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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 27300 | . . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
5 | dchrghm.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 4, 5 | sselid 3993 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
7 | 1, 3 | dchrrcl 27299 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
8 | 5, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
9 | 8 | nnnn0d 12585 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
10 | 2 | zncrng 21581 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
12 | crngring 20263 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
14 | dchrghm.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑍) | |
15 | eqid 2735 | . . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
16 | 14, 15 | unitsubm 20403 | . . . . 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 18846 | . . . 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 21421 | . . . . 5 ⊢ ℂfld ∈ Ring | |
22 | cnfldbas 21386 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
23 | cnfld0 21423 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
24 | cndrng 21429 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
25 | 22, 23, 24 | drngui 20752 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
26 | eqid 2735 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
27 | 25, 26 | unitsubm 20403 | . . . . 5 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
28 | 21, 27 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
29 | df-ima 5702 | . . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈) | |
30 | eqid 2735 | . . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
31 | 1, 2, 3, 30, 5 | dchrf 27301 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
32 | 30, 14 | unitss 20393 | . . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍) |
33 | 32 | sseli 3991 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍)) |
34 | ffvelcdm 7101 | . . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ) | |
35 | 31, 33, 34 | syl2an 596 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ) |
36 | simpr 484 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ 𝑈) | |
37 | 5 | adantr 480 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑋 ∈ 𝐷) |
38 | 33 | adantl 481 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → 𝑥 ∈ (Base‘𝑍)) |
39 | 1, 2, 3, 30, 14, 37, 38 | dchrn0 27309 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈)) |
40 | 36, 39 | mpbird 257 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0) |
41 | eldifsn 4791 | . . . . . . . 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 3144 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
44 | 31 | ffund 6741 | . . . . . . 7 ⊢ (𝜑 → Fun 𝑋) |
45 | 31 | fdmd 6747 | . . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍)) |
46 | 32, 45 | sseqtrrid 4049 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋) |
47 | funimass4 6973 | . . . . . . 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 257 | . . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0})) |
50 | 29, 49 | eqsstrrid 4045 | . . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) |
51 | dchrghm.m | . . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
52 | 51 | resmhm2b 18848 | . . . 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 232 | . 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)) |
55 | 14, 18 | unitgrp 20400 | . . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
56 | 13, 55 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Grp) |
57 | 51 | cnmgpabl 21464 | . . . 4 ⊢ 𝑀 ∈ Abel |
58 | ablgrp 19818 | . . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
59 | 57, 58 | ax-mp 5 | . . 3 ⊢ 𝑀 ∈ Grp |
60 | ghmmhmb 19258 | . . 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 2842 | 1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∀wral 3059 ∖ cdif 3960 ⊆ wss 3963 {csn 4631 dom cdm 5689 ran crn 5690 ↾ cres 5691 “ cima 5692 Fun wfun 6557 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℂcc 11151 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 Basecbs 17245 ↾s cress 17274 MndHom cmhm 18807 SubMndcsubmnd 18808 Grpcgrp 18964 GrpHom cghm 19243 Abelcabl 19814 mulGrpcmgp 20152 Ringcrg 20251 CRingccrg 20252 Unitcui 20372 ℂfldccnfld 21382 ℤ/nℤczn 21531 DChrcdchr 27291 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 ax-addf 11232 ax-mulf 11233 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-tp 4636 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-tpos 8250 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-ec 8746 df-qs 8750 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-sup 9480 df-inf 9481 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-nn 12265 df-2 12327 df-3 12328 df-4 12329 df-5 12330 df-6 12331 df-7 12332 df-8 12333 df-9 12334 df-n0 12525 df-z 12612 df-dec 12732 df-uz 12877 df-fz 13545 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-0g 17488 df-imas 17555 df-qus 17556 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-mhm 18809 df-submnd 18810 df-grp 18967 df-minusg 18968 df-sbg 18969 df-subg 19154 df-nsg 19155 df-eqg 19156 df-ghm 19244 df-cmn 19815 df-abl 19816 df-mgp 20153 df-rng 20171 df-ur 20200 df-ring 20253 df-cring 20254 df-oppr 20351 df-dvdsr 20374 df-unit 20375 df-invr 20405 df-dvr 20418 df-subrng 20563 df-subrg 20587 df-drng 20748 df-lmod 20877 df-lss 20948 df-lsp 20988 df-sra 21190 df-rgmod 21191 df-lidl 21236 df-rsp 21237 df-2idl 21278 df-cnfld 21383 df-zring 21476 df-zn 21535 df-dchr 27292 |
This theorem is referenced by: dchrabs 27319 sum2dchr 27333 |
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