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Mathbox for Thierry Arnoux |
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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 23386 | . . . 4 ⊢ ℂfld ∈ NrmRing | |
2 | eqid 2798 | . . . . 5 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
3 | 2 | zhmnrg 31318 | . . . 4 ⊢ (ℂfld ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmRing) |
4 | nrgngp 23268 | . . . 4 ⊢ ((ℤMod‘ℂfld) ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmGrp) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmGrp |
6 | nrgring 23269 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
7 | ringabl 19326 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Abel) | |
8 | 1, 6, 7 | mp2b 10 | . . . 4 ⊢ ℂfld ∈ Abel |
9 | 2 | zlmlmod 20216 | . . . 4 ⊢ (ℂfld ∈ Abel ↔ (ℤMod‘ℂfld) ∈ LMod) |
10 | 8, 9 | mpbi 233 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ LMod |
11 | zringnrg 23393 | . . 3 ⊢ ℤring ∈ NrmRing | |
12 | 5, 10, 11 | 3pm3.2i 1336 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
13 | simpl 486 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℤ) | |
14 | 13 | zcnd 12076 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℂ) |
15 | simpr 488 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
16 | 14, 15 | absmuld 14806 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
17 | cnfldmulg 20123 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
18 | 17 | fveq2d 6649 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (abs‘(𝑧 · 𝑥))) |
19 | fvres 6664 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
20 | 19 | adantr 484 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) |
21 | 20 | oveq1d 7150 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
22 | 16, 18, 21 | 3eqtr4d 2843 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥))) |
23 | 22 | rgen2 3168 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) |
24 | cnfldbas 20095 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
25 | 2, 24 | zlmbas 20211 | . . 3 ⊢ ℂ = (Base‘(ℤMod‘ℂfld)) |
26 | cnfldex 20094 | . . . 4 ⊢ ℂfld ∈ V | |
27 | cnfldnm 23384 | . . . . 5 ⊢ abs = (norm‘ℂfld) | |
28 | 2, 27 | zlmnm 31317 | . . . 4 ⊢ (ℂfld ∈ V → abs = (norm‘(ℤMod‘ℂfld))) |
29 | 26, 28 | ax-mp 5 | . . 3 ⊢ abs = (norm‘(ℤMod‘ℂfld)) |
30 | eqid 2798 | . . . 4 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
31 | 2, 30 | zlmvsca 20215 | . . 3 ⊢ (.g‘ℂfld) = ( ·𝑠 ‘(ℤMod‘ℂfld)) |
32 | 2 | zlmsca 20214 | . . . 4 ⊢ (ℂfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℂfld))) |
33 | 26, 32 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℂfld)) |
34 | zringbas 20169 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
35 | zringnm 31311 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
36 | 35 | eqcomi 2807 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
37 | 25, 29, 31, 33, 34, 36 | isnlm 23281 | . 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 710 | 1 ⊢ (ℤMod‘ℂfld) ∈ NrmMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ↾ cres 5521 ‘cfv 6324 (class class class)co 7135 ℂcc 10524 · cmul 10531 ℤcz 11969 abscabs 14585 Scalarcsca 16560 .gcmg 18216 Abelcabl 18899 Ringcrg 19290 LModclmod 19627 ℂfldccnfld 20091 ℤringzring 20163 ℤModczlm 20194 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-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-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-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-fin 8496 df-sup 8890 df-inf 8891 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-ico 12732 df-fz 12886 df-fzo 13029 df-seq 13365 df-exp 13426 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-tset 16576 df-ple 16577 df-ds 16579 df-unif 16580 df-rest 16688 df-topn 16689 df-0g 16707 df-topgen 16709 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-grp 18098 df-minusg 18099 df-sbg 18100 df-mulg 18217 df-subg 18268 df-cmn 18900 df-abl 18901 df-mgp 19233 df-ur 19245 df-ring 19292 df-cring 19293 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-cnfld 20092 df-zring 20164 df-zlm 20198 df-top 21499 df-topon 21516 df-topsp 21538 df-bases 21551 df-xms 22927 df-ms 22928 df-nm 23189 df-ngp 23190 df-nrg 23192 df-nlm 23193 |
This theorem is referenced by: cnrrext 31361 |
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