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Theorem dchrisum 27384

Description: If 𝑛 ∈ [𝑀, +∞) ↦ 𝐴(𝑛) is a positive decreasing function approaching zero, then the infinite sum Σ𝑛, 𝑋(𝑛)𝐴(𝑛) is convergent, with the partial sum Σ𝑛 ≤ 𝑥, 𝑋(𝑛)𝐴(𝑛) within 𝑂(𝐴(𝑀)) of the limit 𝑇. Lemma 9.4.1 of [Shapiro], p. 377. (Contributed by Mario Carneiro, 2-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 )
dchrisum.2	⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵)
dchrisum.3	⊢ (𝜑 → 𝑀 ∈ ℕ)
dchrisum.4	⊢ ((𝜑 ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
dchrisum.5	⊢ ((𝜑 ∧ (𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴)
dchrisum.6	⊢ (𝜑 → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝_𝑟 0)
dchrisum.7	⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴))

**Assertion**
Ref	Expression
dchrisum	⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))

Distinct variable groups: 𝑥,𝑛,𝑐,𝑡, 1 𝐹,𝑐,𝑛,𝑡,𝑥 𝐴,𝑐,𝑡,𝑥 𝑁,𝑐,𝑛,𝑡,𝑥 𝜑,𝑐,𝑛,𝑡,𝑥 𝐵,𝑐,𝑛 𝑛,𝑍,𝑥 𝐷,𝑐,𝑛,𝑡,𝑥 𝐿,𝑐,𝑛,𝑡,𝑥 𝑀,𝑐,𝑛,𝑥 𝑋,𝑐,𝑛,𝑡,𝑥 Allowed substitution hints: 𝐴(𝑛) 𝐵(𝑥,𝑡) 𝐺(𝑥,𝑡,𝑛,𝑐) 𝑀(𝑡) 𝑍(𝑡,𝑐)

