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Theorem dchrmusumlem 25410

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 12966	. . . . . . 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 764	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑋 ∈ 𝐷)
8		elfzelz 12535	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℤ)
9	8	adantl 473	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℤ)
10	2, 3, 4, 5, 7, 9	dchrzrhcl 25169	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (𝑋‘(𝐿‘𝑛)) ∈ ℂ)
11		elfznn 12563	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ (1...(⌊‘𝑥)) → 𝑛 ∈ ℕ)
12	11	adantl 473	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → 𝑛 ∈ ℕ)
13		mucl 25066	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ ℕ → (μ‘𝑛) ∈ ℤ)
14	12, 13	syl 17	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℤ)
15	14	zred 11674	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → (μ‘𝑛) ∈ ℝ)
16	15, 12	nndivred 11261	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℝ)
17	16	recnd 10260	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((μ‘𝑛) / 𝑛) ∈ ℂ)
18	10, 17	mulcld 10252	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑥 ∈ ℝ⁺) ∧ 𝑛 ∈ (1...(⌊‘𝑥))) → ((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ)
19	1, 18	fsumcl 14663	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) ∈ ℂ)
20		dchrmusum.t	. . . . . . . 8 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑇)
21		climcl 14429	. . . . . . . 8 ⊢ (seq1( + , 𝐹) ⇝ 𝑇 → 𝑇 ∈ ℂ)
22	20, 21	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇 ∈ ℂ)
23	22	adantr 472	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ∈ ℂ)
24	19, 23	mulcld 10252	. . . . 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 25409	. . . . . 6 ⊢ (𝜑 → 𝑇 ≠ 0)
32	31	adantr 472	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → 𝑇 ≠ 0)
33	24, 23, 32	divrecd 10996	. . . 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 10999	. . . 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 2796	. . 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 4896	. 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 10986	. . . 4 ⊢ (𝜑 → (1 / 𝑇) ∈ ℂ)
38	37	adantr 472	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ⁺) → (1 / 𝑇) ∈ ℂ)
39	3, 5, 25, 2, 4, 26, 6, 27, 28, 29, 20, 30	dchrmusum2 25382	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇)) ∈ 𝑂(1))
40		rpssre 12036	. . . 4 ⊢ ℝ⁺ ⊆ ℝ
41		o1const 14549	. . . 4 ⊢ ((ℝ⁺ ⊆ ℝ ∧ (1 / 𝑇) ∈ ℂ) → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑇)) ∈ 𝑂(1))
42	40, 37, 41	sylancr 698	. . 3 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (1 / 𝑇)) ∈ 𝑂(1))
43	24, 38, 39, 42	o1mul2 14554	. 2 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ ((Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((μ‘𝑛) / 𝑛)) · 𝑇) · (1 / 𝑇))) ∈ 𝑂(1))
44	36, 43	eqeltrrd 2840	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 383 = wceq 1632 ∈ wcel 2139 ≠ wne 2932 ∀wral 3050 ⊆ wss 3715 class class class wbr 4804 ↦ cmpt 4881 ‘cfv 6049 (class class class)co 6813 ℂcc 10126 ℝcr 10127 0cc0 10128 1c1 10129 + caddc 10131 · cmul 10133 +∞cpnf 10263 ≤ cle 10267 − cmin 10458 / cdiv 10876 ℕcn 11212 ℤcz 11569 ℝ⁺crp 12025 [,)cico 12370 ...cfz 12519 ⌊cfl 12785 seqcseq 12995 abscabs 14173 ⇝ cli 14414 𝑂(1)co1 14416 Σcsu 14615 Basecbs 16059 0_gc0g 16302 ℤRHomczrh 20050 ℤ/nℤczn 20053 μcmu 25020 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-iin 4675 df-disj 4773 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-of 7062 df-rpss 7102 df-om 7231 df-1st 7333 df-2nd 7334 df-supp 7464 df-tpos 7521 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-1o 7729 df-2o 7730 df-oadd 7733 df-omul 7734 df-er 7911 df-ec 7913 df-qs 7917 df-map 8025 df-pm 8026 df-ixp 8075 df-en 8122 df-dom 8123 df-sdom 8124 df-fin 8125 df-fsupp 8441 df-fi 8482 df-sup 8513 df-inf 8514 df-oi 8580 df-card 8955 df-acn 8958 df-cda 9182 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-xnn0 11556 df-z 11570 df-dec 11686 df-uz 11880 df-q 11982 df-rp 12026 df-xneg 12139 df-xadd 12140 df-xmul 12141 df-ioo 12372 df-ioc 12373 df-ico 12374 df-icc 12375 df-fz 12520 df-fzo 12660 df-fl 12787 df-mod 12863 df-seq 12996 df-exp 13055 df-fac 13255 df-bc 13284 df-hash 13312 df-word 13485 df-concat 13487 df-s1 13488 df-shft 14006 df-cj 14038 df-re 14039 df-im 14040 df-sqrt 14174 df-abs 14175 df-limsup 14401 df-clim 14418 df-rlim 14419 df-o1 14420 df-lo1 14421 df-sum 14616 df-ef 14997 df-e 14998 df-sin 14999 df-cos 15000 df-pi 15002 df-dvds 15183 df-gcd 15419 df-prm 15588 df-numer 15645 df-denom 15646 df-phi 15673 df-pc 15744 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-hom 16168 df-cco 16169 df-rest 16285 df-topn 16286 df-0g 16304 df-gsum 16305 df-topgen 16306 df-pt 16307 df-prds 16310 df-xrs 16364 df-qtop 16369 df-imas 16370 df-qus 16371 df-xps 16372 df-mre 16448 df-mrc 16449 df-acs 16451 df-mgm 17443 df-sgrp 17485 df-mnd 17496 df-mhm 17536 df-submnd 17537 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-gim 17902 df-ga 17923 df-cntz 17950 df-oppg 17976 df-od 18148 df-gex 18149 df-pgp 18150 df-lsm 18251 df-pj1 18252 df-cmn 18395 df-abl 18396 df-cyg 18480 df-dprd 18594 df-dpj 18595 df-mgp 18690 df-ur 18702 df-ring 18749 df-cring 18750 df-oppr 18823 df-dvdsr 18841 df-unit 18842 df-invr 18872 df-dvr 18883 df-rnghom 18917 df-drng 18951 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-psmet 19940 df-xmet 19941 df-met 19942 df-bl 19943 df-mopn 19944 df-fbas 19945 df-fg 19946 df-cnfld 19949 df-zring 20021 df-zrh 20054 df-zn 20057 df-top 20901 df-topon 20918 df-topsp 20939 df-bases 20952 df-cld 21025 df-ntr 21026 df-cls 21027 df-nei 21104 df-lp 21142 df-perf 21143 df-cn 21233 df-cnp 21234 df-haus 21321 df-cmp 21392 df-tx 21567 df-hmeo 21760 df-fil 21851 df-fm 21943 df-flim 21944 df-flf 21945 df-xms 22326 df-ms 22327 df-tms 22328 df-cncf 22882 df-0p 23636 df-limc 23829 df-dv 23830 df-ply 24143 df-idp 24144 df-coe 24145 df-dgr 24146 df-quot 24245 df-log 24502 df-cxp 24503 df-em 24918 df-cht 25022 df-vma 25023 df-chp 25024 df-ppi 25025 df-mu 25026 df-dchr 25157

This theorem is referenced by: dchrmusum 25412

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