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Mirrors > Home > MPE Home > Th. List > dchrsum | Structured version Visualization version GIF version |
Description: An orthogonality relation for Dirichlet characters: the sum of all the values of a Dirichlet character π is 0 if π is non-principal and Ο(π) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dchrsum.g | β’ πΊ = (DChrβπ) |
dchrsum.z | β’ π = (β€/nβ€βπ) |
dchrsum.d | β’ π· = (BaseβπΊ) |
dchrsum.1 | β’ 1 = (0gβπΊ) |
dchrsum.x | β’ (π β π β π·) |
dchrsum.b | β’ π΅ = (Baseβπ) |
Ref | Expression |
---|---|
dchrsum | β’ (π β Ξ£π β π΅ (πβπ) = if(π = 1 , (Οβπ), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrsum.b | . . . . 5 β’ π΅ = (Baseβπ) | |
2 | eqid 2728 | . . . . 5 β’ (Unitβπ) = (Unitβπ) | |
3 | 1, 2 | unitss 20322 | . . . 4 β’ (Unitβπ) β π΅ |
4 | 3 | a1i 11 | . . 3 β’ (π β (Unitβπ) β π΅) |
5 | dchrsum.g | . . . . 5 β’ πΊ = (DChrβπ) | |
6 | dchrsum.z | . . . . 5 β’ π = (β€/nβ€βπ) | |
7 | dchrsum.d | . . . . 5 β’ π· = (BaseβπΊ) | |
8 | dchrsum.x | . . . . 5 β’ (π β π β π·) | |
9 | 5, 6, 7, 1, 8 | dchrf 27195 | . . . 4 β’ (π β π:π΅βΆβ) |
10 | 3 | sseli 3978 | . . . 4 β’ (π β (Unitβπ) β π β π΅) |
11 | ffvelcdm 7096 | . . . 4 β’ ((π:π΅βΆβ β§ π β π΅) β (πβπ) β β) | |
12 | 9, 10, 11 | syl2an 594 | . . 3 β’ ((π β§ π β (Unitβπ)) β (πβπ) β β) |
13 | eldif 3959 | . . . 4 β’ (π β (π΅ β (Unitβπ)) β (π β π΅ β§ Β¬ π β (Unitβπ))) | |
14 | 8 | adantr 479 | . . . . . . . 8 β’ ((π β§ π β π΅) β π β π·) |
15 | simpr 483 | . . . . . . . 8 β’ ((π β§ π β π΅) β π β π΅) | |
16 | 5, 6, 7, 1, 2, 14, 15 | dchrn0 27203 | . . . . . . 7 β’ ((π β§ π β π΅) β ((πβπ) β 0 β π β (Unitβπ))) |
17 | 16 | biimpd 228 | . . . . . 6 β’ ((π β§ π β π΅) β ((πβπ) β 0 β π β (Unitβπ))) |
18 | 17 | necon1bd 2955 | . . . . 5 β’ ((π β§ π β π΅) β (Β¬ π β (Unitβπ) β (πβπ) = 0)) |
19 | 18 | impr 453 | . . . 4 β’ ((π β§ (π β π΅ β§ Β¬ π β (Unitβπ))) β (πβπ) = 0) |
20 | 13, 19 | sylan2b 592 | . . 3 β’ ((π β§ π β (π΅ β (Unitβπ))) β (πβπ) = 0) |
21 | 5, 7 | dchrrcl 27193 | . . . 4 β’ (π β π· β π β β) |
22 | 6, 1 | znfi 21500 | . . . 4 β’ (π β β β π΅ β Fin) |
23 | 8, 21, 22 | 3syl 18 | . . 3 β’ (π β π΅ β Fin) |
24 | 4, 12, 20, 23 | fsumss 15711 | . 2 β’ (π β Ξ£π β (Unitβπ)(πβπ) = Ξ£π β π΅ (πβπ)) |
25 | dchrsum.1 | . . 3 β’ 1 = (0gβπΊ) | |
26 | 5, 6, 7, 25, 8, 2 | dchrsum2 27221 | . 2 β’ (π β Ξ£π β (Unitβπ)(πβπ) = if(π = 1 , (Οβπ), 0)) |
27 | 24, 26 | eqtr3d 2770 | 1 β’ (π β Ξ£π β π΅ (πβπ) = if(π = 1 , (Οβπ), 0)) |
Colors of variables: wff setvar class |
Syntax hints: Β¬ wn 3 β wi 4 β§ wa 394 = wceq 1533 β wcel 2098 β wne 2937 β cdif 3946 β wss 3949 ifcif 4532 βΆwf 6549 βcfv 6553 Fincfn 8970 βcc 11144 0cc0 11146 βcn 12250 Ξ£csu 15672 Οcphi 16740 Basecbs 17187 0gc0g 17428 Unitcui 20301 β€/nβ€czn 21435 DChrcdchr 27185 |
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 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 ax-addf 11225 ax-mulf 11226 |
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 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-tp 4637 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-of 7691 df-om 7877 df-1st 7999 df-2nd 8000 df-tpos 8238 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-oadd 8497 df-er 8731 df-ec 8733 df-qs 8737 df-map 8853 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-xnn0 12583 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-fz 13525 df-fzo 13668 df-fl 13797 df-mod 13875 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 df-dvds 16239 df-gcd 16477 df-phi 16742 df-struct 17123 df-sets 17140 df-slot 17158 df-ndx 17170 df-base 17188 df-ress 17217 df-plusg 17253 df-mulr 17254 df-starv 17255 df-sca 17256 df-vsca 17257 df-ip 17258 df-tset 17259 df-ple 17260 df-ds 17262 df-unif 17263 df-0g 17430 df-imas 17497 df-qus 17498 df-mgm 18607 df-sgrp 18686 df-mnd 18702 df-mhm 18747 df-grp 18900 df-minusg 18901 df-sbg 18902 df-mulg 19031 df-subg 19085 df-nsg 19086 df-eqg 19087 df-ghm 19175 df-cmn 19744 df-abl 19745 df-mgp 20082 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-oppr 20280 df-dvdsr 20303 df-unit 20304 df-invr 20334 df-rhm 20418 df-subrng 20490 df-subrg 20515 df-lmod 20752 df-lss 20823 df-lsp 20863 df-sra 21065 df-rgmod 21066 df-lidl 21111 df-rsp 21112 df-2idl 21151 df-cnfld 21287 df-zring 21380 df-zrh 21436 df-zn 21439 df-dchr 27186 |
This theorem is referenced by: dchrhash 27224 dchr2sum 27226 dchrisumlem1 27442 |
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