esumpmono - Mathbox for Thierry Arnoux

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Theorem esumpmono 30442

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 12441	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
2		ovexd 6835	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V)
3		elfznn 12555	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
4		icossicc 12445	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
5		esumpmono.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6	4, 5	sseldi 3734	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
7	3, 6	sylan2 492	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞))
8	7	ralrimiva 3096	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
9		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀)
10	9	esumcl 30393	. . . . . 6 ⊢ (((1...𝑀) ∈ V ∧ ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
11	2, 8, 10	syl2anc 696	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
12	1, 11	sseldi 3734	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^)
13		xrleid 12168	. . . 4 ⊢ (Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^ → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴)
14	12, 13	syl 17	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴)
15		ovexd 6835	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ V)
16		esumpmono.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
17	16	adantr 472	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ)
18		peano2nn 11216	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
19		nnuz 11908	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
20	18, 19	syl6eleq 2841	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ_≥‘1))
21		fzss1 12565	. . . . . . . . . 10 ⊢ ((𝑀 + 1) ∈ (ℤ_≥‘1) → ((𝑀 + 1)...𝑁) ⊆ (1...𝑁))
22	17, 20, 21	3syl 18	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → ((𝑀 + 1)...𝑁) ⊆ (1...𝑁))
23		simpr 479	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ((𝑀 + 1)...𝑁))
24	22, 23	sseldd 3737	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁))
25		elfznn 12555	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
26	24, 25	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ)
27	26, 6	syldan 488	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ (0[,]+∞))
28	27	ralrimiva 3096	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
29		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁)
30	29	esumcl 30393	. . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ V ∧ ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
31	15, 28, 30	syl2anc 696	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
32		elxrge0 12466	. . . . 5 ⊢ (Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞) ↔ (Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^* ∧ 0 ≤ Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
33	32	simprbi 483	. . . 4 ⊢ (Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞) → 0 ≤ Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)
34	31, 33	syl 17	. . 3 ⊢ (𝜑 → 0 ≤ Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)
35		0xr 10270	. . . . 5 ⊢ 0 ∈ ℝ^*
36	35	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ^*)
37	1, 31	sseldi 3734	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^)
38		xle2add 12274	. . . 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)))
39	12, 36, 12, 37, 38	syl22anc 1474	. . 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)))
40	14, 34, 39	mp2and 717	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) ≤ (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
41		xaddid1 12257	. . . 4 ⊢ (Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^ → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) = Σ^𝑘 ∈ (1...𝑀)𝐴)
42	12, 41	syl 17	. . 3 ⊢ (𝜑 → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) = Σ^𝑘 ∈ (1...𝑀)𝐴)
43	42	eqcomd 2758	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0))
44	16, 19	syl6eleq 2841	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
45		esumpmono.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
46		eluzfz 12522	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ (1...𝑁))
47	44, 45, 46	syl2anc 696	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
48		fzsplit 12552	. . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
49		esumeq1 30397	. . . 4 ⊢ ((1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)) → Σ^𝑘 ∈ (1...𝑁)𝐴 = Σ^𝑘 ∈ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))𝐴)
50	47, 48, 49	3syl 18	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = Σ^𝑘 ∈ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))𝐴)
51		nfv 1984	. . . 4 ⊢ Ⅎ𝑘𝜑
52		nnre 11211	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
53	52	ltp1d 11138	. . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1))
54		fzdisj 12553	. . . . 5 ⊢ (𝑀 < (𝑀 + 1) → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
55	16, 53, 54	3syl 18	. . . 4 ⊢ (𝜑 → ((1...𝑀) ∩ ((𝑀 + 1)...𝑁)) = ∅)
56	51, 9, 29, 2, 15, 55, 7, 27	esumsplit 30416	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
57	50, 56	eqtrd 2786	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
58	40, 43, 57	3brtr4d 4828	1 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑁)𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 383 = wceq 1624 ∈ wcel 2131 ∀wral 3042 Vcvv 3332 ∪ cun 3705 ∩ cin 3706 ⊆ wss 3707 ∅c0 4050 class class class wbr 4796 ‘cfv 6041 (class class class)co 6805 0cc0 10120 1c1 10121 + caddc 10123 +∞cpnf 10255 ℝ^*cxr 10257 < clt 10258 ≤ cle 10259 ℕcn 11204 ℤ_≥cuz 11871 +_𝑒 cxad 12129 [,)cico 12362 [,]cicc 12363 ...cfz 12511 Σ^*cesum 30390

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-8 2133 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-rep 4915 ax-sep 4925 ax-nul 4933 ax-pow 4984 ax-pr 5047 ax-un 7106 ax-inf2 8703 ax-cnex 10176 ax-resscn 10177 ax-1cn 10178 ax-icn 10179 ax-addcl 10180 ax-addrcl 10181 ax-mulcl 10182 ax-mulrcl 10183 ax-mulcom 10184 ax-addass 10185 ax-mulass 10186 ax-distr 10187 ax-i2m1 10188 ax-1ne0 10189 ax-1rid 10190 ax-rnegex 10191 ax-rrecex 10192 ax-cnre 10193 ax-pre-lttri 10194 ax-pre-lttrn 10195 ax-pre-ltadd 10196 ax-pre-mulgt0 10197 ax-pre-sup 10198 ax-addf 10199 ax-mulf 10200

