Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > dchrisumn0 | Structured version Visualization version GIF version |
Description: The sum Σ𝑛 ∈ ℕ, 𝑋(𝑛) / 𝑛 is nonzero for all non-principal Dirichlet characters (i.e. the assumption 𝑋 ∈ 𝑊 is contradictory). This is the key result that allows us to eliminate the conditionals from dchrmusum2 26347 and dchrvmasumif 26356. Lemma 9.4.4 of [Shapiro], p. 382. (Contributed by Mario Carneiro, 12-May-2016.) |
Ref | Expression |
---|---|
rpvmasum.z | ⊢ 𝑍 = (ℤ/nℤ‘𝑁) |
rpvmasum.l | ⊢ 𝐿 = (ℤRHom‘𝑍) |
rpvmasum.a | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
dchrmusum.g | ⊢ 𝐺 = (DChr‘𝑁) |
dchrmusum.d | ⊢ 𝐷 = (Base‘𝐺) |
dchrmusum.1 | ⊢ 1 = (0g‘𝐺) |
dchrmusum.b | ⊢ (𝜑 → 𝑋 ∈ 𝐷) |
dchrmusum.n1 | ⊢ (𝜑 → 𝑋 ≠ 1 ) |
dchrmusum.f | ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) |
dchrmusum.c | ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) |
dchrmusum.t | ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) |
dchrmusum.2 | ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) |
Ref | Expression |
---|---|
dchrisumn0 | ⊢ (𝜑 → 𝑇 ≠ 0) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rpvmasum.z | . . . 4 ⊢ 𝑍 = (ℤ/nℤ‘𝑁) | |
2 | rpvmasum.l | . . . 4 ⊢ 𝐿 = (ℤRHom‘𝑍) | |
3 | rpvmasum.a | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
4 | 3 | adantr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝑁 ∈ ℕ) |
5 | dchrmusum.g | . . . 4 ⊢ 𝐺 = (DChr‘𝑁) | |
6 | dchrmusum.d | . . . 4 ⊢ 𝐷 = (Base‘𝐺) | |
7 | dchrmusum.1 | . . . 4 ⊢ 1 = (0g‘𝐺) | |
8 | eqid 2734 | . . . 4 ⊢ {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} = {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} | |
9 | dchrmusum.b | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐷) | |
10 | dchrmusum.n1 | . . . . . 6 ⊢ (𝜑 → 𝑋 ≠ 1 ) | |
11 | dchrmusum.f | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | |
12 | dchrmusum.c | . . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞)) | |
13 | dchrmusum.t | . . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇) | |
14 | dchrmusum.2 | . . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦)) | |
15 | 1, 2, 3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 8 | dchrvmaeq0 26357 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0} ↔ 𝑇 = 0)) |
16 | 15 | biimpar 481 | . . . 4 ⊢ ((𝜑 ∧ 𝑇 = 0) → 𝑋 ∈ {𝑦 ∈ (𝐷 ∖ { 1 }) ∣ Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = 0}) |
17 | 1, 2, 4, 5, 6, 7, 8, 16 | dchrisum0 26373 | . . 3 ⊢ ¬ (𝜑 ∧ 𝑇 = 0) |
18 | 17 | imnani 404 | . 2 ⊢ (𝜑 → ¬ 𝑇 = 0) |
19 | 18 | neqned 2942 | 1 ⊢ (𝜑 → 𝑇 ≠ 0) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∈ wcel 2110 ≠ wne 2935 ∀wral 3054 {crab 3058 ∖ cdif 3854 {csn 4531 class class class wbr 5043 ↦ cmpt 5124 ‘cfv 6369 (class class class)co 7202 0cc0 10712 1c1 10713 + caddc 10715 +∞cpnf 10847 ≤ cle 10851 − cmin 11045 / cdiv 11472 ℕcn 11813 [,)cico 12920 ⌊cfl 13348 seqcseq 13557 abscabs 14780 ⇝ cli 15028 Σcsu 15232 Basecbs 16684 0gc0g 16916 ℤRHomczrh 20438 ℤ/nℤczn 20441 DChrcdchr 26085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-inf2 9245 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 ax-pre-sup 10790 ax-addf 10791 ax-mulf 10792 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-iin 4897 df-disj 5009 