rezh - Mathbox for Thierry Arnoux

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Theorem rezh 34227

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 24828	. . . . 5 ⊢ ℂ_fld ∈ NrmRing
2		resubdrg 21648	. . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) ∧ ℝ_fld ∈ DivRing)
3	2	simpli 487	. . . . 5 ⊢ ℝ ∈ (SubRing‘ℂ_fld)
4		df-refld 21645	. . . . . 6 ⊢ ℝ_fld = (ℂ_fld ↾_s ℝ)
5	4	subrgnrg 24721	. . . . 5 ⊢ ((ℂ_fld ∈ NrmRing ∧ ℝ ∈ (SubRing‘ℂ_fld)) → ℝ_fld ∈ NrmRing)
6	1, 3, 5	mp2an 702	. . . 4 ⊢ ℝ_fld ∈ NrmRing
7		eqid 2761	. . . . 5 ⊢ (ℤMod‘ℝ_fld) = (ℤMod‘ℝ_fld)
8	7	zhmnrg 34223	. . . 4 ⊢ (ℝ_fld ∈ NrmRing → (ℤMod‘ℝ_fld) ∈ NrmRing)
9		nrgngp 24710	. . . 4 ⊢ ((ℤMod‘ℝ_fld) ∈ NrmRing → (ℤMod‘ℝ_fld) ∈ NrmGrp)
10	6, 8, 9	mp2b 10	. . 3 ⊢ (ℤMod‘ℝ_fld) ∈ NrmGrp
11		nrgring 24711	. . . . 5 ⊢ (ℝ_fld ∈ NrmRing → ℝ_fld ∈ Ring)
12		ringabl 20318	. . . . 5 ⊢ (ℝ_fld ∈ Ring → ℝ_fld ∈ Abel)
13	6, 11, 12	mp2b 10	. . . 4 ⊢ ℝ_fld ∈ Abel
14	7	zlmlmod 21562	. . . 4 ⊢ (ℝ_fld ∈ Abel ↔ (ℤMod‘ℝ_fld) ∈ LMod)
15	13, 14	mpbi 232	. . 3 ⊢ (ℤMod‘ℝ_fld) ∈ LMod
16		zringnrg 24836	. . 3 ⊢ ℤ_ring ∈ NrmRing
17	10, 15, 16	3pm3.2i 1352	. 2 ⊢ ((ℤMod‘ℝ_fld) ∈ NrmGrp ∧ (ℤMod‘ℝ_fld) ∈ LMod ∧ ℤ_ring ∈ NrmRing)
18		simpl 486	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℤ)
19	18	zcnd 12672	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℂ)
20		simpr 488	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ)
21	20	recnd 11204	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ)
22	19, 21	absmuld 15475	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥)))
23		subrgsubg 20614	. . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂ_fld) → ℝ ∈ (SubGrp‘ℂ_fld))
24	3, 23	ax-mp 5	. . . . . . . 8 ⊢ ℝ ∈ (SubGrp‘ℂ_fld)
25		eqid 2761	. . . . . . . . 9 ⊢ (._g‘ℂ_fld) = (._g‘ℂ_fld)
26		eqid 2761	. . . . . . . . . . 11 ⊢ (._g‘ℝ_fld) = (._g‘ℝ_fld)
27	7, 26	zlmvsca 21561	. . . . . . . . . 10 ⊢ (._g‘ℝ_fld) = ( ·_𝑠 ‘(ℤMod‘ℝ_fld))
28	27	eqcomi 2770	. . . . . . . . 9 ⊢ ( ·_𝑠 ‘(ℤMod‘ℝ_fld)) = (._g‘ℝ_fld)
29	25, 4, 28	subgmulg 19173	. . . . . . . 8 ⊢ ((ℝ ∈ (SubGrp‘ℂ_fld) ∧ 𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥))
30	24, 29	mp3an1 1468	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥))
31		cnfldmulg 21444	. . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧 · 𝑥))
