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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 24304 | . . . . 5 ⊢ ℂfld ∈ NrmRing | |
2 | resubdrg 21167 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
3 | 2 | simpli 484 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
4 | df-refld 21164 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
5 | 4 | subrgnrg 24197 | . . . . 5 ⊢ ((ℂfld ∈ NrmRing ∧ ℝ ∈ (SubRing‘ℂfld)) → ℝfld ∈ NrmRing) |
6 | 1, 3, 5 | mp2an 690 | . . . 4 ⊢ ℝfld ∈ NrmRing |
7 | eqid 2732 | . . . . 5 ⊢ (ℤMod‘ℝfld) = (ℤMod‘ℝfld) | |
8 | 7 | zhmnrg 33016 | . . . 4 ⊢ (ℝfld ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmRing) |
9 | nrgngp 24186 | . . . 4 ⊢ ((ℤMod‘ℝfld) ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmGrp) | |
10 | 6, 8, 9 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ NrmGrp |
11 | nrgring 24187 | . . . . 5 ⊢ (ℝfld ∈ NrmRing → ℝfld ∈ Ring) | |
12 | ringabl 20100 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Abel) | |
13 | 6, 11, 12 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ Abel |
14 | 7 | zlmlmod 21082 | . . . 4 ⊢ (ℝfld ∈ Abel ↔ (ℤMod‘ℝfld) ∈ LMod) |
15 | 13, 14 | mpbi 229 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ LMod |
16 | zringnrg 24311 | . . 3 ⊢ ℤring ∈ NrmRing | |
17 | 10, 15, 16 | 3pm3.2i 1339 | . 2 ⊢ ((ℤMod‘ℝfld) ∈ NrmGrp ∧ (ℤMod‘ℝfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
18 | simpl 483 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℤ) | |
19 | 18 | zcnd 12669 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℂ) |
20 | simpr 485 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
21 | 20 | recnd 11244 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
22 | 19, 21 | absmuld 15403 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
23 | subrgsubg 20329 | . . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
24 | 3, 23 | ax-mp 5 | . . . . . . . 8 ⊢ ℝ ∈ (SubGrp‘ℂfld) |
25 | eqid 2732 | . . . . . . . . 9 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
26 | eqid 2732 | . . . . . . . . . . 11 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
27 | 7, 26 | zlmvsca 21081 | . . . . . . . . . 10 ⊢ (.g‘ℝfld) = ( ·𝑠 ‘(ℤMod‘ℝfld)) |
28 | 27 | eqcomi 2741 | . . . . . . . . 9 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = (.g‘ℝfld) |
29 | 25, 4, 28 | subgmulg 19022 | . . . . . . . 8 ⊢ ((ℝ ∈ (SubGrp‘ℂfld) ∧ 𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
30 | 24, 29 | mp3an1 1448 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
31 | cnfldmulg 20983 | . . . . . . . 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 2774 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥) = (𝑧 · 𝑥)) |
34 | 33 | fveq2d 6895 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = ((abs ↾ ℝ)‘(𝑧 · 𝑥))) |
35 | zre 12564 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → 𝑧 ∈ ℝ) | |
36 | remulcl 11197 | . . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧 · 𝑥) ∈ ℝ) | |
37 | fvres 6910 | . . . . . . 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 2772 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (abs‘(𝑧 · 𝑥))) |
41 | fvres 6910 | . . . . 5 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
42 | fvres 6910 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ((abs ↾ ℝ)‘𝑥) = (abs‘𝑥)) | |
43 | 41, 42 | oveqan12d 7430 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
44 | 22, 40, 43 | 3eqtr4d 2782 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))) |
45 | 44 | rgen2 3197 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) |
46 | rebase 21165 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
47 | 7, 46 | zlmbas 21074 | . . 3 ⊢ ℝ = (Base‘(ℤMod‘ℝfld)) |
48 | recusp 24906 | . . . . 5 ⊢ ℝfld ∈ CUnifSp | |
49 | 48 | elexi 3493 | . . . 4 ⊢ ℝfld ∈ V |
50 | cnring 20973 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
51 | ringmnd 20068 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
52 | 50, 51 | ax-mp 5 | . . . . . 6 ⊢ ℂfld ∈ Mnd |
53 | 0re 11218 | . . . . . 6 ⊢ 0 ∈ ℝ | |
54 | ax-resscn 11169 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
55 | cnfldbas 20954 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
56 | cnfld0 20975 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
57 | cnfldnm 24302 | . . . . . . 7 ⊢ abs = (norm‘ℂfld) | |
58 | 4, 55, 56, 57 | ressnm 32166 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ ℝ ∧ ℝ ⊆ ℂ) → (abs ↾ ℝ) = (norm‘ℝfld)) |
59 | 52, 53, 54, 58 | mp3an 1461 | . . . . 5 ⊢ (abs ↾ ℝ) = (norm‘ℝfld) |
60 | 7, 59 | zlmnm 33015 | . . . 4 ⊢ (ℝfld ∈ V → (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld))) |
61 | 49, 60 | ax-mp 5 | . . 3 ⊢ (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld)) |
62 | eqid 2732 | . . 3 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = ( ·𝑠 ‘(ℤMod‘ℝfld)) | |
63 | 7 | zlmsca 21080 | . . . 4 ⊢ (ℝfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℝfld))) |
64 | 49, 63 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℝfld)) |
65 | zringbas 21029 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
66 | zringnm 33007 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
67 | 66 | eqcomi 2741 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
68 | 47, 61, 62, 64, 65, 67 | isnlm 24199 | . 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 709 | 1 ⊢ (ℤMod‘ℝfld) ∈ NrmMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ⊆ wss 3948 ↾ cres 5678 ‘cfv 6543 (class class class)co 7411 ℂcc 11110 ℝcr 11111 0cc0 11112 · cmul 11117 ℤcz 12560 abscabs 15183 Scalarcsca 17202 ·𝑠 cvsca 17203 Mndcmnd 18627 .gcmg 18952 SubGrpcsubg 19002 Abelcabl 19651 Ringcrg 20058 SubRingcsubrg 20319 DivRingcdr 20361 LModclmod 20475 ℂfldccnfld 20950 ℤringczring 21023 ℤModczlm 21056 ℝfldcrefld 21163 CUnifSpccusp 23809 normcnm 24092 NrmGrpcngp 24093 NrmRingcnrg 24095 NrmModcnlm 24096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-tpos 8213 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-er 8705 df-map 8824 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-seq 13969 df-exp 14030 df-hash 14293 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-oppr 20154 df-dvdsr 20175 df-unit 20176 df-invr 20206 df-dvr 20219 df-subrg 20321 df-drng 20363 df-abv 20429 df-lmod 20477 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-metu 20949 df-cnfld 20951 df-zring 21024 df-zlm 21060 df-refld 21164 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-cn 22738 df-cnp 22739 df-haus 22826 df-cmp 22898 df-tx 23073 df-hmeo 23266 df-fil 23357 df-flim 23450 df-fcls 23452 df-ust 23712 df-utop 23743 df-uss 23768 df-usp 23769 df-cfilu 23799 df-cusp 23810 df-xms 23833 df-ms 23834 df-tms 23835 df-nm 24098 df-ngp 24099 df-nrg 24101 df-nlm 24102 df-cncf 24401 df-cfil 24779 df-cmet 24781 df-cms 24859 |
This theorem is referenced by: rerrext 33058 |
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