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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 4260 | . . . 4 ⊢ (𝑋 ∈ (𝐷 ∖ { 1 }) ↔ (𝑋 ∈ 𝐷 ∧ 𝑋 ≠ 1 )) | |
4 | 1, 2, 3 | sylanbrc 695 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∖ { 1 })) |
5 | fveq1 6102 | . . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑦‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑚))) | |
6 | 5 | oveq1d 6564 | . . . . . . 7 ⊢ (𝑦 = 𝑋 → ((𝑦‘(𝐿‘𝑚)) / 𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
7 | 6 | sumeq2sdv 14282 | . . . . . 6 ⊢ (𝑦 = 𝑋 → Σ𝑚 ∈ ℕ ((𝑦‘(𝐿‘𝑚)) / 𝑚) = Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
8 | 7 | eqeq1d 2612 | . . . . 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 3333 | . . . 4 ⊢ (𝑋 ∈ 𝑊 ↔ (𝑋 ∈ (𝐷 ∖ { 1 }) ∧ Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0)) |
11 | 10 | baib 942 | . . 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 11599 | . . . 4 ⊢ ℕ = (ℤ≥‘1) | |
14 | 1zzd 11285 | . . . 4 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | fveq2 6103 | . . . . . . . 8 ⊢ (𝑎 = 𝑚 → (𝐿‘𝑎) = (𝐿‘𝑚)) | |
16 | 15 | fveq2d 6107 | . . . . . . 7 ⊢ (𝑎 = 𝑚 → (𝑋‘(𝐿‘𝑎)) = (𝑋‘(𝐿‘𝑚))) |
17 | id 22 | . . . . . . 7 ⊢ (𝑎 = 𝑚 → 𝑎 = 𝑚) | |
18 | 16, 17 | oveq12d 6567 | . . . . . 6 ⊢ (𝑎 = 𝑚 → ((𝑋‘(𝐿‘𝑎)) / 𝑎) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
19 | dchrvmasumif.f | . . . . . 6 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) | |
20 | ovex 6577 | . . . . . 6 ⊢ ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ V | |
21 | 18, 19, 20 | fvmpt 6191 | . . . . 5 ⊢ (𝑚 ∈ ℕ → (𝐹‘𝑚) = ((𝑋‘(𝐿‘𝑚)) / 𝑚)) |
22 | 21 | adantl 481 | . . . 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 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑋 ∈ 𝐷) |
28 | nnz 11276 | . . . . . . 7 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℤ) | |
29 | 28 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
30 | 23, 24, 25, 26, 27, 29 | dchrzrhcl 24770 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ) |
31 | nncn 10905 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ∈ ℂ) | |
32 | 31 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) |
33 | nnne0 10930 | . . . . . 6 ⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | |
34 | 33 | adantl 481 | . . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) |
35 | 30, 32, 34 | divcld 10680 | . . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑋‘(𝐿‘𝑚)) / 𝑚) ∈ ℂ) |
36 | dchrvmasumif.s | . . . 4 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆) | |
37 | 13, 14, 22, 35, 36 | isumclim 14330 | . . 3 ⊢ (𝜑 → Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 𝑆) |
38 | 37 | eqeq1d 2612 | . 2 ⊢ (𝜑 → (Σ𝑚 ∈ ℕ ((𝑋‘(𝐿‘𝑚)) / 𝑚) = 0 ↔ 𝑆 = 0)) |
39 | 12, 38 | bitrd 267 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑊 ↔ 𝑆 = 0)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∖ cdif 3537 {csn 4125 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 +∞cpnf 9950 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 ℤcz 11254 [,)cico 12048 ⌊cfl 12453 seqcseq 12663 abscabs 13822 ⇝ cli 14063 Σcsu 14264 Basecbs 15695 0gc0g 15923 ℤRHomczrh 19667 ℤ/nℤczn 19670 DChrcdchr 24757 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-tpos 7239 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-ec 7631 df-qs 7635 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-0g 15925 df-imas 15991 df-qus 15992 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-nsg 17415 df-eqg 17416 df-ghm 17481 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-oppr 18446 df-dvdsr 18464 df-unit 18465 df-rnghom 18538 df-subrg 18601 df-lmod 18688 df-lss 18754 df-lsp 18793 df-sra 18993 df-rgmod 18994 df-lidl 18995 df-rsp 18996 df-2idl 19053 df-cnfld 19568 df-zring 19638 df-zrh 19671 df-zn 19674 df-dchr 24758 |
This theorem is referenced by: rpvmasum2 25001 dchrisum0re 25002 dchrisum0lem2 25007 dchrisumn0 25010 |
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