32	21, 31	syldan 600	. . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(._g‘ℂ_fld)𝑥) = (𝑧 · 𝑥))
33	30, 32	eqtr3d 2798	. . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥) = (𝑧 · 𝑥))
34	33	fveq2d 6866	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = ((abs ↾ ℝ)‘(𝑧 · 𝑥)))
35		zre 12566	. . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ)
36		remulcl 11152	. . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧 · 𝑥) ∈ ℝ)
37		fvres 6881	. . . . . . 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 589	. . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥)))
40	34, 39	eqtrd 2796	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (abs‘(𝑧 · 𝑥)))
41		fvres 6881	. . . . 5 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧))
42		fvres 6881	. . . . 5 ⊢ (𝑥 ∈ ℝ → ((abs ↾ ℝ)‘𝑥) = (abs‘𝑥))
43	41, 42	oveqan12d 7410	. . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥)))
44	22, 40, 43	3eqtr4d 2806	. . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)))
45	44	rgen2 3201	. 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·_𝑠 ‘(ℤMod‘ℝ_fld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))
46		rebase 21646	. . . 4 ⊢ ℝ = (Base‘ℝ_fld)
47	7, 46	zlmbas 21557	. . 3 ⊢ ℝ = (Base‘(ℤMod‘ℝ_fld))
48		recusp 25432	. . . . 5 ⊢ ℝ_fld ∈ CUnifSp
49	48	elexi 3475	. . . 4 ⊢ ℝ_fld ∈ V
50		cnring 21434	. . . . . . 7 ⊢ ℂ_fld ∈ Ring
51		ringmnd 20280	. . . . . . 7 ⊢ (ℂ_fld ∈ Ring → ℂ_fld ∈ Mnd)
52	50, 51	ax-mp 5	. . . . . 6 ⊢ ℂ_fld ∈ Mnd
53		0re 11177	. . . . . 6 ⊢ 0 ∈ ℝ
54		ax-resscn 11124	. . . . . 6 ⊢ ℝ ⊆ ℂ
55		cnfldbas 21416	. . . . . . 7 ⊢ ℂ = (Base‘ℂ_fld)
56		cnfld0 21436	. . . . . . 7 ⊢ 0 = (0_g‘ℂ_fld)
57		cnfldnm 24826	. . . . . . 7 ⊢ abs = (norm‘ℂ_fld)
58	4, 55, 56, 57	ressnm 33103	. . . . . 6 ⊢ ((ℂ_fld ∈ Mnd ∧ 0 ∈ ℝ ∧ ℝ ⊆ ℂ) → (abs ↾ ℝ) = (norm‘ℝ_fld))
59	52, 53, 54, 58	mp3an 1481	. . . . 5 ⊢ (abs ↾ ℝ) = (norm‘ℝ_fld)
60	7, 59	zlmnm 34222	. . . 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 2761	. . 3 ⊢ ( ·_𝑠 ‘(ℤMod‘ℝ_fld)) = ( ·_𝑠 ‘(ℤMod‘ℝ_fld))
63	7	zlmsca 21560	. . . 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 21493	. . 3 ⊢ ℤ = (Base‘ℤ_ring)
66		zringnm 34216	. . . 4 ⊢ (norm‘ℤ_ring) = (abs ↾ ℤ)
67	66	eqcomi 2770	. . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤ_ring)
68	47, 61, 62, 64, 65, 67	isnlm 24723	. 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 721	1 ⊢ (ℤMod‘ℝ_fld) ∈ NrmMod

