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Theorem dchrisum 27473

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 13927	. . 3 ⊢ (0..^𝑁) ∈ Fin
2		fzofi 13927	. . . . . . 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 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → 𝑋 ∈ 𝐷)
10		elfzoelz 13604	. . . . . . . 8 ⊢ (𝑚 ∈ (0..^𝑢) → 𝑚 ∈ ℤ)
11	10	adantl 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → 𝑚 ∈ ℤ)
12	4, 5, 6, 7, 9, 11	dchrzrhcl 27226	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
13	3, 12	fsumcl 15686	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) ∈ ℂ)
14	13	abscld 15392	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
15	14	ralrimivw 3135	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
16		fimaxre3 12093	. . 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 593	. 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 481	. . 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 481	. . 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 481	. . 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 481	. . 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 721	. . 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 1184	. . 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 481	. . 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 776	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑟 ∈ ℝ)
35		simprr 778	. . . 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 6832	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑋‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑛)))
37	36	cbvsumv 15649	. . . . . . . 8 ⊢ Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) = Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛))
38		oveq2 7364	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (0..^𝑢) = (0..^𝑖))
39	38	sumeq1d 15653	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛)) = Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛)))
40	37, 39	eqtrid 2786	. . . . . . 7 ⊢ (𝑢 = 𝑖 → Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) = Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛)))
41	40	fveq2d 6831	. . . . . 6 ⊢ (𝑢 = 𝑖 → (abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) = (abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))))
42	41	breq1d 5082	. . . . 5 ⊢ (𝑢 = 𝑖 → ((abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟 ↔ (abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))) ≤ 𝑟))
43	42	cbvralvw 3217	. . . 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 219	. . 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 27472	. 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 3146	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 396 ∧ w3a 1092 = wceq 1547 ∃wex 1786 ∈ wcel 2119 ≠ wne 2934 ∀wral 3053 ∃wrex 3063 class class class wbr 5072 ↦ cmpt 5153 ‘cfv 6485 (class class class)co 7356 Fincfn 8883 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 +∞cpnf 11167 ≤ cle 11171 − cmin 11368 ℕcn 12165 ℤcz 12515 ℝ⁺crp 12933 [,)cico 13291 ..^cfzo 13599 ⌊cfl 13740 seqcseq 13954 abscabs 15187 ⇝ cli 15437 ⇝_𝑟 crli 15438 Σcsu 15639 Basecbs 17170 0_gc0g 17393 ℤRHomczrh 21474 ℤ/nℤczn 21477 DChrcdchr 27213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-rep 5199 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-inf2 9553 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106 ax-pre-sup 11107 ax-addf 11108 ax-mulf 11109

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-rmo 3344 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-tp 4560 df-op 4562 df-uni 4839 df-int 4878 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-se 5572 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-isom 6494 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-of 7620 df-om 7807 df-1st 7931 df-2nd 7932 df-tpos 8166 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-1o 8395 df-oadd 8399 df-er 8633 df-ec 8635 df-qs 8639 df-map 8765 df-pm 8766 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9854 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-div 11799 df-nn 12166 df-2 12235 df-3 12236 df-4 12237 df-5 12238 df-6 12239 df-7 12240 df-8 12241 df-9 12242 df-n0 12429 df-xnn0 12502 df-z 12516 df-dec 12636 df-uz 12780 df-rp 12934 df-ico 13295 df-fz 13453 df-fzo 13600 df-fl 13742 df-mod 13820 df-seq 13955 df-exp 14015 df-hash 14284 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15424 df-clim 15441 df-rlim 15442 df-sum 15640 df-dvds 16213 df-gcd 16455 df-phi 16727 df-struct 17108 df-sets 17125 df-slot 17143 df-ndx 17155 df-base 17171 df-ress 17192 df-plusg 17224 df-mulr 17225 df-starv 17226 df-sca 17227 df-vsca 17228 df-ip 17229 df-tset 17230 df-ple 17231 df-ds 17233 df-unif 17234 df-0g 17395 df-imas 17463 df-qus 17464 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18742 df-grp 18903 df-minusg 18904 df-sbg 18905 df-mulg 19035 df-subg 19090 df-nsg 19091 df-eqg 19092 df-ghm 19179 df-cmn 19748 df-abl 19749 df-mgp 20113 df-rng 20125 df-ur 20154 df-ring 20207 df-cring 20208 df-oppr 20308 df-dvdsr 20328 df-unit 20329 df-invr 20359 df-rhm 20443 df-subrng 20518 df-subrg 20542 df-lmod 20852 df-lss 20922 df-lsp 20962 df-sra 21163 df-rgmod 21164 df-lidl 21201 df-rsp 21202 df-2idl 21243 df-cnfld 21348 df-zring 21422 df-zrh 21478 df-zn 21481 df-dchr 27214

This theorem is referenced by: dchrmusumlema 27474 dchrvmasumlema 27481 dchrisum0lema 27495

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