Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dchrn0 | Structured version Visualization version Unicode version |
Description: A Dirichlet character is nonzero on the units of ℤ/nℤ. (Contributed by Mario Carneiro, 18-Apr-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | DChr |
dchrmhm.z | ℤ/nℤ |
dchrmhm.b | |
dchrn0.b | |
dchrn0.u | Unit |
dchrn0.x | |
dchrn0.a |
Ref | Expression |
---|---|
dchrn0 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrn0.a | . . . 4 | |
2 | dchrn0.x | . . . . . 6 | |
3 | dchrmhm.g | . . . . . . 7 DChr | |
4 | dchrmhm.z | . . . . . . 7 ℤ/nℤ | |
5 | dchrn0.b | . . . . . . 7 | |
6 | dchrn0.u | . . . . . . 7 Unit | |
7 | dchrmhm.b | . . . . . . . . 9 | |
8 | 3, 7 | dchrrcl 24965 | . . . . . . . 8 |
9 | 2, 8 | syl 17 | . . . . . . 7 |
10 | 3, 4, 5, 6, 9, 7 | dchrelbas2 24962 | . . . . . 6 mulGrp MndHom mulGrpℂfld |
11 | 2, 10 | mpbid 222 | . . . . 5 mulGrp MndHom mulGrpℂfld |
12 | 11 | simprd 479 | . . . 4 |
13 | fveq2 6191 | . . . . . . 7 | |
14 | 13 | neeq1d 2853 | . . . . . 6 |
15 | eleq1 2689 | . . . . . 6 | |
16 | 14, 15 | imbi12d 334 | . . . . 5 |
17 | 16 | rspcv 3305 | . . . 4 |
18 | 1, 12, 17 | sylc 65 | . . 3 |
19 | 18 | imp 445 | . 2 |
20 | ax-1ne0 10005 | . . . . 5 | |
21 | 20 | a1i 11 | . . . 4 |
22 | 9 | nnnn0d 11351 | . . . . . . . 8 |
23 | 4 | zncrng 19893 | . . . . . . . 8 |
24 | crngring 18558 | . . . . . . . 8 | |
25 | 22, 23, 24 | 3syl 18 | . . . . . . 7 |
26 | eqid 2622 | . . . . . . . 8 | |
27 | eqid 2622 | . . . . . . . 8 | |
28 | eqid 2622 | . . . . . . . 8 | |
29 | 6, 26, 27, 28 | unitrinv 18678 | . . . . . . 7 |
30 | 25, 29 | sylan 488 | . . . . . 6 |
31 | 30 | fveq2d 6195 | . . . . 5 |
32 | 11 | simpld 475 | . . . . . . 7 mulGrp MndHom mulGrpℂfld |
33 | 32 | adantr 481 | . . . . . 6 mulGrp MndHom mulGrpℂfld |
34 | 1 | adantr 481 | . . . . . 6 |
35 | 6, 26, 5 | ringinvcl 18676 | . . . . . . 7 |
36 | 25, 35 | sylan 488 | . . . . . 6 |
37 | eqid 2622 | . . . . . . . 8 mulGrp mulGrp | |
38 | 37, 5 | mgpbas 18495 | . . . . . . 7 mulGrp |
39 | 37, 27 | mgpplusg 18493 | . . . . . . 7 mulGrp |
40 | eqid 2622 | . . . . . . . 8 mulGrpℂfld mulGrpℂfld | |
41 | cnfldmul 19752 | . . . . . . . 8 ℂfld | |
42 | 40, 41 | mgpplusg 18493 | . . . . . . 7 mulGrpℂfld |
43 | 38, 39, 42 | mhmlin 17342 | . . . . . 6 mulGrp MndHom mulGrpℂfld |
44 | 33, 34, 36, 43 | syl3anc 1326 | . . . . 5 |
45 | 37, 28 | ringidval 18503 | . . . . . . 7 mulGrp |
46 | cnfld1 19771 | . . . . . . . 8 ℂfld | |
47 | 40, 46 | ringidval 18503 | . . . . . . 7 mulGrpℂfld |
48 | 45, 47 | mhm0 17343 | . . . . . 6 mulGrp MndHom mulGrpℂfld |
49 | 33, 48 | syl 17 | . . . . 5 |
50 | 31, 44, 49 | 3eqtr3d 2664 | . . . 4 |
51 | cnfldbas 19750 | . . . . . . . . 9 ℂfld | |
52 | 40, 51 | mgpbas 18495 | . . . . . . . 8 mulGrpℂfld |
53 | 38, 52 | mhmf 17340 | . . . . . . 7 mulGrp MndHom mulGrpℂfld |
54 | 33, 53 | syl 17 | . . . . . 6 |
55 | 54, 36 | ffvelrnd 6360 | . . . . 5 |
56 | 55 | mul02d 10234 | . . . 4 |
57 | 21, 50, 56 | 3netr4d 2871 | . . 3 |
58 | oveq1 6657 | . . . 4 | |
59 | 58 | necon3i 2826 | . . 3 |
60 | 57, 59 | syl 17 | . 2 |
61 | 19, 60 | impbida 877 | 1 |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 wne 2794 wral 2912 wf 5884 cfv 5888 (class class class)co 6650 cc 9934 cc0 9936 c1 9937 cmul 9941 cn 11020 cn0 11292 cbs 15857 cmulr 15942 MndHom cmhm 17333 mulGrpcmgp 18489 cur 18501 crg 18547 ccrg 18548 Unitcui 18639 cinvr 18671 ℂfldccnfld 19746 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-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-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-subg 17591 df-nsg 17592 df-eqg 17593 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-oppr 18623 df-dvdsr 18641 df-unit 18642 df-invr 18672 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-zn 19855 df-dchr 24958 |
This theorem is referenced by: dchrinvcl 24978 dchrfi 24980 dchrghm 24981 dchreq 24983 dchrabs 24985 dchrabs2 24987 dchr1re 24988 dchrpt 24992 dchrsum 24994 sum2dchr 24999 rpvmasumlem 25176 dchrisum0flblem1 25197 |
Copyright terms: Public domain | W3C validator |