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Theorem dchrmusumlem 24955

Description: The sum of the Möbius function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded. Equation 9.4.16 of [Shapiro], p. 379. (Contributed by Mario Carneiro, 12-May-2016.)

**Hypotheses**
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 = (0_g‘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))

**Assertion**
Ref	Expression
dchrmusumlem	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))

Distinct variable groups: 𝑥,𝑛,𝑦, 1 𝐶,𝑛,𝑥,𝑦 𝑛,𝐹,𝑥,𝑦 𝑥,𝑎,𝑦 𝑛,𝑁,𝑥,𝑦 𝜑,𝑛,𝑥 𝑇,𝑛,𝑥,𝑦 𝑛,𝑍,𝑥,𝑦 𝐷,𝑛,𝑥,𝑦 𝑛,𝑎,𝐿,𝑥,𝑦 𝑋,𝑎,𝑛,𝑥,𝑦 Allowed substitution hints: 𝜑(𝑦,𝑎) 𝐶(𝑎) 𝐷(𝑎) 𝑇(𝑎) 1 (𝑎) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑛,𝑎) 𝑁(𝑎) 𝑍(𝑎)

**Proof of Theorem dchrmusumlem**
Step	Hyp	Ref	Expression
1		fzfid 12591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1...(⌊‘𝑥)) ∈ Fin)
2		dchrmusum.g	. . . . . . . . 9 ⊢ 𝐺 = (DChr‘𝑁)
3		rpvmasum.z	. . . . . . . . 9 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
4		dchrmusum.d	. . . . . . . . 9 ⊢ 𝐷 = (Base‘𝐺)
5		rpvmasum.l	. . . . . . . . 9 ⊢ 𝐿 = (ℤRHom‘𝑍)
6		dchrmusum.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
7	6	ad2antrr 757	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
8		elfzelz 12170	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
9	8	adantl 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
10	2, 3, 4, 5, 7, 9	dchrzrhcl 24714	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
11		elfznn 12198	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
12	11	adantl 480	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
13		mucl 24611	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
15	14	zred 11316	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
16	15, 12	nndivred 10918	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
17	16	recnd 9924	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
18	10, 17	mulcld 9916	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ)
19	1, 18	fsumcl 14259	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ)
20		dchrmusum.t	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
21		climcl 14026	. . . . . . . 8 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ)
23	22	adantr 479	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
24	19, 23	mulcld 9916	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) ∈ ℂ)
25		rpvmasum.a	. . . . . . 7 ⊢ (𝜑 → 𝑁 ∈ ℕ)
26		dchrmusum.1	. . . . . . 7 ⊢ 1 = (0_g‘𝐺)
27		dchrmusum.n1	. . . . . . 7 ⊢ (𝜑 → 𝑋 ≠ 1 )
28		dchrmusum.f	. . . . . . 7 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
29		dchrmusum.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
30		dchrmusum.2	. . . . . . 7 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑇)) ≤ (𝐶 / 𝑦))
31	3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30	dchrisumn0 24954	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 0)
32	31	adantr 479	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ≠ 0)
33	24, 23, 32	divrecd 10655	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) / 𝑇) = ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇)))
34	19, 23, 32	divcan4d 10658	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) / 𝑇) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)))
35	33, 34	eqtr3d 2645	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇)) = Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)))
36	35	mpteq2dva 4666	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) = (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))))
37	22, 31	reccld 10645	. . . 4 ⊢ (𝜑 → (1 / 𝑇) ∈ ℂ)
38	37	adantr 479	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑇) ∈ ℂ)
39	3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30	dchrmusum2 24927	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇)) ∈ 𝑂(1))
40		rpssre 11677	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
41		o1const 14146	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (1 / 𝑇) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑇)) ∈ 𝑂(1))
42	40, 37, 41	sylancr 693	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑇)) ∈ 𝑂(1))
43	24, 38, 39, 42	o1mul2 14151	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) ∈ 𝑂(1))
44	36, 43	eqeltrrd 2688	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 382 = wceq 1474 ∈ wcel 1976 ≠ wne 2779 ∀wral 2895 ⊆ wss 3539 class class class wbr 4577 ↦ cmpt 4637 ‘cfv 5789 (class class class)co 6526 ℂcc 9790 ℝcr 9791 0cc0 9792 1c1 9793 + caddc 9795 · cmul 9797 +∞cpnf 9927 ≤ cle 9931 − cmin 10117 / cdiv 10535 ℕcn 10869 ℤcz 11212 ℝ⁺crp 11666 [,)cico 12006 ...cfz 12154 ⌊cfl 12410 seqcseq 12620 abscabs 13770 ⇝ cli 14011 𝑂(1)co1 14013 Σcsu 14212 Basecbs 15643 0_gc0g 15871 ℤRHomczrh 19614 ℤ/nℤczn 19617 μcmu 24565 DChrcdchr 24701

