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Mirrors > Home > MPE Home > Th. List > dchrvmaeq0 | Structured version Visualization version GIF version |
Description: The set 𝑊 is the collection of all non-principal Dirichlet characters such that the sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is equal to zero. (Contributed by Mario Carneiro, 5-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
rpvmasum.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
rpvmasum.a | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
rpvmasum.g | ⊢ 𝐺 = (DChr‘𝑁) |
rpvmasum.d | ⊢ 𝐷 = (Base‘𝐺) |
rpvmasum.1 | ⊢ 1 = (0g‘𝐺) |
dchrisum.b | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrisum.n1 | ⊢ (𝜑 → 𝑋 ≠ 1 ) |
dchrvmasumif.f | ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) |
dchrvmasumif.c | ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
dchrvmasumif.s | ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) |
dchrvmasumif.1 | ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦)) |
dchrvmaeq0.w | ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} |
Ref | Expression |
---|---|
dchrvmaeq0 | ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dchrisum.b | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
2 | dchrisum.n1 | . . . 4 ⊢ (𝜑 → 𝑋 ≠ 1 ) | |
3 | eldifsn 4462 | . . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 )) | |
4 | 1, 2, 3 | sylanbrc 701 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 })) |
5 | fveq1 6351 | . . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚))) | |
6 | 5 | oveq1d 6828 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
7 | 6 | sumeq2sdv 14634 | . . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
8 | 7 | eqeq1d 2762 | . . . . 5 ⊢ (𝑦 = 𝑋 → (Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0)) |
9 | dchrvmaeq0.w | . . . . 5 ⊢ 𝑊 = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} | |
10 | 8, 9 | elrab2 3507 | . . . 4 ⊢ (𝑋 ∈ 𝑊 ↔ (𝑋 ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0)) |
11 | 10 | baib 982 | . . 3 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0)) |
12 | 4, 11 | syl 17 | . 2 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0)) |
13 | nnuz 11916 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
14 | 1zzd 11600 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | fveq2 6352 | . . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝐿‘𝑎) = (𝐿‘𝑚)) | |
16 | 15 | fveq2d 6356 | . . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) |
17 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) | |
18 | 16, 17 | oveq12d 6831 | . . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
19 | dchrvmasumif.f | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | |
20 | ovex 6841 | . . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V | |
21 | 18, 19, 20 | fvmpt 6444 | . . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
22 | 21 | adantl 473 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
23 | rpvmasum.g | . . . . . 6 ⊢ 𝐺 = (DChr‘𝑁) | |
24 | rpvmasum.z | . . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
25 | rpvmasum.d | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
26 | rpvmasum.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
27 | 1 | adantr 472 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷) |
28 | nnz 11591 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ) | |
29 | 28 | adantl 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
30 | 23, 24, 25, 26, 27, 29 | dchrzrhcl 25169 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
31 | nncn 11220 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
32 | 31 | adantl 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
33 | nnne0 11245 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
34 | 33 | adantl 473 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
35 | 30, 32, 34 | divcld 10993 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
36 | dchrvmasumif.s | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) | |
37 | 13, 14, 22, 35, 36 | isumclim 14687 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆) |
38 | 37 | eqeq1d 2762 | . 2 ⊢ (𝜑 → (Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ 𝑆 = 0)) |
39 | 12, 38 | bitrd 268 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 {crab 3054 ∖ cdif 3712 {csn 4321 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 0cc0 10128 1c1 10129 + caddc 10131 +∞cpnf 10263 ≤ cle 10267 − cmin 10458 / cdiv 10876 ℕcn 11212 ℤcz 11569 [,)cico 12370 ⌊cfl 12785 seqcseq 12995 abscabs 14173 ⇝ cli 14414 Σcsu 14615 Basecbs 16059 0gc0g 16302 ℤRHomczrh 20050 ℤ/nℤczn 20053 DChrcdchr 25156 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 ax-inf2 8711 ax-cnex 10184 ax-resscn 10185 ax-1cn 10186 ax-icn 10187 ax-addcl 10188 ax-addrcl 10189 ax-mulcl 10190 ax-mulrcl 10191 ax-mulcom 10192 ax-addass 10193 ax-mulass 10194 ax-distr 10195 ax-i2m1 10196 ax-1ne0 10197 ax-1rid 10198 ax-rnegex 10199 ax-rrecex 10200 ax-cnre 10201 ax-pre-lttri 10202 ax-pre-lttrn 10203 ax-pre-ltadd 10204 ax-pre-mulgt0 10205 ax-pre-sup 10206 ax-addf 10207 ax-mulf 10208 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-fal 1638 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-nel 3036 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-se 5226 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-isom 6058 df-riota 6774 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-1st 7333 df-2nd 7334 df-tpos 7521 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-oadd 7733 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-pnf 10268 df-mnf 10269 df-xr 10270 df-ltxr 10271 df-le 10272 df-sub 10460 df-neg 10461 df-div 10877 df-nn 11213 df-2 11271 df-3 11272 df-4 11273 df-5 11274 df-6 11275 df-7 11276 df-8 11277 df-9 11278 df-n0 11485 df-z 11570 df-dec 11686 df-uz 11880 df-rp 12026 df-fz 12520 df-fzo 12660 df-seq 12996 df-exp 13055 df-hash 13312 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-clim 14418 df-sum 14616 df-struct 16061 df-ndx 16062 df-slot 16063 df-base 16065 df-sets 16066 df-ress 16067 df-plusg 16156 df-mulr 16157 df-starv 16158 df-sca 16159 df-vsca 16160 df-ip 16161 df-tset 16162 df-ple 16163 df-ds 16166 df-unif 16167 df-0g 16304 df-imas 16370 df-qus 16371 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-grp 17626 df-minusg 17627 df-sbg 17628 df-mulg 17742 df-subg 17792 df-nsg 17793 df-eqg 17794 df-ghm 17859 df-cmn 18395 df-abl 18396 df-mgp 18690 df-ur 18702 df-ring 18749 df-cring 18750 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-rnghom 18917 df-subrg 18980 df-lmod 19067 df-lss 19135 df-lsp 19174 df-sra 19374 df-rgmod 19375 df-lidl 19376 df-rsp 19377 df-2idl 19434 df-cnfld 19949 df-zring 20021 df-zrh 20054 df-zn 20057 df-dchr 25157 |
This theorem is referenced by: rpvmasum2 25400 dchrisum0re 25401 dchrisum0lem2 25406 dchrisumn0 25409 |
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