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 23383 | . . . 4 ⊢ ℂfld ∈ NrmRing | |
2 | eqid 2821 | . . . . 5 ⊢ (ℤMod‘ℂfld) = (ℤMod‘ℂfld) | |
3 | 2 | zhmnrg 31203 | . . . 4 ⊢ (ℂfld ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmRing) |
4 | nrgngp 23265 | . . . 4 ⊢ ((ℤMod‘ℂfld) ∈ NrmRing → (ℤMod‘ℂfld) ∈ NrmGrp) | |
5 | 1, 3, 4 | mp2b 10 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ NrmGrp |
6 | nrgring 23266 | . . . . 5 ⊢ (ℂfld ∈ NrmRing → ℂfld ∈ Ring) | |
7 | ringabl 19324 | . . . . 5 ⊢ (ℂfld ∈ Ring → ℂfld ∈ Abel) | |
8 | 1, 6, 7 | mp2b 10 | . . . 4 ⊢ ℂfld ∈ Abel |
9 | 2 | zlmlmod 20664 | . . . 4 ⊢ (ℂfld ∈ Abel ↔ (ℤMod‘ℂfld) ∈ LMod) |
10 | 8, 9 | mpbi 232 | . . 3 ⊢ (ℤMod‘ℂfld) ∈ LMod |
11 | zringnrg 23390 | . . 3 ⊢ ℤring ∈ NrmRing | |
12 | 5, 10, 11 | 3pm3.2i 1335 | . 2 ⊢ ((ℤMod‘ℂfld) ∈ NrmGrp ∧ (ℤMod‘ℂfld) ∈ LMod ∧ ℤring ∈ NrmRing) |
13 | simpl 485 | . . . . . 6 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℤ) | |
14 | 13 | zcnd 12082 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑧 ∈ ℂ) |
15 | simpr 487 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → 𝑥 ∈ ℂ) | |
16 | 14, 15 | absmuld 14808 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧 · 𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
17 | cnfldmulg 20571 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (𝑧(.g‘ℂfld)𝑥) = (𝑧 · 𝑥)) | |
18 | 17 | fveq2d 6668 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (abs‘(𝑧 · 𝑥))) |
19 | fvres 6683 | . . . . . 6 ⊢ (𝑧 ∈ ℤ → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) | |
20 | 19 | adantr 483 | . . . . 5 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → ((abs ↾ ℤ)‘𝑧) = (abs‘𝑧)) |
21 | 20 | oveq1d 7165 | . . . 4 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) = ((abs‘𝑧) · (abs‘𝑥))) |
22 | 16, 18, 21 | 3eqtr4d 2866 | . . 3 ⊢ ((𝑧 ∈ ℤ ∧ 𝑥 ∈ ℂ) → (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥))) |
23 | 22 | rgen2 3203 | . 2 ⊢ ∀𝑧 ∈ ℤ ∀𝑥 ∈ ℂ (abs‘(𝑧(.g‘ℂfld)𝑥)) = (((abs ↾ ℤ)‘𝑧) · (abs‘𝑥)) |
24 | cnfldbas 20543 | . . . 4 ⊢ ℂ = (Base‘ℂfld) | |
25 | 2, 24 | zlmbas 20659 | . . 3 ⊢ ℂ = (Base‘(ℤMod‘ℂfld)) |
26 | cnfldex 20542 | . . . 4 ⊢ ℂfld ∈ V | |
27 | cnfldnm 23381 | . . . . 5 ⊢ abs = (norm‘ℂfld) | |
28 | 2, 27 | zlmnm 31202 | . . . 4 ⊢ (ℂfld ∈ V → abs = (norm‘(ℤMod‘ℂfld))) |
29 | 26, 28 | ax-mp 5 | . . 3 ⊢ abs = (norm‘(ℤMod‘ℂfld)) |
30 | eqid 2821 | . . . 4 ⊢ (.g‘ℂfld) = (.g‘ℂfld) | |
31 | 2, 30 | zlmvsca 20663 | . . 3 ⊢ (.g‘ℂfld) = ( ·𝑠 ‘(ℤMod‘ℂfld)) |
32 | 2 | zlmsca 20662 | . . . 4 ⊢ (ℂfld ∈ V → ℤring = (Scalar‘(ℤMod‘ℂfld))) |
33 | 26, 32 | ax-mp 5 | . . 3 ⊢ ℤring = (Scalar‘(ℤMod‘ℂfld)) |
34 | zringbas 20617 | . . 3 ⊢ ℤ = (Base‘ℤring) | |
35 | zringnm 31196 | . . . 4 ⊢ (norm‘ℤring) = (abs ↾ ℤ) | |
36 | 35 | eqcomi 2830 | . . 3 ⊢ (abs ↾ ℤ) = (norm‘ℤring) |
37 | 25, 29, 31, 33, 34, 36 | isnlm 23278 | . 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 709 | 1 ⊢ (ℤMod‘ℂfld) ∈ NrmMod |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3494 ↾ cres 5551 ‘cfv 6349 (class class class)co 7150 ℂcc 10529 · cmul 10536 ℤcz 11975 abscabs 14587 Scalarcsca 16562 .gcmg 18218 Abelcabl 18901 Ringcrg 19291 LModclmod 19628 ℂfldccnfld 20539 ℤringzring 20611 ℤModczlm 20642 normcnm 23180 NrmGrpcngp 23181 NrmRingcnrg 23183 NrmModcnlm 23184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-map 8402 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ico 12738 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-rest 16690 df-topn 16691 df-0g 16709 df-topgen 16711 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-abv 19582 df-lmod 19630 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-cnfld 20540 df-zring 20612 df-zlm 20646 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-xms 22924 df-ms 22925 df-nm 23186 df-ngp 23187 df-nrg 23189 df-nlm 23190 |
This theorem is referenced by: cnrrext 31246 |
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