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Theorem dchrvmasumif 27410

Description: An asymptotic approximation for the sum of 𝑋(𝑛)Λ(𝑛) / 𝑛 conditional on the value of the infinite sum 𝑆. (We will later show that the case 𝑆 = 0 is impossible, and hence establish dchrvmasum 27432.) (Contributed by Mario Carneiro, 5-May-2016.)

**Hypotheses**
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 = (0_g‘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))

**Assertion**
Ref	Expression
dchrvmasumif	⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 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 𝑆,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) 𝑆(𝑎) 1 (𝑎) 𝐹(𝑎) 𝐺(𝑥,𝑦,𝑛,𝑎) 𝑁(𝑎) 𝑍(𝑎)

Proof of Theorem dchrvmasumif Dummy variables 𝑐 𝑡 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		rpvmasum.g	. . 3 ⊢ 𝐺 = (DChr‘𝑁)
5		rpvmasum.d	. . 3 ⊢ 𝐷 = (Base‘𝐺)
6		rpvmasum.1	. . 3 ⊢ 1 = (0_g‘𝐺)
7		dchrisum.b	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
8		dchrisum.n1	. . 3 ⊢ (𝜑 → 𝑋 ≠ 1 )
9		eqid 2727	. . 3 ⊢ (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))) = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))
10	1, 2, 3, 4, 5, 6, 7, 8, 9	dchrvmasumlema 27407	. 2 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))
11	3	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → 𝑁 ∈ ℕ)
12	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → 𝑋 ∈ 𝐷)
13	8	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → 𝑋 ≠ 1 )
14		dchrvmasumif.f	. . . . 5 ⊢ 𝐹 = (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) / 𝑎))
15		dchrvmasumif.c	. . . . . 6 ⊢ (𝜑 → 𝐶 ∈ (0[,)+∞))
16	15	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → 𝐶 ∈ (0[,)+∞))
17		dchrvmasumif.s	. . . . . 6 ⊢ (𝜑 → seq1( + , 𝐹) ⇝ 𝑆)
18	17	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → seq1( + , 𝐹) ⇝ 𝑆)
19		dchrvmasumif.1	. . . . . 6 ⊢ (𝜑 → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
20	19	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → ∀𝑦 ∈ (1[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑦)) − 𝑆)) ≤ (𝐶 / 𝑦))
21		simprl 770	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → 𝑐 ∈ (0[,)+∞))
22		simprrl 780	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡)
23		simprrr 781	. . . . 5 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦)))
24	1, 2, 11, 4, 5, 6, 12, 13, 14, 16, 18, 20, 9, 21, 22, 23	dchrvmasumiflem2 27409	. . . 4 ⊢ ((𝜑 ∧ (𝑐 ∈ (0[,)+∞) ∧ (seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))))) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))
25	24	rexlimdvaa 3151	. . 3 ⊢ (𝜑 → (∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1)))
26	25	exlimdv 1929	. 2 ⊢ (𝜑 → (∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎)))) ⇝ 𝑡 ∧ ∀𝑦 ∈ (3[,)+∞)(abs‘((seq1( + , (𝑎 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑎)) · ((log‘𝑎) / 𝑎))))‘(⌊‘𝑦)) − 𝑡)) ≤ (𝑐 · ((log‘𝑦) / 𝑦))) → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1)))
27	10, 26	mpd 15	1 ⊢ (𝜑 → (𝑥 ∈ ℝ⁺ ↦ (Σ𝑛 ∈ (1...(⌊‘𝑥))((𝑋‘(𝐿‘𝑛)) · ((Λ‘𝑛) / 𝑛)) + if(𝑆 = 0, (log‘𝑥), 0))) ∈ 𝑂(1))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2935 ∀wral 3056 ∃wrex 3065 ifcif 4524 class class class wbr 5142 ↦ cmpt 5225 ‘cfv 6542 (class class class)co 7414 0cc0 11124 1c1 11125 + caddc 11127 · cmul 11129 +∞cpnf 11261 ≤ cle 11265 − cmin 11460 / cdiv 11887 ℕcn 12228 3c3 12284 ℝ⁺crp 12992 [,)cico 13344 ...cfz 13502 ⌊cfl 13773 seqcseq 13984 abscabs 15199 ⇝ cli 15446 𝑂(1)co1 15448 Σcsu 15650 Basecbs 17165 0_gc0g 17406 ℤRHomczrh 21405 ℤ/nℤczn 21408 logclog 26462 Λcvma 26998 DChrcdchr 27139

