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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 2823 | . . . . 5 ⊢ (Unit‘𝑍) = (Unit‘𝑍) | |
3 | 1, 2 | unitss 19412 | . . . 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 25820 | . . . 4 ⊢ (𝜑 → 𝑋:𝐵⟶ℂ) |
10 | 3 | sseli 3965 | . . . 4 ⊢ (𝑎 ∈ (Unit‘𝑍) → 𝑎 ∈ 𝐵) |
11 | ffvelrn 6851 | . . . 4 ⊢ ((𝑋:𝐵⟶ℂ ∧ 𝑎 ∈ 𝐵) → (𝑋‘𝑎) ∈ ℂ) | |
12 | 9, 10, 11 | syl2an 597 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (Unit‘𝑍)) → (𝑋‘𝑎) ∈ ℂ) |
13 | eldif 3948 | . . . 4 ⊢ (𝑎 ∈ (𝐵 ∖ (Unit‘𝑍)) ↔ (𝑎 ∈ 𝐵 ∧ ¬ 𝑎 ∈ (Unit‘𝑍))) | |
14 | 8 | adantr 483 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑋 ∈ 𝐷) |
15 | simpr 487 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → 𝑎 ∈ 𝐵) | |
16 | 5, 6, 7, 1, 2, 14, 15 | dchrn0 25828 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) ≠ 0 ↔ 𝑎 ∈ (Unit‘𝑍))) |
17 | 16 | biimpd 231 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → ((𝑋‘𝑎) ≠ 0 → 𝑎 ∈ (Unit‘𝑍))) |
18 | 17 | necon1bd 3036 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ 𝐵) → (¬ 𝑎 ∈ (Unit‘𝑍) → (𝑋‘𝑎) = 0)) |
19 | 18 | impr 457 | . . . 4 ⊢ ((𝜑 ∧ (𝑎 ∈ 𝐵 ∧ ¬ 𝑎 ∈ (Unit‘𝑍))) → (𝑋‘𝑎) = 0) |
20 | 13, 19 | sylan2b 595 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ (𝐵 ∖ (Unit‘𝑍))) → (𝑋‘𝑎) = 0) |
21 | 5, 7 | dchrrcl 25818 | . . . 4 ⊢ (𝑋 ∈ 𝐷 → 𝑁 ∈ ℕ) |
22 | 6, 1 | znfi 20708 | . . . 4 ⊢ (𝑁 ∈ ℕ → 𝐵 ∈ Fin) |
23 | 8, 21, 22 | 3syl 18 | . . 3 ⊢ (𝜑 → 𝐵 ∈ Fin) |
24 | 4, 12, 20, 23 | fsumss 15084 | . 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 25846 | . 2 ⊢ (𝜑 → Σ𝑎 ∈ (Unit‘𝑍)(𝑋‘𝑎) = if(𝑋 = 1 , (ϕ‘𝑁), 0)) |
27 | 24, 26 | eqtr3d 2860 | 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 398 = wceq 1537 ∈ wcel 2114 ≠ wne 3018 ∖ cdif 3935 ⊆ wss 3938 ifcif 4469 ⟶wf 6353 ‘cfv 6357 Fincfn 8511 ℂcc 10537 0cc0 10539 ℕcn 11640 Σcsu 15044 ϕcphi 16103 Basecbs 16485 0gc0g 16715 Unitcui 19391 ℤ/nℤczn 20652 DChrcdchr 25810 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-of 7411 df-om 7583 df-1st 7691 df-2nd 7692 df-tpos 7894 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-ec 8293 df-qs 8297 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-xnn0 11971 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-fl 13165 df-mod 13241 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-clim 14847 df-sum 15045 df-dvds 15610 df-gcd 15846 df-phi 16105 df-struct 16487 df-ndx 16488 df-slot 16489 df-base 16491 df-sets 16492 df-ress 16493 df-plusg 16580 df-mulr 16581 df-starv 16582 df-sca 16583 df-vsca 16584 df-ip 16585 df-tset 16586 df-ple 16587 df-ds 16589 df-unif 16590 df-0g 16717 df-imas 16783 df-qus 16784 df-mgm 17854 df-sgrp 17903 df-mnd 17914 df-mhm 17958 df-grp 18108 df-minusg 18109 df-sbg 18110 df-mulg 18227 df-subg 18278 df-nsg 18279 df-eqg 18280 df-ghm 18358 df-cmn 18910 df-abl 18911 df-mgp 19242 df-ur 19254 df-ring 19301 df-cring 19302 df-oppr 19375 df-dvdsr 19393 df-unit 19394 df-invr 19424 df-rnghom 19469 df-subrg 19535 df-lmod 19638 df-lss 19706 df-lsp 19746 df-sra 19946 df-rgmod 19947 df-lidl 19948 df-rsp 19949 df-2idl 20007 df-cnfld 20548 df-zring 20620 df-zrh 20653 df-zn 20656 df-dchr 25811 |
This theorem is referenced by: dchrhash 25849 dchr2sum 25851 dchrisumlem1 26067 |
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