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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 4786 | . . . 4 β’ (π β (π· β { 1 }) β (π β π· β§ π β 1 )) | |
4 | 1, 2, 3 | sylanbrc 581 | . . 3 β’ (π β π β (π· β { 1 })) |
5 | fveq1 6890 | . . . . . . . 8 β’ (π¦ = π β (π¦β(πΏβπ)) = (πβ(πΏβπ))) | |
6 | 5 | oveq1d 7430 | . . . . . . 7 β’ (π¦ = π β ((π¦β(πΏβπ)) / π) = ((πβ(πΏβπ)) / π)) |
7 | 6 | sumeq2sdv 15680 | . . . . . 6 β’ (π¦ = π β Ξ£π β β ((π¦β(πΏβπ)) / π) = Ξ£π β β ((πβ(πΏβπ)) / π)) |
8 | 7 | eqeq1d 2727 | . . . . 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 3678 | . . . 4 β’ (π β π β (π β (π· β { 1 }) β§ Ξ£π β β ((πβ(πΏβπ)) / π) = 0)) |
11 | 10 | baib 534 | . . 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 12893 | . . . 4 β’ β = (β€β₯β1) | |
14 | 1zzd 12621 | . . . 4 β’ (π β 1 β β€) | |
15 | 2fveq3 6896 | . . . . . . 7 β’ (π = π β (πβ(πΏβπ)) = (πβ(πΏβπ))) | |
16 | id 22 | . . . . . . 7 β’ (π = π β π = π) | |
17 | 15, 16 | oveq12d 7433 | . . . . . 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 7448 | . . . . . 6 β’ ((πβ(πΏβπ)) / π) β V | |
20 | 17, 18, 19 | fvmpt 6999 | . . . . 5 β’ (π β β β (πΉβπ) = ((πβ(πΏβπ)) / π)) |
21 | 20 | adantl 480 | . . . 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 479 | . . . . . 6 β’ ((π β§ π β β) β π β π·) |
27 | nnz 12607 | . . . . . . 7 β’ (π β β β π β β€) | |
28 | 27 | adantl 480 | . . . . . 6 β’ ((π β§ π β β) β π β β€) |
29 | 22, 23, 24, 25, 26, 28 | dchrzrhcl 27194 | . . . . 5 β’ ((π β§ π β β) β (πβ(πΏβπ)) β β) |
30 | nncn 12248 | . . . . . 6 β’ (π β β β π β β) | |
31 | 30 | adantl 480 | . . . . 5 β’ ((π β§ π β β) β π β β) |
32 | nnne0 12274 | . . . . . 6 β’ (π β β β π β 0) | |
33 | 32 | adantl 480 | . . . . 5 β’ ((π β§ π β β) β π β 0) |
34 | 29, 31, 33 | divcld 12018 | . . . 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 15733 | . . 3 β’ (π β Ξ£π β β ((πβ(πΏβπ)) / π) = π) |
37 | 36 | eqeq1d 2727 | . 2 β’ (π β (Ξ£π β β ((πβ(πΏβπ)) / π) = 0 β π = 0)) |
38 | 12, 37 | bitrd 278 | 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 394 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 {crab 3419 β cdif 3937 {csn 4624 class class class wbr 5143 β¦ cmpt 5226 βcfv 6542 (class class class)co 7415 βcc 11134 0cc0 11136 1c1 11137 + caddc 11139 +βcpnf 11273 β€ cle 11277 β cmin 11472 / cdiv 11899 βcn 12240 β€cz 12586 [,)cico 13356 βcfl 13785 seqcseq 13996 abscabs 15211 β cli 15458 Ξ£csu 15662 Basecbs 17177 0gc0g 17418 β€RHomczrh 21427 β€/nβ€czn 21430 DChrcdchr 27181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7737 ax-inf2 9662 ax-cnex 11192 ax-resscn 11193 ax-1cn 11194 ax-icn 11195 ax-addcl 11196 ax-addrcl 11197 ax-mulcl 11198 ax-mulrcl 11199 ax-mulcom 11200 ax-addass 11201 ax-mulass 11202 ax-distr 11203 ax-i2m1 11204 ax-1ne0 11205 ax-1rid 11206 ax-rnegex 11207 ax-rrecex 11208 ax-cnre 11209 ax-pre-lttri 11210 ax-pre-lttrn 11211 ax-pre-ltadd 11212 ax-pre-mulgt0 11213 ax-pre-sup 11214 ax-addf 11215 ax-mulf 11216 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3960 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7371 df-ov 7418 df-oprab 7419 df-mpo 7420 df-om 7868 df-1st 7989 df-2nd 7990 df-tpos 8228 df-frecs 8283 df-wrecs 8314 df-recs 8388 df-rdg 8427 df-1o 8483 df-er 8721 df-ec 8723 df-qs 8727 df-map 8843 df-en 8961 df-dom 8962 df-sdom 8963 df-fin 8964 df-sup 9463 df-inf 9464 df-oi 9531 df-card 9960 df-pnf 11278 df-mnf 11279 df-xr 11280 df-ltxr 11281 df-le 11282 df-sub 11474 df-neg 11475 df-div 11900 df-nn 12241 df-2 12303 df-3 12304 df-4 12305 df-5 12306 df-6 12307 df-7 12308 df-8 12309 df-9 12310 df-n0 12501 df-z 12587 df-dec 12706 df-uz 12851 df-rp 13005 df-fz 13515 df-fzo 13658 df-seq 13997 df-exp 14057 df-hash 14320 df-cj 15076 df-re 15077 df-im 15078 df-sqrt 15212 df-abs 15213 df-clim 15462 df-sum 15663 df-struct 17113 df-sets 17130 df-slot 17148 df-ndx 17160 df-base 17178 df-ress 17207 df-plusg 17243 df-mulr 17244 df-starv 17245 df-sca 17246 df-vsca 17247 df-ip 17248 df-tset 17249 df-ple 17250 df-ds 17252 df-unif 17253 df-0g 17420 df-imas 17487 df-qus 17488 df-mgm 18597 df-sgrp 18676 df-mnd 18692 df-mhm 18737 df-grp 18895 df-minusg 18896 df-sbg 18897 df-mulg 19026 df-subg 19080 df-nsg 19081 df-eqg 19082 df-ghm 19170 df-cmn 19739 df-abl 19740 df-mgp 20077 df-rng 20095 df-ur 20124 df-ring 20177 df-cring 20178 df-oppr 20275 df-dvdsr 20298 df-unit 20299 df-rhm 20413 df-subrng 20485 df-subrg 20510 df-lmod 20747 df-lss 20818 df-lsp 20858 df-sra 21060 df-rgmod 21061 df-lidl 21106 df-rsp 21107 df-2idl 21146 df-cnfld 21282 df-zring 21375 df-zrh 21431 df-zn 21434 df-dchr 27182 |
This theorem is referenced by: rpvmasum2 27461 dchrisum0re 27462 dchrisum0lem2 27467 dchrisumn0 27470 |
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