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Mirrors > Home > MPE Home > Th. List > dchrrcl | Structured version Visualization version GIF version |
Description: Reverse closure for a Dirichlet character. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
dchrrcl.g | β’ πΊ = (DChrβπ) |
dchrrcl.b | β’ π· = (BaseβπΊ) |
Ref | Expression |
---|---|
dchrrcl | β’ (π β π· β π β β) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-dchr 26597 | . . 3 β’ DChr = (π β β β¦ β¦(β€/nβ€βπ) / π§β¦β¦{π₯ β ((mulGrpβπ§) MndHom (mulGrpββfld)) β£ (((Baseβπ§) β (Unitβπ§)) Γ {0}) β π₯} / πβ¦{β¨(Baseβndx), πβ©, β¨(+gβndx), ( βf Β· βΎ (π Γ π))β©}) | |
2 | 1 | dmmptss 6198 | . 2 β’ dom DChr β β |
3 | n0i 4298 | . . 3 β’ (π β π· β Β¬ π· = β ) | |
4 | dchrrcl.g | . . . . 5 β’ πΊ = (DChrβπ) | |
5 | ndmfv 6882 | . . . . 5 β’ (Β¬ π β dom DChr β (DChrβπ) = β ) | |
6 | 4, 5 | eqtrid 2789 | . . . 4 β’ (Β¬ π β dom DChr β πΊ = β ) |
7 | fveq2 6847 | . . . . 5 β’ (πΊ = β β (BaseβπΊ) = (Baseββ )) | |
8 | dchrrcl.b | . . . . 5 β’ π· = (BaseβπΊ) | |
9 | base0 17095 | . . . . 5 β’ β = (Baseββ ) | |
10 | 7, 8, 9 | 3eqtr4g 2802 | . . . 4 β’ (πΊ = β β π· = β ) |
11 | 6, 10 | syl 17 | . . 3 β’ (Β¬ π β dom DChr β π· = β ) |
12 | 3, 11 | nsyl2 141 | . 2 β’ (π β π· β π β dom DChr) |
13 | 2, 12 | sselid 3947 | 1 β’ (π β π· β π β β) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 = wceq 1542 β wcel 2107 {crab 3410 β¦csb 3860 β cdif 3912 β wss 3915 β c0 4287 {csn 4591 {cpr 4593 β¨cop 4597 Γ cxp 5636 dom cdm 5638 βΎ cres 5640 βcfv 6501 (class class class)co 7362 βf cof 7620 0cc0 11058 Β· cmul 11063 βcn 12160 ndxcnx 17072 Basecbs 17090 +gcplusg 17140 MndHom cmhm 18606 mulGrpcmgp 19903 Unitcui 20075 βfldccnfld 20812 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-sep 5261 ax-nul 5268 ax-pow 5325 ax-pr 5389 ax-un 7677 ax-cnex 11114 ax-1cn 11116 ax-addcl 11118 |
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-ral 3066 df-rex 3075 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-op 4598 df-uni 4871 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-ov 7365 df-om 7808 df-2nd 7927 df-frecs 8217 df-wrecs 8248 df-recs 8322 df-rdg 8361 df-nn 12161 df-slot 17061 df-ndx 17073 df-base 17091 df-dchr 26597 |
This theorem is referenced by: dchrmhm 26605 dchrf 26606 dchrelbas4 26607 dchrzrh1 26608 dchrzrhcl 26609 dchrzrhmul 26610 dchrmul 26612 dchrmulcl 26613 dchrn0 26614 dchrmulid2 26616 dchrinvcl 26617 dchrghm 26620 dchrabs 26624 dchrinv 26625 dchrsum2 26632 dchrsum 26633 dchr2sum 26637 |
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