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Theorem dchrvmasum 27028

Description: The sum of the von Mangoldt function multiplied by a non-principal Dirichlet character, divided by 𝑛, is bounded. Equation 9.4.8 of [Shapiro], p. 376. (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 )

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

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

Proof of Theorem dchrvmasum Dummy variables 𝑦 𝑐 𝑡 𝑎 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rpvmasum.z	. . 3 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
2		rpvmasum.l	. . 3 ⊢ 𝐿 = (ℤRHom‘𝑍)
3		rpvmasum.a	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
4		dchrmusum.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
5		dchrmusum.d	. . 3 ⊢ 𝐷 = (Base‘𝐺)
6		dchrmusum.1	. . 3 ⊢ 1 = (0_g‘𝐺)
7		dchrmusum.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8		dchrmusum.n1	. . 3 ⊢ (𝜑 → 𝑋 ≠ 1 )
9		eqid 2733	. . 3 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dchrmusumlema 26996	. 2 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))
11	3	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑁 ∈ ℕ)
12	7	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ∈ 𝐷)
13	8	adantr 482	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑋 ≠ 1 )
14		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → 𝑐 ∈ (0[,)+∞))
15		simprrl 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡)
16		simprrr 781	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦))
17	1, 2, 11, 4, 5, 6, 12, 13, 9, 14, 15, 16	dchrvmasumlem 27026	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)))) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))
18	17	rexlimdvaa 3157	. . 3 ⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1)))
19	18	exlimdv 1937	. 2 ⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎)))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 / 𝑦)) → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1)))
20	10, 19	mpd 15	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 class class class wbr 5149 ↦ cmpt 5232 ‘cfv 6544 (class class class)co 7409 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 +∞cpnf 11245 ≤ cle 11249 − cmin 11444 / cdiv 11871 ℕcn 12212 ℝ⁺crp 12974 [,)cico 13326 ...cfz 13484 ⌊cfl 13755 seqcseq 13966 abscabs 15181 ⇝ cli 15428 𝑂(1)co1 15430 Σcsu 15632 Basecbs 17144 0_gc0g 17385 ℤRHomczrh 21049 ℤ/nℤczn 21052 Λcvma 26596 DChrcdchr 26735

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-rpss 7713 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-tpos 8211 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-omul 8471 df-er 8703 df-ec 8705 df-qs 8709 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-dju 9896 df-card 9934 df-acn 9937 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-word 14465 df-concat 14521 df-s1 14546 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-o1 15434 df-lo1 15435 df-sum 15633 df-ef 16011 df-e 16012 df-sin 16013 df-cos 16014 df-tan 16015 df-pi 16016 df-dvds 16198 df-gcd 16436 df-prm 16609 df-numer 16671 df-denom 16672 df-phi 16699 df-pc 16770 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-xrs 17448 df-qtop 17453 df-imas 17454 df-qus 17455 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-nsg 19004 df-eqg 19005 df-ghm 19090 df-gim 19133 df-ga 19154 df-cntz 19181 df-oppg 19210 df-od 19396 df-gex 19397 df-pgp 19398 df-lsm 19504 df-pj1 19505 df-cmn 19650 df-abl 19651 df-cyg 19746 df-dprd 19865 df-dpj 19866 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-oppr 20150 df-dvdsr 20171 df-unit 20172 df-invr 20202 df-dvr 20215 df-rnghom 20251 df-subrg 20317 df-drng 20359 df-lmod 20473 df-lss 20543 df-lsp 20583 df-sra 20785 df-rgmod 20786 df-lidl 20787 df-rsp 20788 df-2idl 20857 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-zring 21018 df-zrh 21053 df-zn 21056 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-cmp 22891 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-xms 23826 df-ms 23827 df-tms 23828 df-cncf 24394 df-0p 25187 df-limc 25383 df-dv 25384 df-ply 25702 df-idp 25703 df-coe 25704 df-dgr 25705 df-quot 25804 df-ulm 25889 df-log 26065 df-cxp 26066 df-atan 26372 df-em 26497 df-cht 26601 df-vma 26602 df-chp 26603 df-ppi 26604 df-mu 26605 df-dchr 26736

This theorem is referenced by: (None)

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