Proof of Theorem dchrisum Dummy variables 𝑚 𝑢 𝑖 𝑟 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		fzofi 13869	. . 3 ⊢ (0..^𝑁) ∈ Fin
2		fzofi 13869	. . . . . . 7 ⊢ (0..^𝑢) ∈ Fin
3	2	a1i 11	. . . . . 6 ⊢ (𝜑 → (0..^𝑢) ∈ Fin)
4		rpvmasum.g	. . . . . . 7 ⊢ 𝐺 = (DChr‘𝑁)
5		rpvmasum.z	. . . . . . 7 ⊢ 𝑍 = (ℤ/nℤ‘𝑁)
6		rpvmasum.d	. . . . . . 7 ⊢ 𝐷 = (Base‘𝐺)
7		rpvmasum.l	. . . . . . 7 ⊢ 𝐿 = (ℤRHom‘𝑍)
8		dchrisum.b	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
9	8	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → 𝑋 ∈ 𝐷)
10		elfzoelz 13550	. . . . . . . 8 ⊢ (𝑚 ∈ (0..^𝑢) → 𝑚 ∈ ℤ)
11	10	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → 𝑚 ∈ ℤ)
12	4, 5, 6, 7, 9, 11	dchrzrhcl 27137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
13	3, 12	fsumcl 15627	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) ∈ ℂ)
14	13	abscld 15333	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
15	14	ralrimivw 3125	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
16		fimaxre3 12059	. . 3 ⊢ (((0..^𝑁) ∈ Fin ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ) → ∃𝑟 ∈ ℝ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)
17	1, 15, 16	sylancr 587	. 2 ⊢ (𝜑 → ∃𝑟 ∈ ℝ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)
18		rpvmasum.a	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ)
19	18	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑁 ∈ ℕ)
20		rpvmasum.1	. . 3 ⊢ 1 = (0_g‘𝐺)
21	8	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑋 ∈ 𝐷)
22		dchrisum.n1	. . . 4 ⊢ (𝜑 → 𝑋 ≠ 1 )
23	22	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑋 ≠ 1 )
24		dchrisum.2	. . 3 ⊢ (𝑛 = 𝑥 → 𝐴 = 𝐵)
25		dchrisum.3	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℕ)
26	25	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑀 ∈ ℕ)
27		dchrisum.4	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
28	27	adantlr 715	. . 3 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) ∧ 𝑛 ∈ ℝ⁺) → 𝐴 ∈ ℝ)
29		dchrisum.5	. . . 4 ⊢ ((𝜑 ∧ (𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴)
30	29	3adant1r 1178	. . 3 ⊢ (((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) ∧ (𝑛 ∈ ℝ⁺ ∧ 𝑥 ∈ ℝ⁺) ∧ (𝑀 ≤ 𝑛 ∧ 𝑛 ≤ 𝑥)) → 𝐵 ≤ 𝐴)
31		dchrisum.6	. . . 4 ⊢ (𝜑 → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝_𝑟 0)
32	31	adantr 480	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → (𝑛 ∈ ℝ⁺ ↦ 𝐴) ⇝_𝑟 0)
33		dchrisum.7	. . 3 ⊢ 𝐹 = (𝑛 ∈ ℕ ↦ ((𝑋‘(𝐿‘𝑛)) · 𝐴))
34		simprl 770	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑟 ∈ ℝ)
35		simprr 772	. . . 4 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)
36		2fveq3 6821	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑋‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑛)))
37	36	cbvsumv 15590	. . . . . . . 8 ⊢ Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) = Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛))
38		oveq2 7348	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (0..^𝑢) = (0..^𝑖))
39	38	sumeq1d 15594	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛)) = Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛)))
40	37, 39	eqtrid 2776	. . . . . . 7 ⊢ (𝑢 = 𝑖 → Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) = Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛)))
41	40	fveq2d 6820	. . . . . 6 ⊢ (𝑢 = 𝑖 → (abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) = (abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))))
42	41	breq1d 5098	. . . . 5 ⊢ (𝑢 = 𝑖 → ((abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟 ↔ (abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))) ≤ 𝑟))
43	42	cbvralvw 3207	. . . 4 ⊢ (∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟 ↔ ∀𝑖 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))) ≤ 𝑟)
44	35, 43	sylib 218	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → ∀𝑖 ∈ (0..^𝑁)(abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))) ≤ 𝑟)
45	5, 7, 19, 4, 6, 20, 21, 23, 24, 26, 28, 30, 32, 33, 34, 44	dchrisumlem3 27383	. 2 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))
46	17, 45	rexlimddv 3136	1 ⊢ (𝜑 → ∃𝑡∃𝑐 ∈ (0[,)+∞)(seq1( + , 𝐹) ⇝ 𝑡 ∧ ∀𝑥 ∈ (𝑀[,)+∞)(abs‘((seq1( + , 𝐹)‘(⌊‘𝑥)) − 𝑡)) ≤ (𝑐 · 𝐵)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 class class class wbr 5088 ↦ cmpt 5169 ‘cfv 6476 (class class class)co 7340 Fincfn 8863 ℝcr 10996 0cc0 10997 1c1 10998 + caddc 11000 · cmul 11002 +∞cpnf 11134 ≤ cle 11138 − cmin 11335 ℕcn 12116 ℤcz 12459 ℝ⁺crp 12881 [,)cico 13238 ..^cfzo 13545 ⌊cfl 13682 seqcseq 13896 abscabs 15128 ⇝ cli 15378 ⇝_𝑟 crli 15379 Σcsu 15580 Basecbs 17107 0_gc0g 17330 ℤRHomczrh 21390 ℤ/nℤczn 21393 DChrcdchr 27124

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5214 ax-sep 5231 ax-nul 5241 ax-pow 5300 ax-pr 5367 ax-un 7662 ax-inf2 9525 ax-cnex 11053 ax-resscn 11054 ax-1cn 11055 ax-icn 11056 ax-addcl 11057 ax-addrcl 11058 ax-mulcl 11059 ax-mulrcl 11060 ax-mulcom 11061 ax-addass 11062 ax-mulass 11063 ax-distr 11064 ax-i2m1 11065 ax-1ne0 11066 ax-1rid 11067 ax-rnegex 11068 ax-rrecex 11069 ax-cnre 11070 ax-pre-lttri 11071 ax-pre-lttrn 11072 ax-pre-ltadd 11073 ax-pre-mulgt0 11074 ax-pre-sup 11075 ax-addf 11076 ax-mulf 11077

