esumpmono - Mathbox for Thierry Arnoux

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Theorem esumpmono 33065

Description: The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.)

**Hypotheses**
Ref	Expression
esumpmono.1	⊢ (𝜑 → 𝑀 ∈ ℕ)
esumpmono.2	⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
esumpmono.3	⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))

**Assertion**
Ref	Expression
esumpmono	⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑁)𝐴)

Distinct variable groups: 𝑘,𝑀 𝑘,𝑁 𝜑,𝑘 Allowed substitution hint: 𝐴(𝑘)

**Proof of Theorem esumpmono**
Step	Hyp	Ref	Expression
1		iccssxr 13403	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
2		ovexd 7440	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V)
3		elfznn 13526	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
4		icossicc 13409	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
5		esumpmono.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6	4, 5	sselid 3979	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
7	3, 6	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞))
8	7	ralrimiva 3146	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
9		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀)
10	9	esumcl 33016	. . . . . 6 ⊢ (((1...𝑀) ∈ V ∧ ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
11	2, 8, 10	syl2anc 584	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
12	1, 11	sselid 3979	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^)
13	12	xrleidd 13127	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴)
14		ovexd 7440	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ V)
15		esumpmono.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
16	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ)
17		peano2nn 12220	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
18		nnuz 12861	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
19	17, 18	eleqtrdi 2843	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ_≥‘1))
20		fzss1 13536	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘1) → ((𝑀 + 1)...𝑁) ⊆ (1...𝑁))
21	16, 19, 20	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((𝑀 + 1)...𝑁) ⊆ (1...𝑁))
22		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ((𝑀 + 1)...𝑁))
23	21, 22	sseldd 3982	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁))
24		elfznn 13526	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
25	23, 24	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ)
26	25, 6	syldan 591	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ (0[,]+∞))
27	26	ralrimiva 3146	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
28		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁)
29	28	esumcl 33016	. . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ V ∧ ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
30	14, 27, 29	syl2anc 584	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
31		elxrge0 13430	. . . . 5 ⊢ (Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞) ↔ (Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^* ∧ 0 ≤ Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
32	31	simprbi 497	. . . 4 ⊢ (Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞) → 0 ≤ Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)
33	30, 32	syl 17	. . 3 ⊢ (𝜑 → 0 ≤ Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)
34		0xr 11257	. . . . 5 ⊢ 0 ∈ ℝ^*
35	34	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ^*)
36	1, 30	sselid 3979	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^)
37		xle2add 13234	. . . 4 ⊢ (((Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^ ∧ 0 ∈ ℝ^) ∧ (Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^* ∧ Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^)) → ((Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴 ∧ 0 ≤ Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) ≤ (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)))
38	12, 35, 12, 36, 37	syl22anc 837	. . 3 ⊢ (𝜑 → ((Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴 ∧ 0 ≤ Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴) → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) ≤ (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)))
39	13, 33, 38	mp2and 697	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) ≤ (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
40		xaddrid 13216	. . . 4 ⊢ (Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^ → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) = Σ^𝑘 ∈ (1...𝑀)𝐴)
41	12, 40	syl 17	. . 3 ⊢ (𝜑 → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) = Σ^𝑘 ∈ (1...𝑀)𝐴)
42	41	eqcomd 2738	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0))
43	15, 18	eleqtrdi 2843	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
44		esumpmono.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
45		eluzfz 13492	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ (1...𝑁))
46	43, 44, 45	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
47		fzsplit 13523	. . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
48		esumeq1 33020	. . . 4 ⊢ ((1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) → Σ^𝑘 ∈ (1...𝑁)𝐴 = Σ^𝑘 ∈ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))𝐴)
49	46, 47, 48	3syl 18	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = Σ^𝑘 ∈ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))𝐴)
50		nfv 1917	. . . 4 ⊢ Ⅎ𝑘𝜑
51		nnre 12215	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
52	51	ltp1d 12140	. . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1))
53		fzdisj 13524	. . . . 5 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
54	15, 52, 53	3syl 18	. . . 4 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
55	50, 9, 28, 2, 14, 54, 7, 26	esumsplit 33039	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
56	49, 55	eqtrd 2772	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
57	39, 42, 56	3brtr4d 5179	1 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑁)𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∀wral 3061 Vcvv 3474 ∪ cun 3945 ∩ cin 3946 ⊆ wss 3947 ∅c0 4321 class class class wbr 5147 ‘cfv 6540 (class class class)co 7405 0cc0 11106 1c1 11107 + caddc 11109 +∞cpnf 11241 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕcn 12208 ℤ_≥cuz 12818 +_𝑒 cxad 13086 [,)cico 13322 [,]cicc 13323 ...cfz 13480 Σ^*cesum 33013

