Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rezh | Structured version Visualization version GIF version |
Description: The ℤ-module of ℝ is a normed module. (Contributed by Thierry Arnoux, 14-Feb-2018.) |
Ref | Expression |
---|---|
rezh | ⊢ (ℤMod‘ℝfld) ∈ NrmMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnnrg 23316 | . . . . 5 ⊢ ℂfld ∈ NrmRing | |
2 | resubdrg 20680 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
3 | 2 | simpli 484 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
4 | df-refld 20677 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
5 | 4 | subrgnrg 23209 | . . . . 5 ⊢ ((ℂfld ∈ NrmRing ∧ ℝ ∈ (SubRing‘ℂfld)) → ℝfld ∈ NrmRing) |
6 | 1, 3, 5 | mp2an 688 | . . . 4 ⊢ ℝfld ∈ NrmRing |
7 | eqid 2818 | . . . . 5 ⊢ (ℤMod‘ℝfld) = (ℤMod‘ℝfld) | |
8 | 7 | zhmnrg 31107 | . . . 4 ⊢ (ℝfld ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmRing) |
9 | nrgngp 23198 | . . . 4 ⊢ ((ℤMod‘ℝfld) ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmGrp) | |
10 | 6, 8, 9 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ NrmGrp |
11 | nrgring 23199 | . . . . 5 ⊢ (ℝfld ∈ NrmRing → ℝfld ∈ Ring) | |
12 | ringabl 19259 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Abel) | |
13 | 6, 11, 12 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ Abel |
14 | 7 | zlmlmod 20598 | . . . 4 ⊢ (ℝfld ∈ Abel ↔ (ℤMod‘ℝfld) ∈ LMod) |
15 | 13, 14 | mpbi 231 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ LMod |
16 | zringnrg 23323 | . . 3 ⊢ ℤring ∈ NrmRing | |
17 | 10, 15, 16 | 3pm3.2i 1331 | . 2 ⊢ ((ℤMod‘ℝfld) ∈ NrmGrp ∧ (ℤMod‘ℝfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
18 | simpl 483 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℤ) | |
19 | 18 | zcnd 12076 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℂ) |
20 | simpr 485 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
21 | 20 | recnd 10657 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
22 | 19, 21 | absmuld 14802 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
23 | subrgsubg 19470 | . . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
24 | 3, 23 | ax-mp 5 | . . . . . . . 8 ⊢ ℝ ∈ (SubGrp‘ℂfld) |
25 | eqid 2818 | . . . . . . . . 9 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
26 | eqid 2818 | . . . . . . . . . . 11 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
27 | 7, 26 | zlmvsca 20597 | . . . . . . . . . 10 ⊢ (.g‘ℝfld) = ( ·𝑠 ‘(ℤMod‘ℝfld)) |
28 | 27 | eqcomi 2827 | . . . . . . . . 9 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = (.g‘ℝfld) |
29 | 25, 4, 28 | subgmulg 18231 | . . . . . . . 8 ⊢ ((ℝ ∈ (SubGrp‘ℂfld) ∧ 𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
30 | 24, 29 | mp3an1 1439 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
31 | cnfldmulg 20505 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
32 | 21, 31 | syldan 591 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) |
33 | 30, 32 | eqtr3d 2855 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥) = (𝑧 · 𝑥)) |
34 | 33 | fveq2d 6667 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = ((abs ↾ ℝ)‘(𝑧 · 𝑥))) |
35 | zre 11973 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ) | |
36 | remulcl 10610 | . . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧 · 𝑥) ∈ ℝ) | |
37 | fvres 6682 | . . . . . . 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 580 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥))) |
40 | 34, 39 | eqtrd 2853 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (abs‘(𝑧 · 𝑥))) |
41 | fvres 6682 | . . . . 5 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
42 | fvres 6682 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ((abs ↾ ℝ)‘𝑥) = (abs‘𝑥)) | |
43 | 41, 42 | oveqan12d 7164 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
44 | 22, 40, 43 | 3eqtr4d 2863 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))) |
45 | 44 | rgen2 3200 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) |
46 | rebase 20678 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
47 | 7, 46 | zlmbas 20593 | . . 3 ⊢ ℝ = (Base‘(ℤMod‘ℝfld)) |
