Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 23386 | . . . . 5 ⊢ ℂfld ∈ NrmRing | |
2 | resubdrg 20297 | . . . . . 6 ⊢ (ℝ ∈ (SubRing‘ℂfld) ∧ ℝfld ∈ DivRing) | |
3 | 2 | simpli 487 | . . . . 5 ⊢ ℝ ∈ (SubRing‘ℂfld) |
4 | df-refld 20294 | . . . . . 6 ⊢ ℝfld = (ℂfld ↾s ℝ) | |
5 | 4 | subrgnrg 23279 | . . . . 5 ⊢ ((ℂfld ∈ NrmRing ∧ ℝ ∈ (SubRing‘ℂfld)) → ℝfld ∈ NrmRing) |
6 | 1, 3, 5 | mp2an 691 | . . . 4 ⊢ ℝfld ∈ NrmRing |
7 | eqid 2798 | . . . . 5 ⊢ (ℤMod‘ℝfld) = (ℤMod‘ℝfld) | |
8 | 7 | zhmnrg 31318 | . . . 4 ⊢ (ℝfld ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmRing) |
9 | nrgngp 23268 | . . . 4 ⊢ ((ℤMod‘ℝfld) ∈ NrmRing → (ℤMod‘ℝfld) ∈ NrmGrp) | |
10 | 6, 8, 9 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ NrmGrp |
11 | nrgring 23269 | . . . . 5 ⊢ (ℝfld ∈ NrmRing → ℝfld ∈ Ring) | |
12 | ringabl 19326 | . . . . 5 ⊢ (ℝfld ∈ Ring → ℝfld ∈ Abel) | |
13 | 6, 11, 12 | mp2b 10 | . . . 4 ⊢ ℝfld ∈ Abel |
14 | 7 | zlmlmod 20216 | . . . 4 ⊢ (ℝfld ∈ Abel ↔ (ℤMod‘ℝfld) ∈ LMod) |
15 | 13, 14 | mpbi 233 | . . 3 ⊢ (ℤMod‘ℝfld) ∈ LMod |
16 | zringnrg 23393 | . . 3 ⊢ ℤring ∈ NrmRing | |
17 | 10, 15, 16 | 3pm3.2i 1336 | . 2 ⊢ ((ℤMod‘ℝfld) ∈ NrmGrp ∧ (ℤMod‘ℝfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
18 | simpl 486 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℤ) | |
19 | 18 | zcnd 12076 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑧 ∈ ℂ) |
20 | simpr 488 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℝ) | |
21 | 20 | recnd 10658 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → 𝑥 ∈ ℂ) |
22 | 19, 21 | absmuld 14806 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
23 | subrgsubg 19534 | . . . . . . . . 9 ⊢ (ℝ ∈ (SubRing‘ℂfld) → ℝ ∈ (SubGrp‘ℂfld)) | |
24 | 3, 23 | ax-mp 5 | . . . . . . . 8 ⊢ ℝ ∈ (SubGrp‘ℂfld) |
25 | eqid 2798 | . . . . . . . . 9 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
26 | eqid 2798 | . . . . . . . . . . 11 ⊢ (.g‘ℝfld) = (.g‘ℝfld) | |
27 | 7, 26 | zlmvsca 20215 | . . . . . . . . . 10 ⊢ (.g‘ℝfld) = ( ·𝑠 ‘(ℤMod‘ℝfld)) |
28 | 27 | eqcomi 2807 | . . . . . . . . 9 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = (.g‘ℝfld) |
29 | 25, 4, 28 | subgmulg 18285 | . . . . . . . 8 ⊢ ((ℝ ∈ (SubGrp‘ℂfld) ∧ 𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
30 | 24, 29 | mp3an1 1445 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) |
31 | cnfldmulg 20123 | . . . . . . . 8 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
32 | 21, 31 | syldan 594 | . . . . . . 7 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) |
33 | 30, 32 | eqtr3d 2835 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥) = (𝑧 · 𝑥)) |
34 | 33 | fveq2d 6649 | . . . . 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 10611 | . . . . . . 7 ⊢ ((𝑧 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑧 · 𝑥) ∈ ℝ) | |
37 | fvres 6664 | . . . . . . 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 583 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧 · 𝑥)) = (abs‘(𝑧 · 𝑥))) |
40 | 34, 39 | eqtrd 2833 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (abs‘(𝑧 · 𝑥))) |
41 | fvres 6664 | . . . . 5 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
42 | fvres 6664 | . . . . 5 ⊢ (𝑥 ∈ ℝ → ((abs ↾ ℝ)‘𝑥) = (abs‘𝑥)) | |
43 | 41, 42 | oveqan12d 7154 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
44 | 22, 40, 43 | 3eqtr4d 2843 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℝ) → ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥))) |
45 | 44 | rgen2 3168 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℝ ((abs ↾ ℝ)‘(𝑧( ·𝑠 ‘(ℤMod‘ℝfld))𝑥)) = (((abs ↾ ℤ)‘𝑧) · ((abs ↾ ℝ)‘𝑥)) |
46 | rebase 20295 | . . . 4 ⊢ ℝ = (Base‘ℝfld) | |
47 | 7, 46 | zlmbas 20211 | . . 3 ⊢ ℝ = (Base‘(ℤMod‘ℝfld)) |
48 | recusp 23986 | . . . . 5 ⊢ ℝfld ∈ CUnifSp | |
49 | 48 | elexi 3460 | . . . 4 ⊢ ℝfld ∈ V |
50 | cnring 20113 | . . . . . . 7 ⊢ ℂfld ∈ Ring | |
51 | ringmnd 19300 | . . . . . . 7 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Mnd) | |
