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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 24167 | . . . . 5 ⊢ ℂfld ∈ NrmRing | |
2 | resubdrg 21035 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
3 | 2 | simpli 485 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
4 | df-refld 21032 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
5 | 4 | subrgnrg 24060 | . . . . 5 ⊢ ((ℂfld ∈ NrmRing ∧ ℝ ∈ (SubRing‘ℂfld)) → ℝfld ∈ NrmRing) |
6 | 1, 3, 5 | mp2an 691 | . . . 4 ⊢ ℝfld ∈ NrmRing |
7 | eqid 2733 | . . . . 5 ⊢ (ℤMod‘ℝfld) = (ℤMod‘ℝfld) | |
8 | 7 | zhmnrg 32612 | . . . 4 ⊢ (ℝfld ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmRing) |
9 | nrgngp 24049 | . . . 4 ⊢ ((ℤMod‘ℝfld) ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmGrp) | |
10 | 6, 8, 9 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ NrmGrp |
11 | nrgring 24050 | . . . . 5 ⊢ (ℝfld ∈ NrmRing → ℝfld ∈ Ring) | |
12 | ringabl 20010 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Abel) | |
13 | 6, 11, 12 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ Abel |
14 | 7 | zlmlmod 20950 | . . . 4 ⊢ (ℝfld ∈ Abel ↔ (ℤMod‘ℝfld) ∈ LMod) |
15 | 13, 14 | mpbi 229 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ LMod |
16 | zringnrg 24174 | . . 3 ⊢ ℤring ∈ NrmRing | |
17 | 10, 15, 16 | 3pm3.2i 1340 | . 2 ⊢ ((ℤMod‘ℝfld) ∈ NrmGrp ∧ (ℤMod‘ℝfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
18 | simpl 484 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℤ) | |
19 | 18 | zcnd 12616 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℂ) |
20 | simpr 486 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
21 | 20 | recnd 11191 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
22 | 19, 21 | absmuld 15348 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
23 | subrgsubg 20270 | . . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
24 | 3, 23 | ax-mp 5 | . . . . . . . 8 ⊢ ℝ ∈ (SubGrp‘ℂfld) |
25 | eqid 2733 | . . . . . . . . 9 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
26 | eqid 2733 | . . . . . . . . . . 11 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
27 | 7, 26 | zlmvsca 20949 | . . . . . . . . . 10 ⊢ (.g‘ℝfld) = ( ·𝑠 ‘(ℤMod‘ℝfld)) |
28 | 27 | eqcomi 2742 | . . . . . . . . 9 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = (.g‘ℝfld) |
29 | 25, 4, 28 | subgmulg 18950 | . . . . . . . 8 ⊢ ((ℝ ∈ (SubGrp‘ℂfld) ∧ 𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
30 | 24, 29 | mp3an1 1449 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
31 | cnfldmulg 20852 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
32 | 21, 31 | syldan 592 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) |
33 | 30, 32 | eqtr3d 2775 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥) = (𝑧 · 𝑥)) |
34 | 33 | fveq2d 6850 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = ((abs ↾ ℝ)‘(𝑧 · 𝑥))) |
35 | zre 12511 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ) | |
36 | remulcl 11144 | . . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧 · 𝑥) ∈ ℝ) | |
37 | fvres 6865 | . . . . . . 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 581 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥))) |
40 | 34, 39 | eqtrd 2773 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (abs‘(𝑧 · 𝑥))) |
41 | fvres 6865 | . . . . 5 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
42 | fvres 6865 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ((abs ↾ ℝ)‘𝑥) = (abs‘𝑥)) | |
43 | 41, 42 | oveqan12d 7380 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
44 | 22, 40, 43 | 3eqtr4d 2783 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))) |
45 | 44 | rgen2 3191 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) |
46 | rebase 21033 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
47 | 7, 46 | zlmbas 20942 | . . 3 ⊢ ℝ = (Base‘(ℤMod‘ℝfld)) |
48 | recusp 24769 | . . . . 5 ⊢ ℝfld ∈ CUnifSp | |
49 | 48 | elexi 3466 | . . . 4 ⊢ ℝfld ∈ V |
50 | cnring 20842 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
51 | ringmnd 19982 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
52 | 50, 51 | ax-mp 5 | . . . . . 6 ⊢ ℂfld ∈ Mnd |
53 | 0re 11165 | . . . . . 6 ⊢ 0 ∈ ℝ | |
54 | ax-resscn 11116 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
55 | cnfldbas 20823 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
56 | cnfld0 20844 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
57 | cnfldnm 24165 | . . . . . . 7 ⊢ abs = (norm‘ℂfld) | |
