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Mirrors > Home > MPE Home > Th. List > dchrvmaeq0 | Structured version Visualization version GIF version |
Description: The set π is the collection of all non-principal Dirichlet characters such that the sum Ξ£π β β, π(π) / π is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | β’ π = (β€/nβ€βπ) |
rpvmasum.l | β’ πΏ = (β€RHomβπ) |
rpvmasum.a | β’ (π β π β β) |
rpvmasum.g | β’ πΊ = (DChrβπ) |
rpvmasum.d | β’ π· = (BaseβπΊ) |
rpvmasum.1 | β’ 1 = (0gβπΊ) |
dchrisum.b | β’ (π β π β π·) |
dchrisum.n1 | β’ (π β π β 1 ) |
dchrvmasumif.f | β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) |
dchrvmasumif.c | β’ (π β πΆ β (0[,)+β)) |
dchrvmasumif.s | β’ (π β seq1( + , πΉ) β π) |
dchrvmasumif.1 | β’ (π β βπ¦ β (1[,)+β)(absβ((seq1( + , πΉ)β(ββπ¦)) β π)) β€ (πΆ / π¦)) |
dchrvmaeq0.w | β’ π = {π¦ β (π· β { 1 }) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} |
Ref | Expression |
---|---|
dchrvmaeq0 | β’ (π β (π β π β π = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrisum.b | . . . 4 β’ (π β π β π·) | |
2 | dchrisum.n1 | . . . 4 β’ (π β π β 1 ) | |
3 | eldifsn 4752 | . . . 4 β’ (π β (π· β { 1 }) β (π β π· β§ π β 1 )) | |
4 | 1, 2, 3 | sylanbrc 584 | . . 3 β’ (π β π β (π· β { 1 })) |
5 | fveq1 6846 | . . . . . . . 8 β’ (π¦ = π β (π¦β(πΏβπ)) = (πβ(πΏβπ))) | |
6 | 5 | oveq1d 7377 | . . . . . . 7 β’ (π¦ = π β ((π¦β(πΏβπ)) / π) = ((πβ(πΏβπ)) / π)) |
7 | 6 | sumeq2sdv 15596 | . . . . . 6 β’ (π¦ = π β Ξ£π β β ((π¦β(πΏβπ)) / π) = Ξ£π β β ((πβ(πΏβπ)) / π)) |
8 | 7 | eqeq1d 2739 | . . . . 5 β’ (π¦ = π β (Ξ£π β β ((π¦β(πΏβπ)) / π) = 0 β Ξ£π β β ((πβ(πΏβπ)) / π) = 0)) |
9 | dchrvmaeq0.w | . . . . 5 β’ π = {π¦ β (π· β { 1 }) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} | |
10 | 8, 9 | elrab2 3653 | . . . 4 β’ (π β π β (π β (π· β { 1 }) β§ Ξ£π β β ((πβ(πΏβπ)) / π) = 0)) |
11 | 10 | baib 537 | . . 3 β’ (π β (π· β { 1 }) β (π β π β Ξ£π β β ((πβ(πΏβπ)) / π) = 0)) |
12 | 4, 11 | syl 17 | . 2 β’ (π β (π β π β Ξ£π β β ((πβ(πΏβπ)) / π) = 0)) |
13 | nnuz 12813 | . . . 4 β’ β = (β€β₯β1) | |
14 | 1zzd 12541 | . . . 4 β’ (π β 1 β β€) | |
15 | 2fveq3 6852 | . . . . . . 7 β’ (π = π β (πβ(πΏβπ)) = (πβ(πΏβπ))) | |
16 | id 22 | . . . . . . 7 β’ (π = π β π = π) | |
17 | 15, 16 | oveq12d 7380 | . . . . . 6 β’ (π = π β ((πβ(πΏβπ)) / π) = ((πβ(πΏβπ)) / π)) |
18 | dchrvmasumif.f | . . . . . 6 β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) | |
19 | ovex 7395 | . . . . . 6 β’ ((πβ(πΏβπ)) / π) β V | |
20 | 17, 18, 19 | fvmpt 6953 | . . . . 5 β’ (π β β β (πΉβπ) = ((πβ(πΏβπ)) / π)) |
21 | 20 | adantl 483 | . . . 4 β’ ((π β§ π β β) β (πΉβπ) = ((πβ(πΏβπ)) / π)) |
22 | rpvmasum.g | . . . . . 6 β’ πΊ = (DChrβπ) | |
23 | rpvmasum.z | . . . . . 6 β’ π = (β€/nβ€βπ) | |
24 | rpvmasum.d | . . . . . 6 β’ π· = (BaseβπΊ) | |
25 | rpvmasum.l | . . . . . 6 β’ πΏ = (β€RHomβπ) | |
26 | 1 | adantr 482 | . . . . . 6 β’ ((π β§ π β β) β π β π·) |
27 | nnz 12527 | . . . . . . 7 β’ (π β β β π β β€) | |
28 | 27 | adantl 483 | . . . . . 6 β’ ((π β§ π β β) β π β β€) |
29 | 22, 23, 24, 25, 26, 28 | dchrzrhcl 26609 | . . . . 5 β’ ((π β§ π β β) β (πβ(πΏβπ)) β β) |
30 | nncn 12168 | . . . . . 6 β’ (π β β β π β β) | |
31 | 30 | adantl 483 | . . . . 5 β’ ((π β§ π β β) β π β β) |
32 | nnne0 12194 | . . . . . 6 β’ (π β β β π β 0) | |
33 | 32 | adantl 483 | . . . . 5 β’ ((π β§ π β β) β π β 0) |
34 | 29, 31, 33 | divcld 11938 | . . . 4 β’ ((π β§ π β β) β ((πβ(πΏβπ)) / π) β β) |
35 | dchrvmasumif.s | . . . 4 β’ (π β seq1( + , πΉ) β π) | |
36 | 13, 14, 21, 34, 35 | isumclim 15649 | . . 3 β’ (π β Ξ£π β β ((πβ(πΏβπ)) / π) = π) |
37 | 36 | eqeq1d 2739 | . 2 β’ (π β (Ξ£π β β ((πβ(πΏβπ)) / π) = 0 β π = 0)) |
38 | 12, 37 | bitrd 279 | 1 β’ (π β (π β π β π = 0)) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β wb 205 β§ wa 397 = wceq 1542 β wcel 2107 β wne 2944 βwral 3065 {crab 3410 β cdif 3912 {csn 4591 class class class wbr 5110 β¦ cmpt 5193 βcfv 6501 (class class class)co 7362 βcc 11056 0cc0 11058 1c1 11059 + caddc 11061 +βcpnf 11193 β€ cle 11197 β cmin 11392 / cdiv 11819 βcn 12160 β€cz 12506 [,)cico 13273 βcfl 13702 seqcseq 13913 abscabs 15126 β cli 15373 Ξ£csu 15577 Basecbs 17090 0gc0g 17328 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-inf2 9584 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-pre-sup 11136 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-se 5594 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-isom 6510 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-oi 9453 df-card 9882 df-pnf 11198 df-mnf 11199 df-xr 11200 df-ltxr 11201 df-le 11202 df-sub 11394 df-neg 11395 df-div 11820 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-rp 12923 df-fz 13432 df-fzo 13575 df-seq 13914 df-exp 13975 df-hash 14238 df-cj 14991 df-re 14992 df-im 14993 df-sqrt 15127 df-abs 15128 df-clim 15377 df-sum 15578 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-dvdsr 20077 df-unit 20078 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: rpvmasum2 26876 dchrisum0re 26877 dchrisum0lem2 26882 dchrisumn0 26885 |
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