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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 4785 | . . . 4 β’ (π β (π· β { 1 }) β (π β π· β§ π β 1 )) | |
4 | 1, 2, 3 | sylanbrc 582 | . . 3 β’ (π β π β (π· β { 1 })) |
5 | fveq1 6884 | . . . . . . . 8 β’ (π¦ = π β (π¦β(πΏβπ)) = (πβ(πΏβπ))) | |
6 | 5 | oveq1d 7420 | . . . . . . 7 β’ (π¦ = π β ((π¦β(πΏβπ)) / π) = ((πβ(πΏβπ)) / π)) |
7 | 6 | sumeq2sdv 15656 | . . . . . 6 β’ (π¦ = π β Ξ£π β β ((π¦β(πΏβπ)) / π) = Ξ£π β β ((πβ(πΏβπ)) / π)) |
8 | 7 | eqeq1d 2728 | . . . . 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 3681 | . . . 4 β’ (π β π β (π β (π· β { 1 }) β§ Ξ£π β β ((πβ(πΏβπ)) / π) = 0)) |
11 | 10 | baib 535 | . . 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 12869 | . . . 4 β’ β = (β€β₯β1) | |
14 | 1zzd 12597 | . . . 4 β’ (π β 1 β β€) | |
15 | 2fveq3 6890 | . . . . . . 7 β’ (π = π β (πβ(πΏβπ)) = (πβ(πΏβπ))) | |
16 | id 22 | . . . . . . 7 β’ (π = π β π = π) | |
17 | 15, 16 | oveq12d 7423 | . . . . . 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 7438 | . . . . . 6 β’ ((πβ(πΏβπ)) / π) β V | |
20 | 17, 18, 19 | fvmpt 6992 | . . . . 5 β’ (π β β β (πΉβπ) = ((πβ(πΏβπ)) / π)) |
21 | 20 | adantl 481 | . . . 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 480 | . . . . . 6 β’ ((π β§ π β β) β π β π·) |
27 | nnz 12583 | . . . . . . 7 β’ (π β β β π β β€) | |
28 | 27 | adantl 481 | . . . . . 6 β’ ((π β§ π β β) β π β β€) |
29 | 22, 23, 24, 25, 26, 28 | dchrzrhcl 27133 | . . . . 5 β’ ((π β§ π β β) β (πβ(πΏβπ)) β β) |
30 | nncn 12224 | . . . . . 6 β’ (π β β β π β β) | |
31 | 30 | adantl 481 | . . . . 5 β’ ((π β§ π β β) β π β β) |
32 | nnne0 12250 | . . . . . 6 β’ (π β β β π β 0) | |
33 | 32 | adantl 481 | . . . . 5 β’ ((π β§ π β β) β π β 0) |
34 | 29, 31, 33 | divcld 11994 | . . . 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 15709 | . . 3 β’ (π β Ξ£π β β ((πβ(πΏβπ)) / π) = π) |
37 | 36 | eqeq1d 2728 | . 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 395 = wceq 1533 β wcel 2098 β wne 2934 βwral 3055 {crab 3426 β cdif 3940 {csn 4623 class class class wbr 5141 β¦ cmpt 5224 βcfv 6537 (class class class)co 7405 βcc 11110 0cc0 11112 1c1 11113 + caddc 11115 +βcpnf 11249 β€ cle 11253 β cmin 11448 / cdiv 11875 βcn 12216 β€cz 12562 [,)cico 13332 βcfl 13761 seqcseq 13972 abscabs 15187 β cli 15434 Ξ£csu 15638 Basecbs 17153 0gc0g 17394 β€RHomczrh 21386 β€/nβ€czn 21389 DChrcdchr 27120 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 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 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-tpos 8212 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-ec 8707 df-qs 8711 df-map 8824 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-rp 12981 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-0g 17396 df-imas 17463 df-qus 17464 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-nsg 19051 df-eqg 19052 df-ghm 19139 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-oppr 20236 df-dvdsr 20259 df-unit 20260 df-rhm 20374 df-subrng 20446 df-subrg 20471 df-lmod 20708 df-lss 20779 df-lsp 20819 df-sra 21021 df-rgmod 21022 df-lidl 21067 df-rsp 21068 df-2idl 21107 df-cnfld 21241 df-zring 21334 df-zrh 21390 df-zn 21393 df-dchr 27121 |
This theorem is referenced by: rpvmasum2 27400 dchrisum0re 27401 dchrisum0lem2 27406 dchrisumn0 27409 |
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