58 | 4, 55, 56, 57 | ressnm 31874 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ ℝ ∧ ℝ ⊆ ℂ) → (abs ↾ ℝ) = (norm‘ℝfld)) |
59 | 52, 53, 54, 58 | mp3an 1462 | . . . . 5 ⊢ (abs ↾ ℝ) = (norm‘ℝfld) |
60 | 7, 59 | zlmnm 32611 | . . . 4 ⊢ (ℝfld ∈ V → (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld))) |
61 | 49, 60 | ax-mp 5 | . . 3 ⊢ (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld)) |
62 | eqid 2733 | . . 3 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = ( ·𝑠 ‘(ℤMod‘ℝfld)) | |
63 | 7 | zlmsca 20948 | . . . 4 ⊢ (ℝfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℝfld))) |
64 | 49, 63 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℝfld)) |
65 | zringbas 20898 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
66 | zringnm 32603 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
67 | 66 | eqcomi 2742 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
68 | 47, 61, 62, 64, 65, 67 | isnlm 24062 | . 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 710 | 1 ⊢ (ℤMod‘ℝfld) ∈ NrmMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3061 Vcvv 3447 ⊆ wss 3914 ↾ cres 5639 ‘cfv 6500 (class class class)co 7361 ℂcc 11057 ℝcr 11058 0cc0 11059 · cmul 11064 ℤcz 12507 abscabs 15128 Scalarcsca 17144 ·𝑠 cvsca 17145 Mndcmnd 18564 .gcmg 18880 SubGrpcsubg 18930 Abelcabl 19571 Ringcrg 19972 DivRingcdr 20219 SubRingcsubrg 20260 LModclmod 20365 ℂfldccnfld 20819 ℤringczring 20892 ℤModczlm 20924 ℝfldcrefld 21031 CUnifSpccusp 23672 normcnm 23955 NrmGrpcngp 23956 NrmRingcnrg 23958 NrmModcnlm 23959 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 ax-cnex 11115 ax-resscn 11116 ax-1cn 11117 ax-icn 11118 ax-addcl 11119 ax-addrcl 11120 ax-mulcl 11121 ax-mulrcl 11122 ax-mulcom 11123 ax-addass 11124 ax-mulass 11125 ax-distr 11126 ax-i2m1 11127 ax-1ne0 11128 ax-1rid 11129 ax-rnegex 11130 ax-rrecex 11131 ax-cnre 11132 ax-pre-lttri 11133 ax-pre-lttrn 11134 ax-pre-ltadd 11135 ax-pre-mulgt0 11136 ax-pre-sup 11137 ax-addf 11138 ax-mulf 11139 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4870 df-int 4912 df-iun 4960 df-iin 4961 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-se 5593 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7317 df-ov 7364 df-oprab 7365 df-mpo 7366 df-of 7621 df-om 7807 df-1st 7925 df-2nd 7926 df-supp 8097 df-tpos 8161 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-1o 8416 df-2o 8417 df-er 8654 df-map 8773 df-ixp 8842 df-en 8890 df-dom 8891 df-sdom 8892 df-fin 8893 df-fsupp 9312 df-fi 9355 df-sup 9386 df-inf 9387 df-oi 9454 df-card 9883 df-pnf 11199 df-mnf 11200 df-xr 11201 df-ltxr 11202 df-le 11203 df-sub 11395 df-neg 11396 df-div 11821 df-nn 12162 df-2 12224 df-3 12225 df-4 12226 df-5 12227 df-6 12228 df-7 12229 df-8 12230 df-9 12231 df-n0 12422 df-z 12508 df-dec 12627 df-uz 12772 df-q 12882 df-rp 12924 df-xneg 13041 df-xadd 13042 df-xmul 13043 df-ioo 13277 df-ico 13279 df-icc 13280 df-fz 13434 df-fzo 13577 df-seq 13916 df-exp 13977 df-hash 14240 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-struct 17027 df-sets 17044 df-slot 17062 df-ndx 17074 df-base 17092 df-ress 17121 df-plusg 17154 df-mulr 17155 df-starv 17156 df-sca 17157 df-vsca 17158 df-ip 17159 df-tset 17160 df-ple 17161 df-ds 17163 df-unif 17164 df-hom 17165 df-cco 17166 df-rest 17312 df-topn 17313 df-0g 17331 df-gsum 17332 df-topgen 17333 df-pt 17334 df-prds 17337 df-xrs 17392 df-qtop 17397 df-imas 17398 df-xps 17400 df-mre 17474 df-mrc 17475 df-acs 17477 df-mgm 18505 df-sgrp 18554 df-mnd 18565 df-submnd 18610 df-grp 18759 df-minusg 18760 df-sbg 18761 df-mulg 18881 df-subg 18933 df-cntz 19105 df-cmn 19572 df-abl 19573 df-mgp 19905 df-ur 19922 df-ring 19974 df-cring 19975 df-oppr 20057 df-dvdsr 20078 df-unit 20079 df-invr 20109 df-dvr 20120 df-drng 20221 df-subrg 20262 df-abv 20319 df-lmod 20367 df-psmet 20811 df-xmet 20812 df-met 20813 df-bl 20814 df-mopn 20815 df-fbas 20816 df-fg 20817 df-metu 20818 df-cnfld 20820 df-zring 20893 df-zlm 20928 df-refld 21032 df-top 22266 df-topon 22283 df-topsp 22305 df-bases 22319 df-cld 22393 df-ntr 22394 df-cls 22395 df-nei 22472 df-cn 22601 df-cnp 22602 df-haus 22689 df-cmp 22761 df-tx 22936 df-hmeo 23129 df-fil 23220 df-flim 23313 df-fcls 23315 df-ust 23575 df-utop 23606 df-uss 23631 df-usp 23632 df-cfilu 23662 df-cusp 23673 df-xms 23696 df-ms 23697 df-tms 23698 df-nm 23961 df-ngp 23962 df-nrg 23964 df-nlm 23965 df-cncf 24264 df-cfil 24642 df-cmet 24644 df-cms 24722 |
This theorem is referenced by: rerrext 32654 |
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