Colors of variables: wff setvar class

Syntax hints: ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 Vcvv 3453 ⊆ wss 3902 ↾ cres 5645 ‘cfv 6516 (class class class)co 7391 ℂcc 11065 ℝcr 11066 0cc0 11067 · cmul 11072 ℤcz 12562 abscabs 15252 Scalarcsca 17280 ·_𝑠 cvsca 17281 Mndcmnd 18759 ._gcmg 19100 SubGrpcsubg 19153 Abelcabl 19812 Ringcrg 20270 SubRingcsubrg 20606 DivRingcdr 20766 LModclmod 20915 ℂ_fldccnfld 21412 ℤ_ringczring 21486 ℤModczlm 21540 ℝ_fldcrefld 21644 CUnifSpccusp 24344 normcnm 24624 NrmGrpcngp 24625 NrmRingcnrg 24627 NrmModcnlm 24628

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144 ax-pre-sup 11145 ax-addf 11146 ax-mulf 11147

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-tp 4584 df-op 4586 df-uni 4863 df-int 4903 df-iun 4948 df-iin 4949 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-se 5597 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-of 7655 df-om 7842 df-1st 7965 df-2nd 7966 df-supp 8135 df-tpos 8200 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-1o 8431 df-2o 8432 df-er 8672 df-map 8804 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9302 df-fi 9351 df-sup 9382 df-inf 9383 df-oi 9452 df-card 9891 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-div 11839 df-nn 12205 df-2 12274 df-3 12275 df-4 12276 df-5 12277 df-6 12278 df-7 12279 df-8 12280 df-9 12281 df-n0 12476 df-z 12563 df-dec 12683 df-uz 12834 df-q 12944 df-rp 12988 df-xneg 13108 df-xadd 13109 df-xmul 13110 df-ioo 13347 df-ico 13349 df-icc 13350 df-fz 13507 df-fzo 13654 df-seq 14009 df-exp 14069 df-hash 14338 df-cj 15117 df-re 15118 df-im 15119 df-sqrt 15253 df-abs 15254 df-struct 17174 df-sets 17191 df-slot 17209 df-ndx 17221 df-base 17237 df-ress 17258 df-plusg 17290 df-mulr 17291 df-starv 17292 df-sca 17293 df-vsca 17294 df-ip 17295 df-tset 17296 df-ple 17297 df-ds 17299 df-unif 17300 df-hom 17301 df-cco 17302 df-rest 17442 df-topn 17443 df-0g 17461 df-gsum 17462 df-topgen 17463 df-pt 17464 df-prds 17467 df-xrs 17523 df-qtop 17528 df-imas 17529 df-xps 17531 df-mre 17605 df-mrc 17606 df-acs 17608 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-submnd 18809 df-grp 18969 df-minusg 18970 df-sbg 18971 df-mulg 19101 df-subg 19156 df-cntz 19348 df-cmn 19813 df-abl 19814 df-mgp 20178 df-rng 20190 df-ur 20219 df-ring 20272 df-cring 20273 df-oppr 20373 df-dvdsr 20393 df-unit 20394 df-invr 20424 df-dvr 20437 df-subrng 20583 df-subrg 20607 df-drng 20768 df-abv 20846 df-lmod 20917 df-psmet 21404 df-xmet 21405 df-met 21406 df-bl 21407 df-mopn 21408 df-fbas 21409 df-fg 21410 df-metu 21411 df-cnfld 21413 df-zring 21487 df-zlm 21544 df-refld 21645 df-top 22942 df-topon 22959 df-topsp 22981 df-bases 22994 df-cld 23067 df-ntr 23068 df-cls 23069 df-nei 23146 df-cn 23275 df-cnp 23276 df-haus 23363 df-cmp 23435 df-tx 23610 df-hmeo 23803 df-fil 23894 df-flim 23987 df-fcls 23989 df-ust 24249 df-utop 24279 df-uss 24304 df-usp 24305 df-cfilu 24334 df-cusp 24345 df-xms 24368 df-ms 24369 df-tms 24370 df-nm 24630 df-ngp 24631 df-nrg 24633 df-nlm 24634 df-cncf 24928 df-cfil 25305 df-cmet 25307 df-cms 25385

This theorem is referenced by: rerrext 34267

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