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 ax-addf 11185 ax-mulf 11186

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-tp 4632 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7666 df-om 7852 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-2o 8463 df-er 8699 df-map 8818 df-pm 8819 df-ixp 8888 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-fsupp 9358 df-fi 9402 df-sup 9433 df-inf 9434 df-oi 9501 df-card 9930 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-4 12273 df-5 12274 df-6 12275 df-7 12276 df-8 12277 df-9 12278 df-n0 12469 df-z 12555 df-dec 12674 df-uz 12819 df-q 12929 df-rp 12971 df-xneg 13088 df-xadd 13089 df-xmul 13090 df-ioo 13324 df-ioc 13325 df-ico 13326 df-icc 13327 df-fz 13481 df-fzo 13624 df-fl 13753 df-mod 13831 df-seq 13963 df-exp 14024 df-fac 14230 df-bc 14259 df-hash 14287 df-shft 15010 df-cj 15042 df-re 15043 df-im 15044 df-sqrt 15178 df-abs 15179 df-limsup 15411 df-clim 15428 df-rlim 15429 df-sum 15629 df-ef 16007 df-sin 16009 df-cos 16010 df-pi 16012 df-struct 17076 df-sets 17093 df-slot 17111 df-ndx 17123 df-base 17141 df-ress 17170 df-plusg 17206 df-mulr 17207 df-starv 17208 df-sca 17209 df-vsca 17210 df-ip 17211 df-tset 17212 df-ple 17213 df-ds 17215 df-unif 17216 df-hom 17217 df-cco 17218 df-rest 17364 df-topn 17365 df-0g 17383 df-gsum 17384 df-topgen 17385 df-pt 17386 df-prds 17389 df-ordt 17443 df-xrs 17444 df-qtop 17449 df-imas 17450 df-xps 17452 df-mre 17526 df-mrc 17527 df-acs 17529 df-ps 18515 df-tsr 18516 df-plusf 18556 df-mgm 18557 df-sgrp 18606 df-mnd 18622 df-mhm 18667 df-submnd 18668 df-grp 18818 df-minusg 18819 df-sbg 18820 df-mulg 18945 df-subg 18997 df-cntz 19175 df-cmn 19644 df-abl 19645 df-mgp 19982 df-ur 19999 df-ring 20051 df-cring 20052 df-subrg 20353 df-abv 20417 df-lmod 20465 df-scaf 20466 df-sra 20777 df-rgmod 20778 df-psmet 20928 df-xmet 20929 df-met 20930 df-bl 20931 df-mopn 20932 df-fbas 20933 df-fg 20934 df-cnfld 20937 df-top 22387 df-topon 22404 df-topsp 22426 df-bases 22440 df-cld 22514 df-ntr 22515 df-cls 22516 df-nei 22593 df-lp 22631 df-perf 22632 df-cn 22722 df-cnp 22723 df-haus 22810 df-tx 23057 df-hmeo 23250 df-fil 23341 df-fm 23433 df-flim 23434 df-flf 23435 df-tmd 23567 df-tgp 23568 df-tsms 23622 df-trg 23655 df-xms 23817 df-ms 23818 df-tms 23819 df-nm 24082 df-ngp 24083 df-nrg 24085 df-nlm 24086 df-ii 24384 df-cncf 24385 df-limc 25374 df-dv 25375 df-log 26056 df-esum 33014

This theorem is referenced by: esumcvg 33072

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