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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 24766 | . . . . 5 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
5 | dchrghm.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
6 | 4, 5 | sseldi 3566 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
7 | 1, 3 | dchrrcl 24765 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
8 | 5, 7 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
9 | 8 | nnnn0d 11228 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
10 | 2 | zncrng 19712 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
11 | 9, 10 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
12 | crngring 18381 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
13 | 11, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
14 | dchrghm.u | . . . . . 6 ⊢ 𝑈 = (Unit‘𝑍) | |
15 | eqid 2610 | . . . . . 6 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
16 | 14, 15 | unitsubm 18493 | . . . . 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 17182 | . . . 4 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ 𝑈 ∈ (SubMnd‘(mulGrp‘𝑍))) → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
20 | 6, 17, 19 | syl2anc 691 | . . 3 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld))) |
21 | cnring 19587 | . . . . 5 ⊢ ℂfld ∈ Ring | |
22 | cnfldbas 19571 | . . . . . . 7 ⊢ ℂ = (Base‘ℂfld) | |
23 | cnfld0 19589 | . . . . . . 7 ⊢ 0 = (0g‘ℂfld) | |
24 | cndrng 19594 | . . . . . . 7 ⊢ ℂfld ∈ DivRing | |
25 | 22, 23, 24 | drngui 18576 | . . . . . 6 ⊢ (ℂ ∖ {0}) = (Unit‘ℂfld) |
26 | eqid 2610 | . . . . . 6 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
27 | 25, 26 | unitsubm 18493 | . . . . 5 ⊢ (ℂfld ∈ Ring → (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld))) |
28 | 21, 27 | ax-mp 5 | . . . 4 ⊢ (ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) |
29 | df-ima 5051 | . . . . 5 ⊢ (𝑋 “ 𝑈) = ran (𝑋 ↾ 𝑈) | |
30 | eqid 2610 | . . . . . . . . . 10 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
31 | 1, 2, 3, 30, 5 | dchrf 24767 | . . . . . . . . 9 ⊢ (𝜑 → 𝑋:(Base‘𝑍)⟶ℂ) |
32 | 30, 14 | unitss 18483 | . . . . . . . . . 10 ⊢ 𝑈 ⊆ (Base‘𝑍) |
33 | 32 | sseli 3564 | . . . . . . . . 9 ⊢ (𝑥 ∈ 𝑈 → 𝑥 ∈ (Base‘𝑍)) |
34 | ffvelrn 6265 | . . . . . . . . 9 ⊢ ((𝑋:(Base‘𝑍)⟶ℂ ∧ 𝑥 ∈ (Base‘𝑍)) → (𝑋‘𝑥) ∈ ℂ) | |
35 | 31, 33, 34 | syl2an 493 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ ℂ) |
36 | simpr 476 | . . . . . . . . 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 24775 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → ((𝑋‘𝑥) ≠ 0 ↔ 𝑥 ∈ 𝑈)) |
40 | 36, 39 | mpbird 246 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ≠ 0) |
41 | eldifsn 4260 | . . . . . . . 8 ⊢ ((𝑋‘𝑥) ∈ (ℂ ∖ {0}) ↔ ((𝑋‘𝑥) ∈ ℂ ∧ (𝑋‘𝑥) ≠ 0)) | |
42 | 35, 40, 41 | sylanbrc 695 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝑈) → (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
43 | 42 | ralrimiva 2949 | . . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0})) |
44 | ffun 5961 | . . . . . . . 8 ⊢ (𝑋:(Base‘𝑍)⟶ℂ → Fun 𝑋) | |
45 | 31, 44 | syl 17 | . . . . . . 7 ⊢ (𝜑 → Fun 𝑋) |
46 | fdm 5964 | . . . . . . . . 9 ⊢ (𝑋:(Base‘𝑍)⟶ℂ → dom 𝑋 = (Base‘𝑍)) | |
47 | 31, 46 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → dom 𝑋 = (Base‘𝑍)) |
48 | 32, 47 | syl5sseqr 3617 | . . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ dom 𝑋) |
49 | funimass4 6157 | . . . . . . 7 ⊢ ((Fun 𝑋 ∧ 𝑈 ⊆ dom 𝑋) → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) | |
50 | 45, 48, 49 | syl2anc 691 | . . . . . 6 ⊢ (𝜑 → ((𝑋 “ 𝑈) ⊆ (ℂ ∖ {0}) ↔ ∀𝑥 ∈ 𝑈 (𝑋‘𝑥) ∈ (ℂ ∖ {0}))) |
51 | 43, 50 | mpbird 246 | . . . . 5 ⊢ (𝜑 → (𝑋 “ 𝑈) ⊆ (ℂ ∖ {0})) |
52 | 29, 51 | syl5eqssr 3613 | . . . 4 ⊢ (𝜑 → ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) |
53 | dchrghm.m | . . . . 5 ⊢ 𝑀 = ((mulGrp‘ℂfld) ↾s (ℂ ∖ {0})) | |
54 | 53 | resmhm2b 17184 | . . . 4 ⊢ (((ℂ ∖ {0}) ∈ (SubMnd‘(mulGrp‘ℂfld)) ∧ ran (𝑋 ↾ 𝑈) ⊆ (ℂ ∖ {0})) → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))) |
55 | 28, 52, 54 | sylancr 694 | . . 3 ⊢ (𝜑 → ((𝑋 ↾ 𝑈) ∈ (𝐻 MndHom (mulGrp‘ℂfld)) ↔ (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀))) |
56 | 20, 55 | mpbid 221 | . 2 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 MndHom 𝑀)) |
57 | 14, 18 | unitgrp 18490 | . . . 4 ⊢ (𝑍 ∈ Ring → 𝐻 ∈ Grp) |
58 | 13, 57 | syl 17 | . . 3 ⊢ (𝜑 → 𝐻 ∈ Grp) |
59 | 53 | cnmgpabl 19626 | . . . 4 ⊢ 𝑀 ∈ Abel |
60 | ablgrp 18021 | . . . 4 ⊢ (𝑀 ∈ Abel → 𝑀 ∈ Grp) | |
61 | 59, 60 | ax-mp 5 | . . 3 ⊢ 𝑀 ∈ Grp |
62 | ghmmhmb 17494 | . . 3 ⊢ ((𝐻 ∈ Grp ∧ 𝑀 ∈ Grp) → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) | |
63 | 58, 61, 62 | sylancl 693 | . 2 ⊢ (𝜑 → (𝐻 GrpHom 𝑀) = (𝐻 MndHom 𝑀)) |
64 | 56, 63 | eleqtrrd 2691 | 1 ⊢ (𝜑 → (𝑋 ↾ 𝑈) ∈ (𝐻 GrpHom 𝑀)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∖ cdif 3537 ⊆ wss 3540 {csn 4125 dom cdm 5038 ran crn 5039 ↾ cres 5040 “ cima 5041 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 ℕcn 10897 ℕ0cn0 11169 Basecbs 15695 ↾s cress 15696 MndHom cmhm 17156 SubMndcsubmnd 17157 Grpcgrp 17245 GrpHom cghm 17480 Abelcabl 18017 mulGrpcmgp 18312 Ringcrg 18370 CRingccrg 18371 Unitcui 18462 ℂfldccnfld 19567 ℤ/nℤczn 19670 DChrcdchr 24757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-fz 12198 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-invr 18495 df-dvr 18506 df-drng 18572 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zn 19674 df-dchr 24758 |
This theorem is referenced by: dchrabs 24785 sum2dchr 24799 |
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