
30	24, 29	mp3an1 1411	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥))
31		cnfldmulg 19778	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧 · 𝑥))
32	21, 31	syldan 487	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧 · 𝑥))
33	30, 32	eqtr3d 2658	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥) = (𝑧 · 𝑥))
34	33	fveq2d 6195	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = ((abs ↾ ℝ)‘(𝑧 · 𝑥)))
35		zre 11381	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ)
36		remulcl 10021	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧 · 𝑥) ∈ ℝ)
37		fvres 6207	. . . . . . 7 ⊢ ((𝑧 · 𝑥) ∈ ℝ → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥)))
38	36, 37	syl 17	. . . . . 6 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥)))
39	35, 38	sylan 488	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥)))
40	34, 39	eqtrd 2656	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (abs‘(𝑧 · 𝑥)))
41		fvres 6207	. . . . 5 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧))
42		fvres 6207	. . . . 5 ⊢ (𝑥 ∈ ℝ → ((abs ↾ ℝ)‘𝑥) = (abs‘𝑥))
43	41, 42	oveqan12d 6669	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥)))
44	22, 40, 43	3eqtr4d 2666	. . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)))
45	44	rgen2 2975	. 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))
46		rebase 19952	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
47	7, 46	zlmbas 19866	. . 3 ⊢ ℝ = (Base‘(ℤMod‘ℝ_fld))
48		recusp 23170	. . . . 5 ⊢ ℝ_fld ∈ CUnifSp
49	48	elexi 3213	. . . 4 ⊢ ℝ_fld ∈ V
50		cnring 19768	. . . . . . 7 ⊢ ℂ_fld ∈ Ring
51		ringmnd 18556	. . . . . . 7 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
52	50, 51	ax-mp 5	. . . . . 6 ⊢ ℂ_fld ∈ Mnd
53		0re 10040	. . . . . 6 ⊢ 0 ∈ ℝ
54		ax-resscn 9993	. . . . . 6 ⊢ ℝ ⊆ ℂ
55		cnfldbas 19750	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
56		cnfld0 19770	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
57		cnfldnm 22582	. . . . . . 7 ⊢ abs = (norm‘ℂ_fld)
58	4, 55, 56, 57	ressnm 29651	. . . . . 6 ⊢ ((ℂ_fld ∈ Mnd ∧ 0 ∈ ℝ ∧ ℝ ⊆ ℂ) → (abs ↾ ℝ) = (norm‘ℝ_fld))
59	52, 53, 54, 58	mp3an 1424	. . . . 5 ⊢ (abs ↾ ℝ) = (norm‘ℝ_fld)
60	7, 59	zlmnm 30010	. . . 4 ⊢ (ℝ_fld ∈ V → (abs ↾ ℝ) = (norm‘(ℤMod‘ℝ_fld)))
61	49, 60	ax-mp 5	. . 3 ⊢ (abs ↾ ℝ) = (norm‘(ℤMod‘ℝ_fld))
62		eqid 2622	. . 3 ⊢ ( ·_𝑠 ‘(ℤMod‘ℝ_fld)) = ( ·_𝑠 ‘(ℤMod‘ℝ_fld))
63	7	zlmsca 19869	. . . 4 ⊢ (ℝ_fld ∈ V → ℤ_ring = (Scalar‘(ℤMod‘ℝ_fld)))
64	49, 63	ax-mp 5	. . 3 ⊢ ℤ_ring = (Scalar‘(ℤMod‘ℝ_fld))
65		zringbas 19824	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
66		zringnm 30004	. . . 4 ⊢ (norm‘ℤ_ring) = (abs ↾ ℤ)
67	66	eqcomi 2631	. . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤ_ring)
68	47, 61, 62, 64, 65, 67	isnlm 22479	. 2 ⊢ ((ℤMod‘ℝ_fld) ∈ NrmMod ↔ (((ℤMod‘ℝ_fld) ∈ NrmGrp ∧ (ℤMod‘ℝ_fld) ∈ LMod ∧ ℤ_ring ∈ NrmRing) ∧ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))))
69	17, 45, 68	mpbir2an 955	1 ⊢ (ℤMod‘ℝ_fld) ∈ NrmMod

Colors of variables: wff setvar class

Syntax hints: ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 ⊆ wss 3574 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 · cmul 9941 ℤcz 11377 abscabs 13974 Scalarcsca 15944 ·_𝑠 cvsca 15945 Mndcmnd 17294 ._gcmg 17540 SubGrpcsubg 17588 Abelcabl 18194 Ringcrg 18547 DivRingcdr 18747 SubRingcsubrg 18776 LModclmod 18863 ℂ_fldccnfld 19746 ℤ_ringzring 19818 ℤModczlm 19849 ℝ_fldcrefld 19950 CUnifSpccusp 22101 normcnm 22381 NrmGrpcngp 22382 NrmRingcnrg 22384 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-iin 4523 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-se 5074 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 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-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 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-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 df-dvr 18683 df-drng 18749 df-subrg 18778 df-abv 18817 df-lmod 18865 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-metu 19745 df-cnfld 19747 df-zring 19819 df-zlm 19853 df-refld 19951 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-cn 21031 df-cnp 21032 df-haus 21119 df-cmp 21190 df-tx 21365 df-hmeo 21558 df-fil 21650 df-flim 21743 df-fcls 21745 df-ust 22004 df-utop 22035 df-uss 22060 df-usp 22061 df-cfilu 22091 df-cusp 22102 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 df-cncf 22681 df-cfil 23053 df-cmet 23055 df-cms 23132

This theorem is referenced by: rerrext 30053

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