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Mirrors > Home > MPE Home > Th. List > dchrisumn0 | Structured version Visualization version GIF version |
Description: The sum Ξ£π β β, π(π) / π is nonzero for all non-principal Dirichlet characters (i.e. the assumption π β π is contradictory). This is the key result that allows to eliminate the conditionals from dchrmusum2 27445 and dchrvmasumif 27454. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-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) |
dchrmusum.g | β’ πΊ = (DChrβπ) |
dchrmusum.d | β’ π· = (BaseβπΊ) |
dchrmusum.1 | β’ 1 = (0gβπΊ) |
dchrmusum.b | β’ (π β π β π·) |
dchrmusum.n1 | β’ (π β π β 1 ) |
dchrmusum.f | β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) |
dchrmusum.c | β’ (π β πΆ β (0[,)+β)) |
dchrmusum.t | β’ (π β seq1( + , πΉ) β π) |
dchrmusum.2 | β’ (π β βπ¦ β (1[,)+β)(absβ((seq1( + , πΉ)β(ββπ¦)) β π)) β€ (πΆ / π¦)) |
Ref | Expression |
---|---|
dchrisumn0 | β’ (π β π β 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . . 4 β’ π = (β€/nβ€βπ) | |
2 | rpvmasum.l | . . . 4 β’ πΏ = (β€RHomβπ) | |
3 | rpvmasum.a | . . . . 5 β’ (π β π β β) | |
4 | 3 | adantr 479 | . . . 4 β’ ((π β§ π = 0) β π β β) |
5 | dchrmusum.g | . . . 4 β’ πΊ = (DChrβπ) | |
6 | dchrmusum.d | . . . 4 β’ π· = (BaseβπΊ) | |
7 | dchrmusum.1 | . . . 4 β’ 1 = (0gβπΊ) | |
8 | eqid 2725 | . . . 4 β’ {π¦ β (π· β { 1 }) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} = {π¦ β (π· β { 1 }) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} | |
9 | dchrmusum.b | . . . . . 6 β’ (π β π β π·) | |
10 | dchrmusum.n1 | . . . . . 6 β’ (π β π β 1 ) | |
11 | dchrmusum.f | . . . . . 6 β’ πΉ = (π β β β¦ ((πβ(πΏβπ)) / π)) | |
12 | dchrmusum.c | . . . . . 6 β’ (π β πΆ β (0[,)+β)) | |
13 | dchrmusum.t | . . . . . 6 β’ (π β seq1( + , πΉ) β π) | |
14 | dchrmusum.2 | . . . . . 6 β’ (π β βπ¦ β (1[,)+β)(absβ((seq1( + , πΉ)β(ββπ¦)) β π)) β€ (πΆ / π¦)) | |
15 | 1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 8 | dchrvmaeq0 27455 | . . . . 5 β’ (π β (π β {π¦ β (π· β { 1 }) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0} β π = 0)) |
16 | 15 | biimpar 476 | . . . 4 β’ ((π β§ π = 0) β π β {π¦ β (π· β { 1 }) β£ Ξ£π β β ((π¦β(πΏβπ)) / π) = 0}) |
17 | 1, 2, 4, 5, 6, 7, 8, 16 | dchrisum0 27471 | . . 3 β’ Β¬ (π β§ π = 0) |
18 | 17 | imnani 399 | . 2 β’ (π β Β¬ π = 0) |
19 | 18 | neqned 2937 | 1 β’ (π β π β 0) |
Colors of variables: wff setvar class |
Syntax hints: β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2930 βwral 3051 {crab 3419 β cdif 3936 {csn 4624 class class class wbr 5143 β¦ cmpt 5226 βcfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 +βcpnf 11275 β€ cle 11279 β cmin 11474 / cdiv 11901 βcn 12242 [,)cico 13358 βcfl 13787 seqcseq 13998 abscabs 15213 β cli 15460 Ξ£csu 15664 Basecbs 17179 0gc0g 17420 β€RHomczrh 21429 β€/nβ€czn 21432 DChrcdchr 27183 |
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 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
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 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 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-iin 4994 df-disj 5109 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 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-rpss 7726 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-tpos 8230 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-oadd 8489 df-omul 8490 df-er 8723 df-ec 8725 df-qs 8729 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-dju 9924 df-card 9962 df-acn 9965 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-xnn0 12575 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-word 14497 df-concat 14553 df-s1 14578 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-o1 15466 df-lo1 15467 df-sum 15665 df-ef 16043 df-e 16044 df-sin 16045 df-cos 16046 df-tan 16047 df-pi 16048 df-dvds 16231 df-gcd 16469 df-prm 16642 df-numer 16706 df-denom 16707 df-phi 16734 df-pc 16805 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-xrs 17483 df-qtop 17488 df-imas 17489 df-qus 17490 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-nsg 19083 df-eqg 19084 df-ghm 19172 df-gim 19217 df-ga 19245 df-cntz 19272 df-oppg 19301 df-od 19487 df-gex 19488 df-pgp 19489 df-lsm 19595 df-pj1 19596 df-cmn 19741 df-abl 19742 df-cyg 19837 df-dprd 19956 df-dpj 19957 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-oppr 20277 df-dvdsr 20300 df-unit 20301 df-invr 20331 df-dvr 20344 df-rhm 20415 df-subrng 20487 df-subrg 20512 df-drng 20630 df-lmod 20749 df-lss 20820 df-lsp 20860 df-sra 21062 df-rgmod 21063 df-lidl 21108 df-rsp 21109 df-2idl 21148 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-zring 21377 df-zrh 21433 df-zn 21436 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-cmp 23309 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-xms 24244 df-ms 24245 df-tms 24246 df-cncf 24816 df-0p 25617 df-limc 25813 df-dv 25814 df-ply 26140 df-idp 26141 df-coe 26142 df-dgr 26143 df-quot 26244 df-ulm 26331 df-log 26508 df-cxp 26509 df-atan 26817 df-em 26943 df-cht 27047 df-vma 27048 df-chp 27049 df-ppi 27050 df-mu 27051 df-dchr 27184 |
This theorem is referenced by: dchrmusumlem 27473 dchrvmasumlem 27474 |
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