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7732 ax-inf2 9650 ax-cnex 11180 ax-resscn 11181 ax-1cn 11182 ax-icn 11183 ax-addcl 11184 ax-addrcl 11185 ax-mulcl 11186 ax-mulrcl 11187 ax-mulcom 11188 ax-addass 11189 ax-mulass 11190 ax-distr 11191 ax-i2m1 11192 ax-1ne0 11193 ax-1rid 11194 ax-rnegex 11195 ax-rrecex 11196 ax-cnre 11197 ax-pre-lttri 11198 ax-pre-lttrn 11199 ax-pre-ltadd 11200 ax-pre-mulgt0 11201 ax-pre-sup 11202 ax-addf 11203 ax-mulf 11204

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-disj 5108 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-isom 6551 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-of 7677 df-om 7863 df-1st 7985 df-2nd 7986 df-supp 8158 df-tpos 8223 df-frecs 8278 df-wrecs 8309 df-recs 8383 df-rdg 8422 df-1o 8478 df-2o 8479 df-oadd 8482 df-omul 8483 df-er 8716 df-ec 8718 df-qs 8722 df-map 8836 df-pm 8837 df-ixp 8906 df-en 8954 df-dom 8955 df-sdom 8956 df-fin 8957 df-fsupp 9376 df-fi 9420 df-sup 9451 df-inf 9452 df-oi 9519 df-dju 9910 df-card 9948 df-acn 9951 df-pnf 11266 df-mnf 11267 df-xr 11268 df-ltxr 11269 df-le 11270 df-sub 11462 df-neg 11463 df-div 11888 df-nn 12229 df-2 12291 df-3 12292 df-4 12293 df-5 12294 df-6 12295 df-7 12296 df-8 12297 df-9 12298 df-n0 12489 df-xnn0 12561 df-z 12575 df-dec 12694 df-uz 12839 df-q 12949 df-rp 12993 df-xneg 13110 df-xadd 13111 df-xmul 13112 df-ioo 13346 df-ioc 13347 df-ico 13348 df-icc 13349 df-fz 13503 df-fzo 13646 df-fl 13775 df-mod 13853 df-seq 13985 df-exp 14045 df-fac 14251 df-bc 14280 df-hash 14308 df-shft 15032 df-cj 15064 df-re 15065 df-im 15066 df-sqrt 15200 df-abs 15201 df-limsup 15433 df-clim 15450 df-rlim 15451 df-o1 15452 df-lo1 15453 df-sum 15651 df-ef 16029 df-e 16030 df-sin 16031 df-cos 16032 df-tan 16033 df-pi 16034 df-dvds 16217 df-gcd 16455 df-prm 16628 df-phi 16720 df-pc 16791 df-struct 17101 df-sets 17118 df-slot 17136 df-ndx 17148 df-base 17166 df-ress 17195 df-plusg 17231 df-mulr 17232 df-starv 17233 df-sca 17234 df-vsca 17235 df-ip 17236 df-tset 17237 df-ple 17238 df-ds 17240 df-unif 17241 df-hom 17242 df-cco 17243 df-rest 17389 df-topn 17390 df-0g 17408 df-gsum 17409 df-topgen 17410 df-pt 17411 df-prds 17414 df-xrs 17469 df-qtop 17474 df-imas 17475 df-qus 17476 df-xps 17477 df-mre 17551 df-mrc 17552 df-acs 17554 df-mgm 18585 df-sgrp 18664 df-mnd 18680 df-mhm 18725 df-submnd 18726 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19008 df-subg 19062 df-nsg 19063 df-eqg 19064 df-ghm 19152 df-cntz 19252 df-od 19467 df-cmn 19721 df-abl 19722 df-mgp 20059 df-rng 20077 df-ur 20106 df-ring 20159 df-cring 20160 df-oppr 20255 df-dvdsr 20278 df-unit 20279 df-invr 20309 df-dvr 20322 df-rhm 20393 df-subrng 20465 df-subrg 20490 df-drng 20608 df-lmod 20727 df-lss 20798 df-lsp 20838 df-sra 21040 df-rgmod 21041 df-lidl 21086 df-rsp 21087 df-2idl 21126 df-psmet 21251 df-xmet 21252 df-met 21253 df-bl 21254 df-mopn 21255 df-fbas 21256 df-fg 21257 df-cnfld 21260 df-zring 21353 df-zrh 21409 df-zn 21412 df-top 22770 df-topon 22787 df-topsp 22809 df-bases 22823 df-cld 22897 df-ntr 22898 df-cls 22899 df-nei 22976 df-lp 23014 df-perf 23015 df-cn 23105 df-cnp 23106 df-haus 23193 df-cmp 23265 df-tx 23440 df-hmeo 23633 df-fil 23724 df-fm 23816 df-flim 23817 df-flf 23818 df-xms 24200 df-ms 24201 df-tms 24202 df-cncf 24772 df-limc 25769 df-dv 25770 df-ulm 26287 df-log 26464 df-cxp 26465 df-atan 26773 df-em 26899 df-vma 27004 df-mu 27007 df-dchr 27140

This theorem is referenced by: rpvmasum2 27419 dchrvmasumlem 27430

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