Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 4722 | . . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 )) | |
4 | 1, 2, 3 | sylanbrc 585 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 })) |
5 | fveq1 6672 | . . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚))) | |
6 | 5 | oveq1d 7174 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
7 | 6 | sumeq2sdv 15064 | . . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
8 | 7 | eqeq1d 2826 | . . . . 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 3686 | . . . 4 ⊢ (𝑋 ∈ 𝑊 ↔ (𝑋 ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0)) |
11 | 10 | baib 538 | . . 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 12284 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
14 | 1zzd 12016 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | 2fveq3 6678 | . . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) | |
16 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) | |
17 | 15, 16 | oveq12d 7177 | . . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
18 | dchrvmasumif.f | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | |
19 | ovex 7192 | . . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V | |
20 | 17, 18, 19 | fvmpt 6771 | . . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
21 | 20 | adantl 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
22 | rpvmasum.g | . . . . . 6 ⊢ 𝐺 = (DChr‘𝑁) | |
23 | rpvmasum.z | . . . . . 6 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
24 | rpvmasum.d | . . . . . 6 ⊢ 𝐷 = (Base‘𝐺) | |
25 | rpvmasum.l | . . . . . 6 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
26 | 1 | adantr 483 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷) |
27 | nnz 12007 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ) | |
28 | 27 | adantl 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
29 | 22, 23, 24, 25, 26, 28 | dchrzrhcl 25824 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
30 | nncn 11649 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
31 | 30 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
32 | nnne0 11674 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
33 | 32 | adantl 484 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
34 | 29, 31, 33 | divcld 11419 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
35 | dchrvmasumif.s | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) | |
36 | 13, 14, 21, 34, 35 | isumclim 15115 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆) |
37 | 36 | eqeq1d 2826 | . 2 ⊢ (𝜑 → (Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ 𝑆 = 0)) |
38 | 12, 37 | bitrd 281 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 {crab 3145 ∖ cdif 3936 {csn 4570 class class class wbr 5069 ↦ cmpt 5149 ‘cfv 6358 (class class class)co 7159 ℂcc 10538 0cc0 10540 1c1 10541 + caddc 10543 +∞cpnf 10675 ≤ cle 10679 − cmin 10873 / cdiv 11300 ℕcn 11641 ℤcz 11984 [,)cico 12743 ⌊cfl 13163 seqcseq 13372 abscabs 14596 ⇝ cli 14844 Σcsu 15045 Basecbs 16486 0gc0g 16716 ℤRHomczrh 20650 ℤ/nℤczn 20653 DChrcdchr 25811 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 ax-inf2 9107 ax-cnex 10596 ax-resscn 10597 ax-1cn 10598 ax-icn 10599 ax-addcl 10600 ax-addrcl 10601 ax-mulcl 10602 ax-mulrcl 10603 ax-mulcom 10604 ax-addass 10605 ax-mulass 10606 ax-distr 10607 ax-i2m1 10608 ax-1ne0 10609 ax-1rid 10610 ax-rnegex 10611 ax-rrecex 10612 ax-cnre 10613 ax-pre-lttri 10614 ax-pre-lttrn 10615 ax-pre-ltadd 10616 ax-pre-mulgt0 10617 ax-pre-sup 10618 ax-addf 10619 ax-mulf 10620 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-nel 3127 df-ral 3146 df-rex 3147 df-reu 3148 df-rmo 3149 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-int 4880 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-se 5518 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-isom 6367 df-riota 7117 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-1st 7692 df-2nd 7693 df-tpos 7895 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-1o 8105 df-oadd 8109 df-er 8292 df-ec 8294 df-qs 8298 df-map 8411 df-en 8513 df-dom 8514 df-sdom 8515 df-fin 8516 df-sup 8909 df-inf 8910 df-oi 8977 df-card 9371 df-pnf 10680 df-mnf 10681 df-xr 10682 df-ltxr 10683 df-le 10684 df-sub 10875 df-neg 10876 df-div 11301 df-nn 11642 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-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-fz 12896 df-fzo 13037 df-seq 13373 df-exp 13433 df-hash 13694 df-cj 14461 df-re 14462 df-im 14463 df-sqrt 14597 df-abs 14598 df-clim 14848 df-sum 15046 df-struct 16488 df-ndx 16489 df-slot 16490 df-base 16492 df-sets 16493 df-ress 16494 df-plusg 16581 df-mulr 16582 df-starv 16583 df-sca 16584 df-vsca 16585 df-ip 16586 df-tset 16587 df-ple 16588 df-ds 16590 df-unif 16591 df-0g 16718 df-imas 16784 df-qus 16785 df-mgm 17855 df-sgrp 17904 df-mnd 17915 df-mhm 17959 df-grp 18109 df-minusg 18110 df-sbg 18111 df-mulg 18228 df-subg 18279 df-nsg 18280 df-eqg 18281 df-ghm 18359 df-cmn 18911 df-abl 18912 df-mgp 19243 df-ur 19255 df-ring 19302 df-cring 19303 df-oppr 19376 df-dvdsr 19394 df-unit 19395 df-rnghom 19470 df-subrg 19536 df-lmod 19639 df-lss 19707 df-lsp 19747 df-sra 19947 df-rgmod 19948 df-lidl 19949 df-rsp 19950 df-2idl 20008 df-cnfld 20549 df-zring 20621 df-zrh 20654 df-zn 20657 df-dchr 25812 |
This theorem is referenced by: rpvmasum2 26091 dchrisum0re 26092 dchrisum0lem2 26097 dchrisumn0 26100 |
Copyright terms: Public domain | W3C validator |