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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 26076 | . . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
5 | dchrghm.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 4, 5 | sseldi 3885 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
7 | 1, 3 | dchrrcl 26075 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
8 | 5, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
9 | 8 | nnnn0d 12115 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
10 | 2 | zncrng 20463 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
12 | crngring 19528 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
14 | dchrghm.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑍) | |
15 | eqid 2736 | . . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
16 | 14, 15 | unitsubm 19642 | . . . . 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 18201 | . . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
20 | 6, 17, 19 | syl2anc 587 | . . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
21 | cnring 20339 | . . . . 5 ⊢ ℂfld ∈ Ring | |
22 | cnfldbas 20321 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
23 | cnfld0 20341 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
24 | cndrng 20346 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
25 | 22, 23, 24 | drngui 19727 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
26 | eqid 2736 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
27 | 25, 26 | unitsubm 19642 | . . . . 5 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
28 | 21, 27 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
29 | df-ima 5549 | . . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈) | |
30 | eqid 2736 | . . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
31 | 1, 2, 3, 30, 5 | dchrf 26077 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
32 | 30, 14 | unitss 19632 | . . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍) |
33 | 32 | sseli 3883 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍)) |
34 | ffvelrn 6880 | . . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ) | |
35 | 31, 33, 34 | syl2an 599 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ) |
36 | simpr 488 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
37 | 5 | adantr 484 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷) |
38 | 33 | adantl 485 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍)) |
39 | 1, 2, 3, 30, 14, 37, 38 | dchrn0 26085 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈)) |
40 | 36, 39 | mpbird 260 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0) |
41 | eldifsn 4686 | . . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0)) | |
42 | 35, 40, 41 | sylanbrc 586 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
43 | 42 | ralrimiva 3095 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
44 | 31 | ffund 6527 | . . . . . . 7 ⊢ (𝜑 → Fun 𝑋) |
45 | 31 | fdmd 6534 | . . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍)) |
46 | 32, 45 | sseqtrrid 3940 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋) |
47 | funimass4 6755 | . . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) | |
48 | 44, 46, 47 | syl2anc 587 | . . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) |
49 | 43, 48 | mpbird 260 | . . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0})) |
50 | 29, 49 | eqsstrrid 3936 | . . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) |
51 | dchrghm.m | . . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
52 | 51 | resmhm2b 18203 | . . . 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 590 | . . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))) |
54 | 20, 53 | mpbid 235 | . 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)) |
55 | 14, 18 | unitgrp 19639 | . . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
56 | 13, 55 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Grp) |
57 | 51 | cnmgpabl 20378 | . . . 4 ⊢ 𝑀 ∈ Abel |
58 | ablgrp 19129 | . . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
59 | 57, 58 | ax-mp 5 | . . 3 ⊢ 𝑀 ∈ Grp |
60 | ghmmhmb 18587 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) | |
61 | 56, 59, 60 | sylancl 589 | . 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) |
62 | 54, 61 | eleqtrrd 2834 | 1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 ∖ cdif 3850 ⊆ wss 3853 {csn 4527 dom cdm 5536 ran crn 5537 ↾ cres 5538 “ cima 5539 Fun wfun 6352 ⟶wf 6354 ‘cfv 6358 (class class class)co 7191 ℂcc 10692 0cc0 10694 ℕcn 11795 ℕ0cn0 12055 Basecbs 16666 ↾s cress 16667 MndHom cmhm 18170 SubMndcsubmnd 18171 Grpcgrp 18319 GrpHom cghm 18573 Abelcabl 19125 mulGrpcmgp 19458 Ringcrg 19516 CRingccrg 19517 Unitcui 19611 ℂfldccnfld 20317 ℤ/nℤczn 20423 DChrcdchr 26067 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 ax-cnex 10750 ax-resscn 10751 ax-1cn 10752 ax-icn 10753 ax-addcl 10754 ax-addrcl 10755 ax-mulcl 10756 ax-mulrcl 10757 ax-mulcom 10758 ax-addass 10759 ax-mulass 10760 ax-distr 10761 ax-i2m1 10762 ax-1ne0 10763 ax-1rid 10764 ax-rnegex 10765 ax-rrecex 10766 ax-cnre 10767 ax-pre-lttri 10768 ax-pre-lttrn 10769 ax-pre-ltadd 10770 ax-pre-mulgt0 10771 ax-addf 10773 ax-mulf 10774 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-nel 3037 df-ral 3056 df-rex 3057 df-reu 3058 df-rmo 3059 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-riota 7148 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-tpos 7946 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-1o 8180 df-er 8369 df-ec 8371 df-qs 8375 df-map 8488 df-en 8605 df-dom 8606 df-sdom 8607 df-fin 8608 df-sup 9036 df-inf 9037 df-pnf 10834 df-mnf 10835 df-xr 10836 df-ltxr 10837 df-le 10838 df-sub 11029 df-neg 11030 df-div 11455 df-nn 11796 df-2 11858 df-3 11859 df-4 11860 df-5 11861 df-6 11862 df-7 11863 df-8 11864 df-9 11865 df-n0 12056 df-z 12142 df-dec 12259 df-uz 12404 df-fz 13061 df-struct 16668 df-ndx 16669 df-slot 16670 df-base 16672 df-sets 16673 df-ress 16674 df-plusg 16762 df-mulr 16763 df-starv 16764 df-sca 16765 df-vsca 16766 df-ip 16767 df-tset 16768 df-ple 16769 df-ds 16771 df-unif 16772 df-0g 16900 df-imas 16967 df-qus 16968 df-mgm 18068 df-sgrp 18117 df-mnd 18128 df-mhm 18172 df-submnd 18173 df-grp 18322 df-minusg 18323 df-sbg 18324 df-subg 18494 df-nsg 18495 df-eqg 18496 df-ghm 18574 df-cmn 19126 df-abl 19127 df-mgp 19459 df-ur 19471 df-ring 19518 df-cring 19519 df-oppr 19595 df-dvdsr 19613 df-unit 19614 df-invr 19644 df-dvr 19655 df-drng 19723 df-subrg 19752 df-lmod 19855 df-lss 19923 df-lsp 19963 df-sra 20163 df-rgmod 20164 df-lidl 20165 df-rsp 20166 df-2idl 20224 df-cnfld 20318 df-zring 20390 df-zn 20427 df-dchr 26068 |
This theorem is referenced by: dchrabs 26095 sum2dchr 26109 |
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