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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 2726 | . . . . 5 β’ (Unitβπ) = (Unitβπ) | |
3 | 1, 2 | unitss 20275 | . . . 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 27125 | . . . 4 β’ (π β π:π΅βΆβ) |
10 | 3 | sseli 3973 | . . . 4 β’ (π β (Unitβπ) β π β π΅) |
11 | ffvelcdm 7076 | . . . 4 β’ ((π:π΅βΆβ β§ π β π΅) β (πβπ) β β) | |
12 | 9, 10, 11 | syl2an 595 | . . 3 β’ ((π β§ π β (Unitβπ)) β (πβπ) β β) |
13 | eldif 3953 | . . . 4 β’ (π β (π΅ β (Unitβπ)) β (π β π΅ β§ Β¬ π β (Unitβπ))) | |
14 | 8 | adantr 480 | . . . . . . . 8 β’ ((π β§ π β π΅) β π β π·) |
15 | simpr 484 | . . . . . . . 8 β’ ((π β§ π β π΅) β π β π΅) | |
16 | 5, 6, 7, 1, 2, 14, 15 | dchrn0 27133 | . . . . . . 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 2952 | . . . . 5 β’ ((π β§ π β π΅) β (Β¬ π β (Unitβπ) β (πβπ) = 0)) |
19 | 18 | impr 454 | . . . 4 β’ ((π β§ (π β π΅ β§ Β¬ π β (Unitβπ))) β (πβπ) = 0) |
20 | 13, 19 | sylan2b 593 | . . 3 β’ ((π β§ π β (π΅ β (Unitβπ))) β (πβπ) = 0) |
21 | 5, 7 | dchrrcl 27123 | . . . 4 β’ (π β π· β π β β) |
22 | 6, 1 | znfi 21449 | . . . 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 15674 | . 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 27151 | . 2 β’ (π β Ξ£π β (Unitβπ)(πβπ) = if(π = 1 , (Οβπ), 0)) |
27 | 24, 26 | eqtr3d 2768 | 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 395 = wceq 1533 β wcel 2098 β wne 2934 β cdif 3940 β wss 3943 ifcif 4523 βΆwf 6532 βcfv 6536 Fincfn 8938 βcc 11107 0cc0 11109 βcn 12213 Ξ£csu 15635 Οcphi 16703 Basecbs 17150 0gc0g 17391 Unitcui 20254 β€/nβ€czn 21384 DChrcdchr 27115 |
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 7721 ax-inf2 9635 ax-cnex 11165 ax-resscn 11166 ax-1cn 11167 ax-icn 11168 ax-addcl 11169 ax-addrcl 11170 ax-mulcl 11171 ax-mulrcl 11172 ax-mulcom 11173 ax-addass 11174 ax-mulass 11175 ax-distr 11176 ax-i2m1 11177 ax-1ne0 11178 ax-1rid 11179 ax-rnegex 11180 ax-rrecex 11181 ax-cnre 11182 ax-pre-lttri 11183 ax-pre-lttrn 11184 ax-pre-ltadd 11185 ax-pre-mulgt0 11186 ax-pre-sup 11187 ax-addf 11188 ax-mulf 11189 |
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 6293 df-ord 6360 df-on 6361 df-lim 6362 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7360 df-ov 7407 df-oprab 7408 df-mpo 7409 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-tpos 8209 df-frecs 8264 df-wrecs 8295 df-recs 8369 df-rdg 8408 df-1o 8464 df-oadd 8468 df-er 8702 df-ec 8704 df-qs 8708 df-map 8821 df-en 8939 df-dom 8940 df-sdom 8941 df-fin 8942 df-sup 9436 df-inf 9437 df-oi 9504 df-card 9933 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255 df-sub 11447 df-neg 11448 df-div 11873 df-nn 12214 df-2 12276 df-3 12277 df-4 12278 df-5 12279 df-6 12280 df-7 12281 df-8 12282 df-9 12283 df-n0 12474 df-xnn0 12546 df-z 12560 df-dec 12679 df-uz 12824 df-rp 12978 df-fz 13488 df-fzo 13631 df-fl 13760 df-mod 13838 df-seq 13970 df-exp 14030 df-hash 14293 df-cj 15049 df-re 15050 df-im 15051 df-sqrt 15185 df-abs 15186 df-clim 15435 df-sum 15636 df-dvds 16202 df-gcd 16440 df-phi 16705 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17151 df-ress 17180 df-plusg 17216 df-mulr 17217 df-starv 17218 df-sca 17219 df-vsca 17220 df-ip 17221 df-tset 17222 df-ple 17223 df-ds 17225 df-unif 17226 df-0g 17393 df-imas 17460 df-qus 17461 df-mgm 18570 df-sgrp 18649 df-mnd 18665 df-mhm 18710 df-grp 18863 df-minusg 18864 df-sbg 18865 df-mulg 18993 df-subg 19047 df-nsg 19048 df-eqg 19049 df-ghm 19136 df-cmn 19699 df-abl 19700 df-mgp 20037 df-rng 20055 df-ur 20084 df-ring 20137 df-cring 20138 df-oppr 20233 df-dvdsr 20256 df-unit 20257 df-invr 20287 df-rhm 20371 df-subrng 20443 df-subrg 20468 df-lmod 20705 df-lss 20776 df-lsp 20816 df-sra 21018 df-rgmod 21019 df-lidl 21064 df-rsp 21065 df-2idl 21104 df-cnfld 21236 df-zring 21329 df-zrh 21385 df-zn 21388 df-dchr 27116 |
This theorem is referenced by: dchrhash 27154 dchr2sum 27156 dchrisumlem1 27372 |
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