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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 𝑋 is 0 if 𝑋 is non-principal and ϕ(𝑛) otherwise. Part of Theorem 6.5.1 of [Shapiro] p. 230. (Contributed by Mario Carneiro, 28-Apr-2016.) |
Ref | Expression |
---|---|
dchrsum.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrsum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
dchrsum.d | ⊢ 𝐷 = (Base‘𝐺) |
dchrsum.1 | ⊢ 1 = (0g‘𝐺) |
dchrsum.x | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrsum.b | ⊢ 𝐵 = (Base‘𝑍) |
Ref | Expression |
---|---|
dchrsum | ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrsum.b | . . . . 5 ⊢ 𝐵 = (Base‘𝑍) | |
2 | eqid 2734 | . . . . 5 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
3 | 1, 2 | unitss 20392 | . . . 4 ⊢ (Unit‘𝑍) ⊆ 𝐵 |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (Unit‘𝑍) ⊆ 𝐵) |
5 | dchrsum.g | . . . . 5 ⊢ 𝐺 = (DChr‘𝑁) | |
6 | dchrsum.z | . . . . 5 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
7 | dchrsum.d | . . . . 5 ⊢ 𝐷 = (Base‘𝐺) | |
8 | dchrsum.x | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
9 | 5, 6, 7, 1, 8 | dchrf 27300 | . . . 4 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
10 | 3 | sseli 3990 | . . . 4 ⊢ (𝑎 ∈ (Unit‘𝑍) → 𝑎 ∈ 𝐵) |
11 | ffvelcdm 7100 | . . . 4 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑎 ∈ 𝐵) → (𝑋‘𝑎) ∈ ℂ) | |
12 | 9, 10, 11 | syl2an 596 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Unit‘𝑍)) → (𝑋‘𝑎) ∈ ℂ) |
13 | eldif 3972 | . . . 4 ⊢ (𝑎 ∈ (𝐵 ∖ (Unit‘𝑍)) ↔ (𝑎 ∈ 𝐵 ∧ ¬ 𝑎 ∈ (Unit‘𝑍))) | |
14 | 8 | adantr 480 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐷) |
15 | simpr 484 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) | |
16 | 5, 6, 7, 1, 2, 14, 15 | dchrn0 27308 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) ≠ 0 ↔ 𝑎 ∈ (Unit‘𝑍))) |
17 | 16 | biimpd 229 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) ≠ 0 → 𝑎 ∈ (Unit‘𝑍))) |
18 | 17 | necon1bd 2955 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (¬ 𝑎 ∈ (Unit‘𝑍) → (𝑋‘𝑎) = 0)) |
19 | 18 | impr 454 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ ¬ 𝑎 ∈ (Unit‘𝑍))) → (𝑋‘𝑎) = 0) |
20 | 13, 19 | sylan2b 594 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐵 ∖ (Unit‘𝑍))) → (𝑋‘𝑎) = 0) |
21 | 5, 7 | dchrrcl 27298 | . . . 4 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
22 | 6, 1 | znfi 21595 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin) |
23 | 8, 21, 22 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐵 ∈ Fin) |
24 | 4, 12, 20, 23 | fsumss 15757 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ (Unit‘𝑍)(𝑋‘𝑎) = Σ𝑎 ∈ 𝐵 (𝑋‘𝑎)) |
25 | dchrsum.1 | . . 3 ⊢ 1 = (0g‘𝐺) | |
26 | 5, 6, 7, 25, 8, 2 | dchrsum2 27326 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ (Unit‘𝑍)(𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) |
27 | 24, 26 | eqtr3d 2776 | 1 ⊢ (𝜑 → Σ𝑎 ∈ 𝐵 (𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 ∖ cdif 3959 ⊆ wss 3962 ifcif 4530 ⟶wf 6558 ‘cfv 6562 Fincfn 8983 ℂcc 11150 0cc0 11152 ℕcn 12263 Σcsu 15718 ϕcphi 16797 Basecbs 17244 0gc0g 17485 Unitcui 20371 ℤ/nℤczn 21530 DChrcdchr 27290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 ax-addf 11231 ax-mulf 11232 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-tp 4635 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-of 7696 df-om 7887 df-1st 8012 df-2nd 8013 df-tpos 8249 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-oadd 8508 df-er 8743 df-ec 8745 df-qs 8749 df-map 8866 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-xnn0 12597 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-fz 13544 df-fzo 13691 df-fl 13828 df-mod 13906 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 df-dvds 16287 df-gcd 16528 df-phi 16799 df-struct 17180 df-sets 17197 df-slot 17215 df-ndx 17227 df-base 17245 df-ress 17274 df-plusg 17310 df-mulr 17311 df-starv 17312 df-sca 17313 df-vsca 17314 df-ip 17315 df-tset 17316 df-ple 17317 df-ds 17319 df-unif 17320 df-0g 17487 df-imas 17554 df-qus 17555 df-mgm 18665 df-sgrp 18744 df-mnd 18760 df-mhm 18808 df-grp 18966 df-minusg 18967 df-sbg 18968 df-mulg 19098 df-subg 19153 df-nsg 19154 df-eqg 19155 df-ghm 19243 df-cmn 19814 df-abl 19815 df-mgp 20152 df-rng 20170 df-ur 20199 df-ring 20252 df-cring 20253 df-oppr 20350 df-dvdsr 20373 df-unit 20374 df-invr 20404 df-rhm 20488 df-subrng 20562 df-subrg 20586 df-lmod 20876 df-lss 20947 df-lsp 20987 df-sra 21189 df-rgmod 21190 df-lidl 21235 df-rsp 21236 df-2idl 21277 df-cnfld 21382 df-zring 21475 df-zrh 21531 df-zn 21534 df-dchr 27291 |
This theorem is referenced by: dchrhash 27329 dchr2sum 27331 dchrisumlem1 27547 |
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