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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 27219 | . . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
5 | dchrghm.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 4, 5 | sselid 3974 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
7 | 1, 3 | dchrrcl 27218 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
8 | 5, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
9 | 8 | nnnn0d 12565 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
10 | 2 | zncrng 21495 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
12 | crngring 20197 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
14 | dchrghm.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑍) | |
15 | eqid 2725 | . . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
16 | 14, 15 | unitsubm 20337 | . . . . 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 18780 | . . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
20 | 6, 17, 19 | syl2anc 582 | . . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
21 | cnring 21335 | . . . . 5 ⊢ ℂfld ∈ Ring | |
22 | cnfldbas 21300 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
23 | cnfld0 21337 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
24 | cndrng 21343 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
25 | 22, 23, 24 | drngui 20642 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
26 | eqid 2725 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
27 | 25, 26 | unitsubm 20337 | . . . . 5 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
28 | 21, 27 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
29 | df-ima 5691 | . . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈) | |
30 | eqid 2725 | . . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
31 | 1, 2, 3, 30, 5 | dchrf 27220 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
32 | 30, 14 | unitss 20327 | . . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍) |
33 | 32 | sseli 3972 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍)) |
34 | ffvelcdm 7090 | . . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ) | |
35 | 31, 33, 34 | syl2an 594 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ) |
36 | simpr 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
37 | 5 | adantr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷) |
38 | 33 | adantl 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍)) |
39 | 1, 2, 3, 30, 14, 37, 38 | dchrn0 27228 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈)) |
40 | 36, 39 | mpbird 256 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0) |
41 | eldifsn 4792 | . . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0)) | |
42 | 35, 40, 41 | sylanbrc 581 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
43 | 42 | ralrimiva 3135 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
44 | 31 | ffund 6727 | . . . . . . 7 ⊢ (𝜑 → Fun 𝑋) |
45 | 31 | fdmd 6733 | . . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍)) |
46 | 32, 45 | sseqtrrid 4030 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋) |
47 | funimass4 6962 | . . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) | |
48 | 44, 46, 47 | syl2anc 582 | . . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) |
49 | 43, 48 | mpbird 256 | . . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0})) |
50 | 29, 49 | eqsstrrid 4026 | . . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) |
51 | dchrghm.m | . . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
52 | 51 | resmhm2b 18782 | . . . 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 585 | . . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))) |
54 | 20, 53 | mpbid 231 | . 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)) |
55 | 14, 18 | unitgrp 20334 | . . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
56 | 13, 55 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Grp) |
57 | 51 | cnmgpabl 21378 | . . . 4 ⊢ 𝑀 ∈ Abel |
58 | ablgrp 19752 | . . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
59 | 57, 58 | ax-mp 5 | . . 3 ⊢ 𝑀 ∈ Grp |
60 | ghmmhmb 19190 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) | |
61 | 56, 59, 60 | sylancl 584 | . 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) |
62 | 54, 61 | eleqtrrd 2828 | 1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ≠ wne 2929 ∀wral 3050 ∖ cdif 3941 ⊆ wss 3944 {csn 4630 dom cdm 5678 ran crn 5679 ↾ cres 5680 “ cima 5681 Fun wfun 6543 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 ℂcc 11138 0cc0 11140 ℕcn 12245 ℕ0cn0 12505 Basecbs 17183 ↾s cress 17212 MndHom cmhm 18741 SubMndcsubmnd 18742 Grpcgrp 18898 GrpHom cghm 19175 Abelcabl 19748 mulGrpcmgp 20086 Ringcrg 20185 CRingccrg 20186 Unitcui 20306 ℂfldccnfld 21296 ℤ/nℤczn 21445 DChrcdchr 27210 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 ax-addf 11219 ax-mulf 11220 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-tpos 8232 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-ec 8727 df-qs 8731 df-map 8847 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9467 df-inf 9468 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-div 11904 df-nn 12246 df-2 12308 df-3 12309 df-4 12310 df-5 12311 df-6 12312 df-7 12313 df-8 12314 df-9 12315 df-n0 12506 df-z 12592 df-dec 12711 df-uz 12856 df-fz 13520 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-mulr 17250 df-starv 17251 df-sca 17252 df-vsca 17253 df-ip 17254 df-tset 17255 df-ple 17256 df-ds 17258 df-unif 17259 df-0g 17426 df-imas 17493 df-qus 17494 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-mhm 18743 df-submnd 18744 df-grp 18901 df-minusg 18902 df-sbg 18903 df-subg 19086 df-nsg 19087 df-eqg 19088 df-ghm 19176 df-cmn 19749 df-abl 19750 df-mgp 20087 df-rng 20105 df-ur 20134 df-ring 20187 df-cring 20188 df-oppr 20285 df-dvdsr 20308 df-unit 20309 df-invr 20339 df-dvr 20352 df-subrng 20495 df-subrg 20520 df-drng 20638 df-lmod 20757 df-lss 20828 df-lsp 20868 df-sra 21070 df-rgmod 21071 df-lidl 21116 df-rsp 21117 df-2idl 21157 df-cnfld 21297 df-zring 21390 df-zn 21449 df-dchr 27211 |
This theorem is referenced by: dchrabs 27238 sum2dchr 27252 |
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