48 | recusp 23912 | . . . . 5 ⊢ ℝfld ∈ CUnifSp | |
49 | 48 | elexi 3511 | . . . 4 ⊢ ℝfld ∈ V |
50 | cnring 20495 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
51 | ringmnd 19235 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
52 | 50, 51 | ax-mp 5 | . . . . . 6 ⊢ ℂfld ∈ Mnd |
53 | 0re 10631 | . . . . . 6 ⊢ 0 ∈ ℝ | |
54 | ax-resscn 10582 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
55 | cnfldbas 20477 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
56 | cnfld0 20497 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
57 | cnfldnm 23314 | . . . . . . 7 ⊢ abs = (norm‘ℂfld) | |
58 | 4, 55, 56, 57 | ressnm 30565 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ ℝ ∧ ℝ ⊆ ℂ) → (abs ↾ ℝ) = (norm‘ℝfld)) |
59 | 52, 53, 54, 58 | mp3an 1452 | . . . . 5 ⊢ (abs ↾ ℝ) = (norm‘ℝfld) |
60 | 7, 59 | zlmnm 31106 | . . . 4 ⊢ (ℝfld ∈ V → (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld))) |
61 | 49, 60 | ax-mp 5 | . . 3 ⊢ (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld)) |
62 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = ( ·𝑠 ‘(ℤMod‘ℝfld)) | |
63 | 7 | zlmsca 20596 | . . . 4 ⊢ (ℝfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℝfld))) |
64 | 49, 63 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℝfld)) |
65 | zringbas 20551 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
66 | zringnm 31100 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
67 | 66 | eqcomi 2827 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
68 | 47, 61, 62, 64, 65, 67 | isnlm 23211 | . 2 ⊢ ((ℤMod‘ℝfld) ∈ NrmMod ↔ (((ℤMod‘ℝfld) ∈ NrmGrp ∧ (ℤMod‘ℝfld) ∈ LMod ∧ ℤring ∈ NrmRing) ∧ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)))) |
69 | 17, 45, 68 | mpbir2an 707 | 1 ⊢ (ℤMod‘ℝfld) ∈ NrmMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 ℝcr 10524 0cc0 10525 · cmul 10530 ℤcz 11969 abscabs 14581 Scalarcsca 16556 ·𝑠 cvsca 16557 Mndcmnd 17899 .gcmg 18162 SubGrpcsubg 18211 Abelcabl 18836 Ringcrg 19226 DivRingcdr 19431 SubRingcsubrg 19460 LModclmod 19563 ℂfldccnfld 20473 ℤringzring 20545 ℤModczlm 20576 ℝfldcrefld 20676 CUnifSpccusp 22833 normcnm 23113 NrmGrpcngp 23114 NrmRingcnrg 23116 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-rep 5181 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-iin 4913 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-se 5508 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-isom 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-of 7398 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-2o 8092 df-oadd 8095 df-er 8278 df-map 8397 df-ixp 8450 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-fsupp 8822 df-fi 8863 df-sup 8894 df-inf 8895 df-oi 8962 df-card 9356 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-q 12337 df-rp 12378 df-xneg 12495 df-xadd 12496 df-xmul 12497 df-ioo 12730 df-ico 12732 df-icc 12733 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 df-hash 13679 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-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-hom 16577 df-cco 16578 df-rest 16684 df-topn 16685 df-0g 16703 df-gsum 16704 df-topgen 16705 df-pt 16706 df-prds 16709 df-xrs 16763 df-qtop 16768 df-imas 16769 df-xps 16771 df-mre 16845 df-mrc 16846 df-acs 16848 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-cntz 18385 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-subrg 19462 df-abv 19517 df-lmod 19565 df-psmet 20465 df-xmet 20466 df-met 20467 df-bl 20468 df-mopn 20469 df-fbas 20470 df-fg 20471 df-metu 20472 df-cnfld 20474 df-zring 20546 df-zlm 20580 df-refld 20677 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-cld 21555 df-ntr 21556 df-cls 21557 df-nei 21634 df-cn 21763 df-cnp 21764 df-haus 21851 df-cmp 21923 df-tx 22098 df-hmeo 22291 df-fil 22382 df-flim 22475 df-fcls 22477 df-ust 22736 df-utop 22767 df-uss 22792 df-usp 22793 df-cfilu 22823 df-cusp 22834 df-xms 22857 df-ms 22858 df-tms 22859 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 df-cncf 23413 df-cfil 23785 df-cmet 23787 df-cms 23865 |
This theorem is referenced by: rerrext 31149 |
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