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Mirrors > Home > MPE Home > Th. List > Mathboxes > cnzh | Structured version Visualization version GIF version |
Description: The ℤ-module of ℂ is a normed module. (Contributed by Thierry Arnoux, 25-Feb-2018.) |
Ref | Expression |
---|---|
cnzh | ⊢ (ℤMod‘ℂfld) ∈ NrmMod |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnnrg 23316 | . . . 4 ⊢ ℂfld ∈ NrmRing | |
2 | eqid 2818 | . . . . 5 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
3 | 2 | zhmnrg 31107 | . . . 4 ⊢ (ℂfld ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmRing) |
4 | nrgngp 23198 | . . . 4 ⊢ ((ℤMod‘ℂfld) ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmGrp) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmGrp |
6 | nrgring 23199 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
7 | ringabl 19259 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Abel) | |
8 | 1, 6, 7 | mp2b 10 | . . . 4 ⊢ ℂfld ∈ Abel |
9 | 2 | zlmlmod 20598 | . . . 4 ⊢ (ℂfld ∈ Abel ↔ (ℤMod‘ℂfld) ∈ LMod) |
10 | 8, 9 | mpbi 231 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ LMod |
11 | zringnrg 23323 | . . 3 ⊢ ℤring ∈ NrmRing | |
12 | 5, 10, 11 | 3pm3.2i 1331 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
13 | simpl 483 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℤ) | |
14 | 13 | zcnd 12076 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℂ) |
15 | simpr 485 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
16 | 14, 15 | absmuld 14802 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
17 | cnfldmulg 20505 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
18 | 17 | fveq2d 6667 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (abs‘(𝑧 · 𝑥))) |
19 | fvres 6682 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
20 | 19 | adantr 481 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) |
21 | 20 | oveq1d 7160 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
22 | 16, 18, 21 | 3eqtr4d 2863 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥))) |
23 | 22 | rgen2 3200 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) |
24 | cnfldbas 20477 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
25 | 2, 24 | zlmbas 20593 | . . 3 ⊢ ℂ = (Base‘(ℤMod‘ℂfld)) |
26 | cnfldex 20476 | . . . 4 ⊢ ℂfld ∈ V | |
27 | cnfldnm 23314 | . . . . 5 ⊢ abs = (norm‘ℂfld) | |
28 | 2, 27 | zlmnm 31106 | . . . 4 ⊢ (ℂfld ∈ V → abs = (norm‘(ℤMod‘ℂfld))) |
29 | 26, 28 | ax-mp 5 | . . 3 ⊢ abs = (norm‘(ℤMod‘ℂfld)) |
30 | eqid 2818 | . . . 4 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
31 | 2, 30 | zlmvsca 20597 | . . 3 ⊢ (.g‘ℂfld) = ( ·𝑠 ‘(ℤMod‘ℂfld)) |
32 | 2 | zlmsca 20596 | . . . 4 ⊢ (ℂfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℂfld))) |
33 | 26, 32 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℂfld)) |
34 | zringbas 20551 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
35 | zringnm 31100 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
36 | 35 | eqcomi 2827 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
37 | 25, 29, 31, 33, 34, 36 | isnlm 23211 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmMod ↔ (((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) ∧ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)))) |
38 | 12, 23, 37 | mpbir2an 707 | 1 ⊢ (ℤMod‘ℂfld) ∈ NrmMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 · cmul 10530 ℤcz 11969 abscabs 14581 Scalarcsca 16556 .gcmg 18162 Abelcabl 18836 Ringcrg 19226 LModclmod 19563 ℂfldccnfld 20473 ℤringzring 20545 ℤModczlm 20576 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-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-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-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 df-sup 8894 df-inf 8895 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-ico 12732 df-fz 12881 df-fzo 13022 df-seq 13358 df-exp 13418 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-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-rest 16684 df-topn 16685 df-0g 16703 df-topgen 16705 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-grp 18044 df-minusg 18045 df-sbg 18046 df-mulg 18163 df-subg 18214 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 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-cnfld 20474 df-zring 20546 df-zlm 20580 df-top 21430 df-topon 21447 df-topsp 21469 df-bases 21482 df-xms 22857 df-ms 22858 df-nm 23119 df-ngp 23120 df-nrg 23122 df-nlm 23123 |
This theorem is referenced by: cnrrext 31150 |
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