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3393 df-v 3435 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4281 df-if 4473 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4857 df-int 4895 df-iun 4940 df-br 5089 df-opab 5151 df-mpt 5170 df-tr 5196 df-id 5508 df-eprel 5513 df-po 5521 df-so 5522 df-fr 5566 df-se 5567 df-we 5568 df-xp 5619 df-rel 5620 df-cnv 5621 df-co 5622 df-dm 5623 df-rn 5624 df-res 5625 df-ima 5626 df-pred 6243 df-ord 6304 df-on 6305 df-lim 6306 df-suc 6307 df-iota 6432 df-fun 6478 df-fn 6479 df-f 6480 df-f1 6481 df-fo 6482 df-f1o 6483 df-fv 6484 df-isom 6485 df-riota 7297 df-ov 7343 df-oprab 7344 df-mpo 7345 df-of 7604 df-om 7791 df-1st 7915 df-2nd 7916 df-tpos 8150 df-frecs 8205 df-wrecs 8236 df-recs 8285 df-rdg 8323 df-1o 8379 df-oadd 8383 df-er 8616 df-ec 8618 df-qs 8622 df-map 8746 df-pm 8747 df-en 8864 df-dom 8865 df-sdom 8866 df-fin 8867 df-sup 9320 df-inf 9321 df-oi 9390 df-card 9823 df-pnf 11139 df-mnf 11140 df-xr 11141 df-ltxr 11142 df-le 11143 df-sub 11337 df-neg 11338 df-div 11766 df-nn 12117 df-2 12179 df-3 12180 df-4 12181 df-5 12182 df-6 12183 df-7 12184 df-8 12185 df-9 12186 df-n0 12373 df-xnn0 12446 df-z 12460 df-dec 12580 df-uz 12724 df-rp 12882 df-ico 13242 df-fz 13399 df-fzo 13546 df-fl 13684 df-mod 13762 df-seq 13897 df-exp 13957 df-hash 14226 df-cj 14993 df-re 14994 df-im 14995 df-sqrt 15129 df-abs 15130 df-limsup 15365 df-clim 15382 df-rlim 15383 df-sum 15581 df-dvds 16151 df-gcd 16393 df-phi 16664 df-struct 17045 df-sets 17062 df-slot 17080 df-ndx 17092 df-base 17108 df-ress 17129 df-plusg 17161 df-mulr 17162 df-starv 17163 df-sca 17164 df-vsca 17165 df-ip 17166 df-tset 17167 df-ple 17168 df-ds 17170 df-unif 17171 df-0g 17332 df-imas 17399 df-qus 17400 df-mgm 18501 df-sgrp 18580 df-mnd 18596 df-mhm 18644 df-grp 18802 df-minusg 18803 df-sbg 18804 df-mulg 18934 df-subg 18989 df-nsg 18990 df-eqg 18991 df-ghm 19079 df-cmn 19648 df-abl 19649 df-mgp 20013 df-rng 20025 df-ur 20054 df-ring 20107 df-cring 20108 df-oppr 20209 df-dvdsr 20229 df-unit 20230 df-invr 20260 df-rhm 20344 df-subrng 20415 df-subrg 20439 df-lmod 20749 df-lss 20819 df-lsp 20859 df-sra 21061 df-rgmod 21062 df-lidl 21099 df-rsp 21100 df-2idl 21141 df-cnfld 21246 df-zring 21338 df-zrh 21394 df-zn 21397 df-dchr 27125

This theorem is referenced by: dchrmusumlema 27385 dchrvmasumlema 27392 dchrisum0lema 27406

W3C validator