52 | 50, 51 | ax-mp 5 | . . . . . 6 ⊢ ℂfld ∈ Mnd |
53 | 0re 10632 | . . . . . 6 ⊢ 0 ∈ ℝ | |
54 | ax-resscn 10583 | . . . . . 6 ⊢ ℝ ⊆ ℂ | |
55 | cnfldbas 20095 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
56 | cnfld0 20115 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
57 | cnfldnm 23384 | . . . . . . 7 ⊢ abs = (norm‘ℂfld) | |
58 | 4, 55, 56, 57 | ressnm 30664 | . . . . . 6 ⊢ ((ℂfld ∈ Mnd ∧ 0 ∈ ℝ ∧ ℝ ⊆ ℂ) → (abs ↾ ℝ) = (norm‘ℝfld)) |
59 | 52, 53, 54, 58 | mp3an 1458 | . . . . 5 ⊢ (abs ↾ ℝ) = (norm‘ℝfld) |
60 | 7, 59 | zlmnm 31317 | . . . 4 ⊢ (ℝfld ∈ V → (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld))) |
61 | 49, 60 | ax-mp 5 | . . 3 ⊢ (abs ↾ ℝ) = (norm‘(ℤMod‘ℝfld)) |
62 | eqid 2798 | . . 3 ⊢ ( ·𝑠 ‘(ℤMod‘ℝfld)) = ( ·𝑠 ‘(ℤMod‘ℝfld)) | |
63 | 7 | zlmsca 20214 | . . . 4 ⊢ (ℝfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℝfld))) |
64 | 49, 63 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℝfld)) |
65 | zringbas 20169 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
66 | zringnm 31311 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
67 | 66 | eqcomi 2807 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
68 | 47, 61, 62, 64, 65, 67 | isnlm 23281 | . 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 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 · cmul 10531 ℤcz 11969 abscabs 14585 Scalarcsca 16560 ·𝑠 cvsca 16561 Mndcmnd 17903 .gcmg 18216 SubGrpcsubg 18265 Abelcabl 18899 Ringcrg 19290 DivRingcdr 19495 SubRingcsubrg 19524 LModclmod 19627 ℂfldccnfld 20091 ℤringzring 20163 ℤModczlm 20194 ℝfldcrefld 20293 CUnifSpccusp 22903 normcnm 23183 NrmGrpcngp 23184 NrmRingcnrg 23186 NrmModcnlm 23187 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 ax-pre-sup 10604 ax-addf 10605 ax-mulf 10606 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-iin 4884 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-se 5479 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-isom 6333 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-of 7389 df-om 7561 df-1st 7671 df-2nd 7672 df-supp 7814 df-tpos 7875 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-2o 8086 df-oadd 8089 df-er 8272 df-map 8391 df-ixp 8445 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-fsupp 8818 df-fi 8859 df-sup 8890 df-inf 8891 df-oi 8958 df-card 9352 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-div 11287 df-nn 11626 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 12886 df-fzo 13029 df-seq 13365 df-exp 13426 df-hash 13687 df-cj 14450 df-re 14451 df-im 14452 df-sqrt 14586 df-abs 14587 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-mulr 16571 df-starv 16572 df-sca 16573 df-vsca 16574 df-ip 16575 df-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-hom 16581 df-cco 16582 df-rest 16688 df-topn 16689 df-0g 16707 df-gsum 16708 df-topgen 16709 df-pt 16710 df-prds 16713 df-xrs 16767 df-qtop 16772 df-imas 16773 df-xps 16775 df-mre 16849 df-mrc 16850 df-acs 16852 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cntz 18439 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 df-oppr 19369 df-dvdsr 19387 df-unit 19388 df-invr 19418 df-dvr 19429 df-drng 19497 df-subrg 19526 df-abv 19581 df-lmod 19629 df-psmet 20083 df-xmet 20084 df-met 20085 df-bl 20086 df-mopn 20087 df-fbas 20088 df-fg 20089 df-metu 20090 df-cnfld 20092 df-zring 20164 df-zlm 20198 df-refld 20294 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-cld 21624 df-ntr 21625 df-cls 21626 df-nei 21703 df-cn 21832 df-cnp 21833 df-haus 21920 df-cmp 21992 df-tx 22167 df-hmeo 22360 df-fil 22451 df-flim 22544 df-fcls 22546 df-ust 22806 df-utop 22837 df-uss 22862 df-usp 22863 df-cfilu 22893 df-cusp 22904 df-xms 22927 df-ms 22928 df-tms 22929 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 df-cncf 23483 df-cfil 23859 df-cmet 23861 df-cms 23939 |
This theorem is referenced by: rerrext 31360 |
Copyright terms: Public domain | W3C validator |