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Mirrors > Home > MPE Home > Th. List > sumdchr | Structured version Visualization version GIF version |
Description: An orthogonality relation for Dirichlet characters: the sum of 𝑥(𝐴) for fixed 𝐴 and all 𝑥 is 0 if 𝐴 = 1 and ϕ(𝑛) otherwise. Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
sumdchr.g | ⊢ 𝐺 = (DChr‘𝑁) |
sumdchr.d | ⊢ 𝐷 = (Base‘𝐺) |
sumdchr.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
sumdchr.1 | ⊢ 1 = (1r‘𝑍) |
sumdchr.b | ⊢ 𝐵 = (Base‘𝑍) |
sumdchr.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
sumdchr.a | ⊢ (𝜑 → 𝐴 ∈ 𝐵) |
Ref | Expression |
---|---|
sumdchr | ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (ϕ‘𝑁), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sumdchr.g | . . 3 ⊢ 𝐺 = (DChr‘𝑁) | |
2 | sumdchr.d | . . 3 ⊢ 𝐷 = (Base‘𝐺) | |
3 | sumdchr.z | . . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
4 | sumdchr.1 | . . 3 ⊢ 1 = (1r‘𝑍) | |
5 | sumdchr.b | . . 3 ⊢ 𝐵 = (Base‘𝑍) | |
6 | sumdchr.n | . . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
7 | sumdchr.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵) | |
8 | 1, 2, 3, 4, 5, 6, 7 | sumdchr2 25845 | . 2 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (♯‘𝐷), 0)) |
9 | 1, 2 | dchrhash 25846 | . . . 4 ⊢ (𝑁 ∈ ℕ → (♯‘𝐷) = (ϕ‘𝑁)) |
10 | 6, 9 | syl 17 | . . 3 ⊢ (𝜑 → (♯‘𝐷) = (ϕ‘𝑁)) |
11 | 10 | ifeq1d 4484 | . 2 ⊢ (𝜑 → if(𝐴 = 1 , (♯‘𝐷), 0) = if(𝐴 = 1 , (ϕ‘𝑁), 0)) |
12 | 8, 11 | eqtrd 2856 | 1 ⊢ (𝜑 → Σ𝑥 ∈ 𝐷 (𝑥‘𝐴) = if(𝐴 = 1 , (ϕ‘𝑁), 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ifcif 4466 ‘cfv 6354 0cc0 10536 ℕcn 11637 ♯chash 13689 Σcsu 15041 ϕcphi 16100 Basecbs 16482 1rcur 19250 ℤ/nℤczn 20649 DChrcdchr 25807 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-inf2 9103 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614 ax-addf 10615 ax-mulf 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-int 4876 df-iun 4920 df-iin 4921 df-disj 5031 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-isom 6363 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-of 7408 df-rpss 7448 df-om 7580 df-1st 7688 df-2nd 7689 df-supp 7830 df-tpos 7891 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-1o 8101 df-2o 8102 df-oadd 8105 df-omul 8106 df-er 8288 df-ec 8290 df-qs 8294 df-map 8407 df-pm 8408 df-ixp 8461 df-en 8509 df-dom 8510 df-sdom 8511 df-fin 8512 df-fsupp 8833 df-fi 8874 df-sup 8905 df-inf 8906 df-oi 8973 df-dju 9329 df-card 9367 df-acn 9370 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-4 11701 df-5 11702 df-6 11703 df-7 11704 df-8 11705 df-9 11706 df-n0 11897 df-xnn0 11967 df-z 11981 df-dec 12098 df-uz 12243 df-q 12348 df-rp 12389 df-xneg 12506 df-xadd 12507 df-xmul 12508 df-ioo 12741 df-ioc 12742 df-ico 12743 df-icc 12744 df-fz 12892 df-fzo 13033 df-fl 13161 df-mod 13237 df-seq 13369 df-exp 13429 df-fac 13633 df-bc 13662 df-hash 13690 df-word 13861 df-concat 13922 df-s1 13949 df-shft 14425 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-limsup 14827 df-clim 14844 df-rlim 14845 df-sum 15042 df-ef 15420 df-sin 15422 df-cos 15423 df-pi 15425 df-dvds 15607 df-gcd 15843 df-prm 16015 df-phi 16102 df-pc 16173 df-struct 16484 df-ndx 16485 df-slot 16486 df-base 16488 df-sets 16489 df-ress 16490 df-plusg 16577 df-mulr 16578 df-starv 16579 df-sca 16580 df-vsca 16581 df-ip 16582 df-tset 16583 df-ple 16584 df-ds 16586 df-unif 16587 df-hom 16588 df-cco 16589 df-rest 16695 df-topn 16696 df-0g 16714 df-gsum 16715 df-topgen 16716 df-pt 16717 df-prds 16720 df-xrs 16774 df-qtop 16779 df-imas 16780 df-qus 16781 df-xps 16782 df-mre 16856 df-mrc 16857 df-acs 16859 df-mgm 17851 df-sgrp 17900 df-mnd 17911 df-mhm 17955 df-submnd 17956 df-grp 18105 df-minusg 18106 df-sbg 18107 df-mulg 18224 df-subg 18275 df-nsg 18276 df-eqg 18277 df-ghm 18355 df-gim 18398 df-ga 18419 df-cntz 18446 df-oppg 18473 df-od 18655 df-gex 18656 df-pgp 18657 df-lsm 18760 df-pj1 18761 df-cmn 18907 df-abl 18908 df-cyg 18996 df-dprd 19116 df-dpj 19117 df-mgp 19239 df-ur 19251 df-ring 19298 df-cring 19299 df-oppr 19372 df-dvdsr 19390 df-unit 19391 df-invr 19421 df-rnghom 19466 df-subrg 19532 df-lmod 19635 df-lss 19703 df-lsp 19743 df-sra 19943 df-rgmod 19944 df-lidl 19945 df-rsp 19946 df-2idl 20004 df-psmet 20536 df-xmet 20537 df-met 20538 df-bl 20539 df-mopn 20540 df-fbas 20541 df-fg 20542 df-cnfld 20545 df-zring 20617 df-zrh 20650 df-zn 20653 df-top 21501 df-topon 21518 df-topsp 21540 df-bases 21553 df-cld 21626 df-ntr 21627 df-cls 21628 df-nei 21705 df-lp 21743 df-perf 21744 df-cn 21834 df-cnp 21835 df-haus 21922 df-tx 22169 df-hmeo 22362 df-fil 22453 df-fm 22545 df-flim 22546 df-flf 22547 df-xms 22929 df-ms 22930 df-tms 22931 df-cncf 23485 df-0p 24270 df-limc 24463 df-dv 24464 df-ply 24777 df-idp 24778 df-coe 24779 df-dgr 24780 df-quot 24879 df-log 25139 df-cxp 25140 df-dchr 25808 |
This theorem is referenced by: sum2dchr 25849 |
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