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1627 df-fal 1630 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-ne 2925 df-nel 3028 df-ral 3047 df-rex 3048 df-reu 3049 df-rmo 3050 df-rab 3051 df-v 3334 df-sbc 3569 df-csb 3667 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-pss 3723 df-nul 4051 df-if 4223 df-pw 4296 df-sn 4314 df-pr 4316 df-tp 4318 df-op 4320 df-uni 4581 df-int 4620 df-iun 4666 df-iin 4667 df-br 4797 df-opab 4857 df-mpt 4874 df-tr 4897 df-id 5166 df-eprel 5171 df-po 5179 df-so 5180 df-fr 5217 df-se 5218 df-we 5219 df-xp 5264 df-rel 5265 df-cnv 5266 df-co 5267 df-dm 5268 df-rn 5269 df-res 5270 df-ima 5271 df-pred 5833 df-ord 5879 df-on 5880 df-lim 5881 df-suc 5882 df-iota 6004 df-fun 6043 df-fn 6044 df-f 6045 df-f1 6046 df-fo 6047 df-f1o 6048 df-fv 6049 df-isom 6050 df-riota 6766 df-ov 6808 df-oprab 6809 df-mpt2 6810 df-of 7054 df-om 7223 df-1st 7325 df-2nd 7326 df-supp 7456 df-wrecs 7568 df-recs 7629 df-rdg 7667 df-1o 7721 df-2o 7722 df-oadd 7725 df-er 7903 df-map 8017 df-pm 8018 df-ixp 8067 df-en 8114 df-dom 8115 df-sdom 8116 df-fin 8117 df-fsupp 8433 df-fi 8474 df-sup 8505 df-inf 8506 df-oi 8572 df-card 8947 df-cda 9174 df-pnf 10260 df-mnf 10261 df-xr 10262 df-ltxr 10263 df-le 10264 df-sub 10452 df-neg 10453 df-div 10869 df-nn 11205 df-2 11263 df-3 11264 df-4 11265 df-5 11266 df-6 11267 df-7 11268 df-8 11269 df-9 11270 df-n0 11477 df-z 11562 df-dec 11678 df-uz 11872 df-q 11974 df-rp 12018 df-xneg 12131 df-xadd 12132 df-xmul 12133 df-ioo 12364 df-ioc 12365 df-ico 12366 df-icc 12367 df-fz 12512 df-fzo 12652 df-fl 12779 df-mod 12855 df-seq 12988 df-exp 13047 df-fac 13247 df-bc 13276 df-hash 13304 df-shft 13998 df-cj 14030 df-re 14031 df-im 14032 df-sqrt 14166 df-abs 14167 df-limsup 14393 df-clim 14410 df-rlim 14411 df-sum 14608 df-ef 14989 df-sin 14991 df-cos 14992 df-pi 14994 df-struct 16053 df-ndx 16054 df-slot 16055 df-base 16057 df-sets 16058 df-ress 16059 df-plusg 16148 df-mulr 16149 df-starv 16150 df-sca 16151 df-vsca 16152 df-ip 16153 df-tset 16154 df-ple 16155 df-ds 16158 df-unif 16159 df-hom 16160 df-cco 16161 df-rest 16277 df-topn 16278 df-0g 16296 df-gsum 16297 df-topgen 16298 df-pt 16299 df-prds 16302 df-ordt 16355 df-xrs 16356 df-qtop 16361 df-imas 16362 df-xps 16364 df-mre 16440 df-mrc 16441 df-acs 16443 df-ps 17393 df-tsr 17394 df-plusf 17434 df-mgm 17435 df-sgrp 17477 df-mnd 17488 df-mhm 17528 df-submnd 17529 df-grp 17618 df-minusg 17619 df-sbg 17620 df-mulg 17734 df-subg 17784 df-cntz 17942 df-cmn 18387 df-abl 18388 df-mgp 18682 df-ur 18694 df-ring 18741 df-cring 18742 df-subrg 18972 df-abv 19011 df-lmod 19059 df-scaf 19060 df-sra 19366 df-rgmod 19367 df-psmet 19932 df-xmet 19933 df-met 19934 df-bl 19935 df-mopn 19936 df-fbas 19937 df-fg 19938 df-cnfld 19941 df-top 20893 df-topon 20910 df-topsp 20931 df-bases 20944 df-cld 21017 df-ntr 21018 df-cls 21019 df-nei 21096 df-lp 21134 df-perf 21135 df-cn 21225 df-cnp 21226 df-haus 21313 df-tx 21559 df-hmeo 21752 df-fil 21843 df-fm 21935 df-flim 21936 df-flf 21937 df-tmd 22069 df-tgp 22070 df-tsms 22123 df-trg 22156 df-xms 22318 df-ms 22319 df-tms 22320 df-nm 22580 df-ngp 22581 df-nrg 22583 df-nlm 22584 df-ii 22873 df-cncf 22874 df-limc 23821 df-dv 23822 df-log 24494 df-esum 30391

This theorem is referenced by: esumcvg 30449

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