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Mirrors > Home > MPE Home > Th. List > dchrzrhmul | Structured version Visualization version GIF version |
Description: A Dirichlet character is completely multiplicative. (Contributed by Mario Carneiro, 4-May-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmhm.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrmhm.b | ⊢ 𝐷 = (Base‘𝐺) |
dchrelbas4.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
dchrzrh1.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrzrh1.a | ⊢ (𝜑 → 𝐴 ∈ ℤ) |
dchrzrh1.c | ⊢ (𝜑 → 𝐶 ∈ ℤ) |
Ref | Expression |
---|---|
dchrzrhmul | ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrzrh1.x | . . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | dchrmhm.g | . . . . . . . . . 10 ⊢ 𝐺 = (DChr‘𝑁) | |
3 | dchrmhm.b | . . . . . . . . . 10 ⊢ 𝐷 = (Base‘𝐺) | |
4 | 2, 3 | dchrrcl 24965 | . . . . . . . . 9 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
5 | 1, 4 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ ℕ) |
6 | 5 | nnnn0d 11351 | . . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ0) |
7 | dchrmhm.z | . . . . . . . 8 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
8 | 7 | zncrng 19893 | . . . . . . 7 ⊢ (𝑁 ∈ ℕ0 → 𝑍 ∈ CRing) |
9 | 6, 8 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑍 ∈ CRing) |
10 | crngring 18558 | . . . . . 6 ⊢ (𝑍 ∈ CRing → 𝑍 ∈ Ring) | |
11 | 9, 10 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝑍 ∈ Ring) |
12 | dchrelbas4.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
13 | 12 | zrhrhm 19860 | . . . . 5 ⊢ (𝑍 ∈ Ring → 𝐿 ∈ (ℤring RingHom 𝑍)) |
14 | 11, 13 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (ℤring RingHom 𝑍)) |
15 | dchrzrh1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℤ) | |
16 | dchrzrh1.c | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℤ) | |
17 | zringbas 19824 | . . . . 5 ⊢ ℤ = (Base‘ℤring) | |
18 | zringmulr 19827 | . . . . 5 ⊢ · = (.r‘ℤring) | |
19 | eqid 2622 | . . . . 5 ⊢ (.r‘𝑍) = (.r‘𝑍) | |
20 | 17, 18, 19 | rhmmul 18727 | . . . 4 ⊢ ((𝐿 ∈ (ℤring RingHom 𝑍) ∧ 𝐴 ∈ ℤ ∧ 𝐶 ∈ ℤ) → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
21 | 14, 15, 16, 20 | syl3anc 1326 | . . 3 ⊢ (𝜑 → (𝐿‘(𝐴 · 𝐶)) = ((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) |
22 | 21 | fveq2d 6195 | . 2 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶)))) |
23 | 2, 7, 3 | dchrmhm 24966 | . . . 4 ⊢ 𝐷 ⊆ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) |
24 | 23, 1 | sseldi 3601 | . . 3 ⊢ (𝜑 → 𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld))) |
25 | eqid 2622 | . . . . . 6 ⊢ (Base‘𝑍) = (Base‘𝑍) | |
26 | 17, 25 | rhmf 18726 | . . . . 5 ⊢ (𝐿 ∈ (ℤring RingHom 𝑍) → 𝐿:ℤ⟶(Base‘𝑍)) |
27 | 14, 26 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐿:ℤ⟶(Base‘𝑍)) |
28 | 27, 15 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → (𝐿‘𝐴) ∈ (Base‘𝑍)) |
29 | 27, 16 | ffvelrnd 6360 | . . 3 ⊢ (𝜑 → (𝐿‘𝐶) ∈ (Base‘𝑍)) |
30 | eqid 2622 | . . . . 5 ⊢ (mulGrp‘𝑍) = (mulGrp‘𝑍) | |
31 | 30, 25 | mgpbas 18495 | . . . 4 ⊢ (Base‘𝑍) = (Base‘(mulGrp‘𝑍)) |
32 | 30, 19 | mgpplusg 18493 | . . . 4 ⊢ (.r‘𝑍) = (+g‘(mulGrp‘𝑍)) |
33 | eqid 2622 | . . . . 5 ⊢ (mulGrp‘ℂfld) = (mulGrp‘ℂfld) | |
34 | cnfldmul 19752 | . . . . 5 ⊢ · = (.r‘ℂfld) | |
35 | 33, 34 | mgpplusg 18493 | . . . 4 ⊢ · = (+g‘(mulGrp‘ℂfld)) |
36 | 31, 32, 35 | mhmlin 17342 | . . 3 ⊢ ((𝑋 ∈ ((mulGrp‘𝑍) MndHom (mulGrp‘ℂfld)) ∧ (𝐿‘𝐴) ∈ (Base‘𝑍) ∧ (𝐿‘𝐶) ∈ (Base‘𝑍)) → (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
37 | 24, 28, 29, 36 | syl3anc 1326 | . 2 ⊢ (𝜑 → (𝑋‘((𝐿‘𝐴)(.r‘𝑍)(𝐿‘𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
38 | 22, 37 | eqtrd 2656 | 1 ⊢ (𝜑 → (𝑋‘(𝐿‘(𝐴 · 𝐶))) = ((𝑋‘(𝐿‘𝐴)) · (𝑋‘(𝐿‘𝐶)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 · cmul 9941 ℕcn 11020 ℕ0cn0 11292 ℤcz 11377 Basecbs 15857 .rcmulr 15942 MndHom cmhm 17333 mulGrpcmgp 18489 Ringcrg 18547 CRingccrg 18548 RingHom crh 18712 ℂfldccnfld 19746 ℤringzring 19818 ℤRHomczrh 19848 ℤ/nℤczn 19851 DChrcdchr 24957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-tpos 7352 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-ec 7744 df-qs 7748 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-fz 12327 df-seq 12802 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-0g 16102 df-imas 16168 df-qus 16169 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-nsg 17592 df-eqg 17593 df-ghm 17658 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-rnghom 18715 df-subrg 18778 df-lmod 18865 df-lss 18933 df-lsp 18972 df-sra 19172 df-rgmod 19173 df-lidl 19174 df-rsp 19175 df-2idl 19232 df-cnfld 19747 df-zring 19819 df-zrh 19852 df-zn 19855 df-dchr 24958 |
This theorem is referenced by: dchrmusum2 25183 dchrvmasumlem1 25184 dchrvmasum2lem 25185 dchrisum0fmul 25195 |
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