dchrisum - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > dchrisum		Structured version Visualization version GIF version

Theorem dchrisum 27656

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 14006	. . 3 ⊢ (0..^𝑁) ∈ Fin
2		fzofi 14006	. . . . . . 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 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → 𝑋 ∈ 𝐷)
10		elfzoelz 13683	. . . . . . . 8 ⊢ (𝑚 ∈ (0..^𝑢) → 𝑚 ∈ ℤ)
11	10	adantl 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → 𝑚 ∈ ℤ)
12	4, 5, 6, 7, 9, 11	dchrzrhcl 27409	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (0..^𝑢)) → (𝑋‘(𝐿‘𝑚)) ∈ ℂ)
13	3, 12	fsumcl 15780	. . . . 5 ⊢ (𝜑 → Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) ∈ ℂ)
14	13	abscld 15486	. . . 4 ⊢ (𝜑 → (abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
15	14	ralrimivw 3161	. . 3 ⊢ (𝜑 → ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ∈ ℝ)
16		fimaxre3 12156	. . 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 598	. 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 485	. . 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 485	. . 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 485	. . 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 485	. . 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 727	. . 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 1196	. . 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 485	. . 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 782	. . 3 ⊢ ((𝜑 ∧ (𝑟 ∈ ℝ ∧ ∀𝑢 ∈ (0..^𝑁)(abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟)) → 𝑟 ∈ ℝ)
35		simprr 784	. . . 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 6886	. . . . . . . . 9 ⊢ (𝑚 = 𝑛 → (𝑋‘(𝐿‘𝑚)) = (𝑋‘(𝐿‘𝑛)))
37	36	cbvsumv 15743	. . . . . . . 8 ⊢ Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) = Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛))
38		oveq2 7418	. . . . . . . . 9 ⊢ (𝑢 = 𝑖 → (0..^𝑢) = (0..^𝑖))
39	38	sumeq1d 15747	. . . . . . . 8 ⊢ (𝑢 = 𝑖 → Σ𝑛 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑛)) = Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛)))
40	37, 39	eqtrid 2810	. . . . . . 7 ⊢ (𝑢 = 𝑖 → Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚)) = Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛)))
41	40	fveq2d 6885	. . . . . 6 ⊢ (𝑢 = 𝑖 → (abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) = (abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))))
42	41	breq1d 5119	. . . . 5 ⊢ (𝑢 = 𝑖 → ((abs‘Σ𝑚 ∈ (0..^𝑢)(𝑋‘(𝐿‘𝑚))) ≤ 𝑟 ↔ (abs‘Σ𝑛 ∈ (0..^𝑖)(𝑋‘(𝐿‘𝑛))) ≤ 𝑟))
43	42	cbvralvw 3243	. . . 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 221	. . 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 27655	. 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 3172	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 400 ∧ w3a 1103 = wceq 1570 ∃wex 1809 ∈ wcel 2143 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 class class class wbr 5109 ↦ cmpt 5192 ‘cfv 6536 (class class class)co 7410 Fincfn 8939 ℝcr 11094 0cc0 11095 1c1 11096 + caddc 11098 · cmul 11100 +∞cpnf 11235 ≤ cle 11239 − cmin 11436 ℕcn 12228 ℤcz 12586 ℝ⁺crp 13011 [,)cico 13369 ..^cfzo 13678 ⌊cfl 13819 seqcseq 14033 abscabs 15281 ⇝ cli 15531 ⇝_𝑟 crli 15532 Σcsu 15733 Basecbs 17264 0_gc0g 17487 ℤRHomczrh 21649 ℤ/nℤczn 21652 DChrcdchr 27396