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1712 ax-4 1727 ax-5 1826 ax-6 1874 ax-7 1921 ax-8 1978 ax-9 1985 ax-10 2005 ax-11 2020 ax-12 2033 ax-13 2233 ax-ext 2589 ax-rep 4693 ax-sep 4703 ax-nul 4711 ax-pow 4763 ax-pr 4827 ax-un 6824 ax-inf2 8398 ax-cnex 9848 ax-resscn 9849 ax-1cn 9850 ax-icn 9851 ax-addcl 9852 ax-addrcl 9853 ax-mulcl 9854 ax-mulrcl 9855 ax-mulcom 9856 ax-addass 9857 ax-mulass 9858 ax-distr 9859 ax-i2m1 9860 ax-1ne0 9861 ax-1rid 9862 ax-rnegex 9863 ax-rrecex 9864 ax-cnre 9865 ax-pre-lttri 9866 ax-pre-lttrn 9867 ax-pre-ltadd 9868 ax-pre-mulgt0 9869 ax-pre-sup 9870 ax-addf 9871 ax-mulf 9872

This theorem depends on definitions: df-bi 195 df-or 383 df-an 384 df-3or 1031 df-3an 1032 df-tru 1477 df-fal 1480 df-ex 1695 df-nf 1700 df-sb 1867 df-eu 2461 df-mo 2462 df-clab 2596 df-cleq 2602 df-clel 2605 df-nfc 2739 df-ne 2781 df-nel 2782 df-ral 2900 df-rex 2901 df-reu 2902 df-rmo 2903 df-rab 2904 df-v 3174 df-sbc 3402 df-csb 3499 df-dif 3542 df-un 3544 df-in 3546 df-ss 3553 df-pss 3555 df-nul 3874 df-if 4036 df-pw 4109 df-sn 4125 df-pr 4127 df-tp 4129 df-op 4131 df-uni 4367 df-int 4405 df-iun 4451 df-iin 4452 df-disj 4548 df-br 4578 df-opab 4638 df-mpt 4639 df-tr 4675 df-eprel 4938 df-id 4942 df-po 4948 df-so 4949 df-fr 4986 df-se 4987 df-we 4988 df-xp 5033 df-rel 5034 df-cnv 5035 df-co 5036 df-dm 5037 df-rn 5038 df-res 5039 df-ima 5040 df-pred 5582 df-ord 5628 df-on 5629 df-lim 5630 df-suc 5631 df-iota 5753 df-fun 5791 df-fn 5792 df-f 5793 df-f1 5794 df-fo 5795 df-f1o 5796 df-fv 5797 df-isom 5798 df-riota 6488 df-ov 6529 df-oprab 6530 df-mpt2 6531 df-of 6772 df-rpss 6812 df-om 6935 df-1st 7036 df-2nd 7037 df-supp 7160 df-tpos 7216 df-wrecs 7271 df-recs 7332 df-rdg 7370 df-1o 7424 df-2o 7425 df-oadd 7428 df-omul 7429 df-er 7606 df-ec 7608 df-qs 7612 df-map 7723 df-pm 7724 df-ixp 7772 df-en 7819 df-dom 7820 df-sdom 7821 df-fin 7822 df-fsupp 8136 df-fi 8177 df-sup 8208 df-inf 8209 df-oi 8275 df-card 8625 df-acn 8628 df-cda 8850 df-pnf 9932 df-mnf 9933 df-xr 9934 df-ltxr 9935 df-le 9936 df-sub 10119 df-neg 10120 df-div 10536 df-nn 10870 df-2 10928 df-3 10929 df-4 10930 df-5 10931 df-6 10932 df-7 10933 df-8 10934 df-9 10935 df-n0 11142 df-z 11213 df-dec 11328 df-uz 11522 df-q 11623 df-rp 11667 df-xneg 11780 df-xadd 11781 df-xmul 11782 df-ioo 12008 df-ioc 12009 df-ico 12010 df-icc 12011 df-fz 