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Mirrors > Home > MPE Home > Th. List > dchrghm | Structured version Visualization version GIF version |
Description: A Dirichlet character restricted to the unit group of ℤ/nℤ is a group homomorphism into the multiplicative group of nonzero complex numbers. (Contributed by Mario Carneiro, 21-Apr-2016.) |
Ref | Expression |
---|---|
dchrghm.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrghm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrghm.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrghm.u | ⊢ 𝑈 = (Unit‘𝑍) |
dchrghm.h | ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) |
dchrghm.m | ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) |
dchrghm.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
Ref | Expression |
---|---|
dchrghm | ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrghm.g | . . . . . 6 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | dchrghm.z | . . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
3 | dchrghm.b | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
4 | 1, 2, 3 | dchrmhm 25744 | . . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
5 | dchrghm.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 4, 5 | sseldi 3962 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
7 | 1, 3 | dchrrcl 25743 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
8 | 5, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
9 | 8 | nnnn0d 11943 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
10 | 2 | zncrng 20619 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
12 | crngring 19237 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
14 | dchrghm.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑍) | |
15 | eqid 2818 | . . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
16 | 14, 15 | unitsubm 19349 | . . . . 5 ⊢ (𝑍 ∈ Ring → 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) |
17 | 13, 16 | syl 17 | . . . 4 ⊢ (𝜑 → 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) |
18 | dchrghm.h | . . . . 5 ⊢ 𝐻 = ((mulGrp‘𝑍) ↾s 𝑈) | |
19 | 18 | resmhm 17973 | . . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
20 | 6, 17, 19 | syl2anc 584 | . . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
21 | cnring 20495 | . . . . 5 ⊢ ℂfld ∈ Ring | |
22 | cnfldbas 20477 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
23 | cnfld0 20497 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
24 | cndrng 20502 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
25 | 22, 23, 24 | drngui 19437 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
26 | eqid 2818 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
27 | 25, 26 | unitsubm 19349 | . . . . 5 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
28 | 21, 27 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
29 | df-ima 5561 | . . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈) | |
30 | eqid 2818 | . . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
31 | 1, 2, 3, 30, 5 | dchrf 25745 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
32 | 30, 14 | unitss 19339 | . . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍) |
33 | 32 | sseli 3960 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍)) |
34 | ffvelrn 6841 | . . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ) | |
35 | 31, 33, 34 | syl2an 595 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ) |
36 | simpr 485 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
37 | 5 | adantr 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷) |
38 | 33 | adantl 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍)) |
39 | 1, 2, 3, 30, 14, 37, 38 | dchrn0 25753 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈)) |
40 | 36, 39 | mpbird 258 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0) |
41 | eldifsn 4711 | . . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0)) | |
42 | 35, 40, 41 | sylanbrc 583 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
43 | 42 | ralrimiva 3179 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
44 | 31 | ffund 6511 | . . . . . . 7 ⊢ (𝜑 → Fun 𝑋) |
45 | 31 | fdmd 6516 | . . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍)) |
46 | 32, 45 | sseqtrrid 4017 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋) |
47 | funimass4 6723 | . . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) | |
48 | 44, 46, 47 | syl2anc 584 | . . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) |
49 | 43, 48 | mpbird 258 | . . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0})) |
50 | 29, 49 | eqsstrrid 4013 | . . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) |
51 | dchrghm.m | . . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
52 | 51 | resmhm2b 17975 | . . . 4 ⊢ (((ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))) |
53 | 28, 50, 52 | sylancr 587 | . . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))) |
54 | 20, 53 | mpbid 233 | . 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)) |
55 | 14, 18 | unitgrp 19346 | . . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
56 | 13, 55 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Grp) |
57 | 51 | cnmgpabl 20534 | . . . 4 ⊢ 𝑀 ∈ Abel |
58 | ablgrp 18840 | . . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
59 | 57, 58 | ax-mp 5 | . . 3 ⊢ 𝑀 ∈ Grp |
60 | ghmmhmb 18307 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) | |
61 | 56, 59, 60 | sylancl 586 | . 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) |
62 | 54, 61 | eleqtrrd 2913 | 1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 ∖ cdif 3930 ⊆ wss 3933 {csn 4557 dom cdm 5548 ran crn 5549 ↾ cres 5550 “ cima 5551 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 0cc0 10525 ℕcn 11626 ℕ0cn0 11885 Basecbs 16471 ↾s cress 16472 MndHom cmhm 17942 SubMndcsubmnd 17943 Grpcgrp 18041 GrpHom cghm 18293 Abelcabl 18836 mulGrpcmgp 19168 Ringcrg 19226 CRingccrg 19227 Unitcui 19318 ℂfldccnfld 20473 ℤ/nℤczn 20578 DChrcdchr 25735 |
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-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-tpos 7881 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-ec 8280 df-qs 8284 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-fz 12881 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-ip 16571 df-tset 16572 df-ple 16573 df-ds 16575 df-unif 16576 df-0g 16703 df-imas 16769 df-qus 16770 df-mgm 17840 df-sgrp 17889 df-mnd 17900 df-mhm 17944 df-submnd 17945 df-grp 18044 df-minusg 18045 df-sbg 18046 df-subg 18214 df-nsg 18215 df-eqg 18216 df-ghm 18294 df-cmn 18837 df-abl 18838 df-mgp 19169 df-ur 19181 df-ring 19228 df-cring 19229 df-oppr 19302 df-dvdsr 19320 df-unit 19321 df-invr 19351 df-dvr 19362 df-drng 19433 df-subrg 19462 df-lmod 19565 df-lss 19633 df-lsp 19673 df-sra 19873 df-rgmod 19874 df-lidl 19875 df-rsp 19876 df-2idl 19933 df-cnfld 20474 df-zring 20546 df-zn 20582 df-dchr 25736 |
This theorem is referenced by: dchrabs 25763 sum2dchr 25777 |
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