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5238 ax-sep 5257 ax-nul 5269 ax-pow 5336 ax-pr 5404 ax-un 7732 ax-inf2 9606 ax-cnex 11151 ax-resscn 11152 ax-1cn 11153 ax-icn 11154 ax-addcl 11155 ax-addrcl 11156 ax-mulcl 11157 ax-mulrcl 11158 ax-mulcom 11159 ax-addass 11160 ax-mulass 11161 ax-distr 11162 ax-i2m1 11163 ax-1ne0 11164 ax-1rid 11165 ax-rnegex 11166 ax-rrecex 11167 ax-cnre 11168 ax-pre-lttri 11169 ax-pre-lttrn 11170 ax-pre-ltadd 11171 ax-pre-mulgt0 11172 ax-pre-sup 11173 ax-addf 11174 ax-mulf 11175

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3745 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4488 df-pw 4564 df-sn 4590 df-pr 4592 df-tp 4594 df-op 4596 df-uni 4873 df-int 4913 df-iun 4958 df-br 5110 df-opab 5174 df-mpt 5193 df-tr 5219 df-id 5556 df-eprel 5561 df-po 5569 df-so 5570 df-fr 5614 df-se 5615 df-we 5616 df-xp 5667 df-rel 5668 df-cnv 5669 df-co 5670 df-dm 5671 df-rn 5672 df-res 5673 df-ima 5674 df-pred 6302 df-ord 6363 df-on 6364 df-lim 6365 df-suc 6366 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-isom 6545 df-riota 7367 df-ov 7413 df-oprab 7414 df-mpo 7415 df-of 7674 df-om 7859 df-1st 7982 df-2nd 7983 df-tpos 8218 df-frecs 8274 df-wrecs 8305 df-recs 8354 df-rdg 8393 df-1o 8449 df-oadd 8453 df-er 8690 df-ec 8692 df-qs 8696 df-map 8822 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9398 df-inf 9399 df-oi 9468 df-card 9921 df-pnf 11240 df-mnf 11241 df-xr 11242 df-ltxr 11243 df-le 11244 df-sub 11438 df-neg 11439 df-div 11867 df-nn 12229 df-2 12298 df-3 12299 df-4 12300 df-5 12301 df-6 12302 df-7 12303 df-8 12304 df-9 12305 df-n0 12500 df-xnn0 12573 df-z 12587 df-dec 12707 df-uz 12858 df-rp 13012 df-ico 13373 df-fz 13531 df-fzo 13679 df-fl 13821 df-mod 13899 df-seq 14034 df-exp 14094 df-hash 14363 df-cj 15146 df-re 15147 df-im 15148 df-sqrt 15282 df-abs 15283 df-limsup 15518 df-clim 15535 df-rlim 15536 df-sum 15734 df-dvds 16306 df-gcd 16548 df-phi 16820 df-struct 17202 df-sets 17219 df-slot 17237 df-ndx 17249 df-base 17265 df-ress 17286 df-plusg 17318 df-mulr 17319 df-starv 17320 df-sca 17321 df-vsca 17322 df-ip 17323 df-tset 17324 df-ple 17325 df-ds 17327 df-unif 17328 df-0g 17489 df-imas 17557 df-qus 17558 df-mgm 18693 df-sgrp 18772 df-mnd 18788 df-mhm 18836 df-grp 18998 df-minusg 18999 df-sbg 19000 df-mulg 19129 df-subg 19184 df-nsg 19185 df-eqg 19186 df-ghm 19279 df-cmn 19847 df-abl 19848 df-mgp 20212 df-rng 20226 df-ur 20259 df-ring 20312 df-cring 20313 df-oppr 20415 df-dvdsr 20435 df-unit 20436 df-invr 20466 df-rhm 20550 df-subrng 20645 df-subrg 20669 df-lmod 20983 df-lss 21053 df-lsp 21093 df-sra 21294 df-rgmod 21295 df-lidl 21332 df-rsp 21333 df-2idl 21389 df-cnfld 21523 df-zring 21597 df-zrh 21653 df-zn 21656 df-dchr 27397

This theorem is referenced by: dchrmusumlema 27657 dchrvmasumlema 27664 dchrisum0lema 27678

W3C validator