12155 df-fzo 12292 df-fl 12412 df-mod 12488 df-seq 12621 df-exp 12680 df-fac 12880 df-bc 12909 df-hash 12937 df-word 13102 df-concat 13104 df-s1 13105 df-shft 13603 df-cj 13635 df-re 13636 df-im 13637 df-sqrt 13771 df-abs 13772 df-limsup 13998 df-clim 14015 df-rlim 14016 df-o1 14017 df-lo1 14018 df-sum 14213 df-ef 14585 df-e 14586 df-sin 14587 df-cos 14588 df-pi 14590 df-dvds 14770 df-gcd 15003 df-prm 15172 df-numer 15229 df-denom 15230 df-phi 15257 df-pc 15328 df-struct 15645 df-ndx 15646 df-slot 15647 df-base 15648 df-sets 15649 df-ress 15650 df-plusg 15729 df-mulr 15730 df-starv 15731 df-sca 15732 df-vsca 15733 df-ip 15734 df-tset 15735 df-ple 15736 df-ds 15739 df-unif 15740 df-hom 15741 df-cco 15742 df-rest 15854 df-topn 15855 df-0g 15873 df-gsum 15874 df-topgen 15875 df-pt 15876 df-prds 15879 df-xrs 15933 df-qtop 15938 df-imas 15939 df-qus 15940 df-xps 15941 df-mre 16017 df-mrc 16018 df-acs 16020 df-mgm 17013 df-sgrp 17055 df-mnd 17066 df-mhm 17106 df-submnd 17107 df-grp 17196 df-minusg 17197 df-sbg 17198 df-mulg 17312 df-subg 17362 df-nsg 17363 df-eqg 17364 df-ghm 17429 df-gim 17472 df-ga 17494 df-cntz 17521 df-oppg 17547 df-od 17719 df-gex 17720 df-pgp 17721 df-lsm 17822 df-pj1 17823 df-cmn 17966 df-abl 17967 df-cyg 18051 df-dprd 18165 df-dpj 18166 df-mgp 18261 df-ur 18273 df-ring 18320 df-cring 18321 df-oppr 18394 df-dvdsr 18412 df-unit 18413 df-invr 18443 df-dvr 18454 df-rnghom 18486 df-drng 18520 df-subrg 18549 df-lmod 18636 df-lss 18702 df-lsp 18741 df-sra 18941 df-rgmod 18942 df-lidl 18943 df-rsp 18944 df-2idl 19001 df-psmet 19507 df-xmet 19508 df-met 19509 df-bl 19510 df-mopn 19511 df-fbas 19512 df-fg 19513 df-cnfld 19516 df-zring 19586 df-zrh 19618 df-zn 19621 df-top 20468 df-bases 20469 df-topon 20470 df-topsp 20471 df-cld 20580 df-ntr 20581 df-cls 20582 df-nei 20659 df-lp 20697 df-perf 20698 df-cn 20788 df-cnp 20789 df-haus 20876 df-cmp 20947 df-tx 21122 df-hmeo 21315 df-fil 21407 df-fm 21499 df-flim 21500 df-flf 21501 df-xms 21882 df-ms 21883 df-tms 21884 df-cncf 22436 df-0p 23187 df-limc 23380 df-dv 23381 df-ply 23692 df-idp 23693 df-coe 23694 df-dgr 23695 df-quot 23794 df-log 24051 df-cxp 24052 df-em 24463 df-cht 24567 df-vma 24568 df-chp 24569 df-ppi 24570 df-mu 24571 df-dchr 24702

This theorem is referenced by: dchrmusum 24957

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