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-se 5499 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-isom 6378 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-of 7458 df-rpss 7500 df-om 7634 df-1st 7750 df-2nd 7751 df-supp 7893 df-tpos 7957 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-2o 8192 df-oadd 8195 df-omul 8196 df-er 8380 df-ec 8382 df-qs 8386 df-map 8499 df-pm 8500 df-ixp 8568 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-fsupp 8975 df-fi 9016 df-sup 9047 df-inf 9048 df-oi 9115 df-dju 9500 df-card 9538 df-acn 9541 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-div 11473 df-nn 11814 df-2 11876 df-3 11877 df-4 11878 df-5 11879 df-6 11880 df-7 11881 df-8 11882 df-9 11883 df-n0 12074 df-xnn0 12146 df-z 12160 df-dec 12277 df-uz 12422 df-q 12528 df-rp 12570 df-xneg 12687 df-xadd 12688 df-xmul 12689 df-ioo 12922 df-ioc 12923 df-ico 12924 df-icc 12925 df-fz 13079 df-fzo 13222 df-fl 13350 df-mod 13426 df-seq 13558 df-exp 13619 df-fac 13823 df-bc 13852 df-hash 13880 df-word 14053 df-concat 14109 df-s1 14136 df-shft 14613 df-cj 14645 df-re 14646 df-im 14647 df-sqrt 14781 df-abs 14782 df-limsup 15015 df-clim 15032 df-rlim 15033 df-o1 15034 df-lo1 15035 df-sum 15233 df-ef 15610 df-e 15611 df-sin 15612 df-cos 15613 df-tan 15614 df-pi 15615 df-dvds 15797 df-gcd 16035 df-prm 16210 df-numer 16272 df-denom 16273 df-phi 16300 df-pc 16371 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-mulr 16781 df-starv 16782 df-sca 16783 df-vsca 16784 df-ip 16785 df-tset 16786 df-ple 16787 df-ds 16789 df-unif 16790 df-hom 16791 df-cco 16792 df-rest 16899 df-topn 16900 df-0g 16918 df-gsum 16919 df-topgen 16920 df-pt 16921 df-prds 16924 df-xrs 16979 df-qtop 16984 df-imas 16985 df-qus 16986 df-xps 16987 df-mre 17061 df-mrc 17062 df-acs 17064 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-mhm 18190 df-submnd 18191 df-grp 18340 df-minusg 18341 df-sbg 18342 df-mulg 18461 df-subg 18512 df-nsg 18513 df-eqg 18514 df-ghm 18592 df-gim 18635 df-ga 18656 df-cntz 18683 df-oppg 18710 df-od 18892 df-gex 18893 df-pgp 18894 df-lsm 18997 df-pj1 18998 df-cmn 19144 df-abl 19145 df-cyg 19234 df-dprd 19354 df-dpj 19355 df-mgp 19477 df-ur 19489 df-ring 19536 df-cring 19537 df-oppr 19613 df-dvdsr 19631 df-unit 19632 df-invr 19662 df-dvr 19673 df-rnghom 19707 df-drng 19741 df-subrg 19770 df-lmod 19873 df-lss 19941 df-lsp 19981 df-sra 20181 df-rgmod 20182 df-lidl 20183 df-rsp 20184 df-2idl 20242 df-psmet 20327 df-xmet 20328 df-met 20329 df-bl 20330 df-mopn 20331 df-fbas 20332 df-fg 20333 df-cnfld 20336 df-zring 20408 df-zrh 20442 df-zn 20445 df-top 21763 df-topon 21780 df-topsp 21802 df-bases 21815 df-cld 21888 df-ntr 21889 df-cls 21890 df-nei 21967 df-lp 22005 df-perf 22006 df-cn 22096 df-cnp 22097 df-haus 22184 df-cmp 22256 df-tx 22431 df-hmeo 22624 df-fil 22715 df-fm 22807 df-flim 22808 df-flf 22809 df-xms 23190 df-ms 23191 df-tms 23192 df-cncf 23747 df-0p 24539 df-limc 24735 df-dv 24736 df-ply 25054 df-idp 25055 df-coe 25056 df-dgr 25057 df-quot 25156 df-ulm 25241 df-log 25417 df-cxp 25418 df-atan 25722 df-em 25847 df-cht 25951 df-vma 25952 df-chp 25953 df-ppi 25954 df-mu 25955 df-dchr 26086 |
This theorem is referenced by: dchrmusumlem 26375 dchrvmasumlem 26376 |
Copyright terms: Public domain | W3C validator |