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Mirrors > Home > MPE Home > Th. List > dchrzrh1 | Structured version Visualization version GIF version |
Description: Value of a Dirichlet character at one. (Contributed by Mario Carneiro, 4-May-2016.) |
Ref | Expression |
---|---|
dchrmhm.g | β’ πΊ = (DChrβπ) |
dchrmhm.z | β’ π = (β€/nβ€βπ) |
dchrmhm.b | β’ π· = (BaseβπΊ) |
dchrelbas4.l | β’ πΏ = (β€RHomβπ) |
dchrzrh1.x | β’ (π β π β π·) |
Ref | Expression |
---|---|
dchrzrh1 | β’ (π β (πβ(πΏβ1)) = 1) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrzrh1.x | . . . . . 6 β’ (π β π β π·) | |
2 | dchrmhm.g | . . . . . . 7 β’ πΊ = (DChrβπ) | |
3 | dchrmhm.b | . . . . . . 7 β’ π· = (BaseβπΊ) | |
4 | 2, 3 | dchrrcl 26604 | . . . . . 6 β’ (π β π· β π β β) |
5 | 1, 4 | syl 17 | . . . . 5 β’ (π β π β β) |
6 | 5 | nnnn0d 12480 | . . . 4 β’ (π β π β β0) |
7 | dchrmhm.z | . . . . 5 β’ π = (β€/nβ€βπ) | |
8 | 7 | zncrng 20967 | . . . 4 β’ (π β β0 β π β CRing) |
9 | crngring 19983 | . . . 4 β’ (π β CRing β π β Ring) | |
10 | dchrelbas4.l | . . . . 5 β’ πΏ = (β€RHomβπ) | |
11 | eqid 2737 | . . . . 5 β’ (1rβπ) = (1rβπ) | |
12 | 10, 11 | zrh1 20929 | . . . 4 β’ (π β Ring β (πΏβ1) = (1rβπ)) |
13 | 6, 8, 9, 12 | 4syl 19 | . . 3 β’ (π β (πΏβ1) = (1rβπ)) |
14 | 13 | fveq2d 6851 | . 2 β’ (π β (πβ(πΏβ1)) = (πβ(1rβπ))) |
15 | 2, 7, 3 | dchrmhm 26605 | . . . 4 β’ π· β ((mulGrpβπ) MndHom (mulGrpββfld)) |
16 | 15, 1 | sselid 3947 | . . 3 β’ (π β π β ((mulGrpβπ) MndHom (mulGrpββfld))) |
17 | eqid 2737 | . . . . 5 β’ (mulGrpβπ) = (mulGrpβπ) | |
18 | 17, 11 | ringidval 19922 | . . . 4 β’ (1rβπ) = (0gβ(mulGrpβπ)) |
19 | eqid 2737 | . . . . 5 β’ (mulGrpββfld) = (mulGrpββfld) | |
20 | cnfld1 20838 | . . . . 5 β’ 1 = (1rββfld) | |
21 | 19, 20 | ringidval 19922 | . . . 4 β’ 1 = (0gβ(mulGrpββfld)) |
22 | 18, 21 | mhm0 18617 | . . 3 β’ (π β ((mulGrpβπ) MndHom (mulGrpββfld)) β (πβ(1rβπ)) = 1) |
23 | 16, 22 | syl 17 | . 2 β’ (π β (πβ(1rβπ)) = 1) |
24 | 14, 23 | eqtrd 2777 | 1 β’ (π β (πβ(πΏβ1)) = 1) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 = wceq 1542 β wcel 2107 βcfv 6501 (class class class)co 7362 1c1 11059 βcn 12160 β0cn0 12420 Basecbs 17090 MndHom cmhm 18606 mulGrpcmgp 19903 1rcur 19920 Ringcrg 19971 CRingccrg 19972 βfldccnfld 20812 β€RHomczrh 20916 β€/nβ€czn 20919 DChrcdchr 26596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-rep 5247 ax-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-resscn 11115 ax-1cn 11116 ax-icn 11117 ax-addcl 11118 ax-addrcl 11119 ax-mulcl 11120 ax-mulrcl 11121 ax-mulcom 11122 ax-addass 11123 ax-mulass 11124 ax-distr 11125 ax-i2m1 11126 ax-1ne0 11127 ax-1rid 11128 ax-rnegex 11129 ax-rrecex 11130 ax-cnre 11131 ax-pre-lttri 11132 ax-pre-lttrn 11133 ax-pre-ltadd 11134 ax-pre-mulgt0 11135 ax-addf 11137 ax-mulf 11138 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ne 2945 df-nel 3051 df-ral 3066 df-rex 3075 df-rmo 3356 df-reu 3357 df-rab 3411 df-v 3450 df-sbc 3745 df-csb 3861 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-tp 4596 df-op 4598 df-uni 4871 df-int 4913 df-iun 4961 df-br 5111 df-opab 5173 df-mpt 5194 df-tr 5228 df-id 5536 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6258 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-riota 7318 df-ov 7365 df-oprab 7366 df-mpo 7367 df-om 7808 df-1st 7926 df-2nd 7927 df-tpos 8162 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-1o 8417 df-er 8655 df-ec 8657 df-qs 8661 df-map 8774 df-en 8891 df-dom 8892 df-sdom 8893 df-fin 8894 df-sup 9385 df-inf 9386 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-nn 12161 df-2 12223 df-3 12224 df-4 12225 df-5 12226 df-6 12227 df-7 12228 df-8 12229 df-9 12230 df-n0 12421 df-z 12507 df-dec 12626 df-uz 12771 df-fz 13432 df-seq 13914 df-struct 17026 df-sets 17043 df-slot 17061 df-ndx 17073 df-base 17091 df-ress 17120 df-plusg 17153 df-mulr 17154 df-starv 17155 df-sca 17156 df-vsca 17157 df-ip 17158 df-tset 17159 df-ple 17160 df-ds 17162 df-unif 17163 df-0g 17330 df-imas 17397 df-qus 17398 df-mgm 18504 df-sgrp 18553 df-mnd 18564 df-mhm 18608 df-grp 18758 df-minusg 18759 df-sbg 18760 df-mulg 18880 df-subg 18932 df-nsg 18933 df-eqg 18934 df-ghm 19013 df-cmn 19571 df-abl 19572 df-mgp 19904 df-ur 19921 df-ring 19973 df-cring 19974 df-oppr 20056 df-rnghom 20155 df-subrg 20236 df-lmod 20340 df-lss 20409 df-lsp 20449 df-sra 20649 df-rgmod 20650 df-lidl 20651 df-rsp 20652 df-2idl 20718 df-cnfld 20813 df-zring 20886 df-zrh 20920 df-zn 20923 df-dchr 26597 |
This theorem is referenced by: dchrmusum2 26858 dchrvmasum2lem 26860 |
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