esumpmono - Mathbox for Thierry Arnoux

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Theorem esumpmono 34236

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 13346	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
2		ovexd 7393	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V)
3		elfznn 13469	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
4		icossicc 13352	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
5		esumpmono.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6	4, 5	sselid 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
7	3, 6	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞))
8	7	ralrimiva 3128	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
9		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀)
10	9	esumcl 34187	. . . . . 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 3931	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^)
13	12	xrleidd 13066	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴)
14		ovexd 7393	. . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ V)
15		esumpmono.1	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ)
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ)
17		peano2nn 12157	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
18		nnuz 12790	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
19	17, 18	eleqtrdi 2846	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ_≥‘1))
20		fzss1 13479	. . . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ((𝑀 + 1)...𝑁))
23	21, 22	sseldd 3934	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁))
24		elfznn 13469	. . . . . . . 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 3128	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
28		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁)
29	28	esumcl 34187	. . . . 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 13373	. . . . 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 496	. . . 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 11179	. . . . 5 ⊢ 0 ∈ ℝ^*
35	34	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ^*)
36	1, 30	sselid 3931	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^)
37		xle2add 13174	. . . 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 838	. . 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 699	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) ≤ (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
40		xaddrid 13156	. . . 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 2742	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0))
43	15, 18	eleqtrdi 2846	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
44		esumpmono.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
45		eluzfz 13435	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ (1...𝑁))
46	43, 44, 45	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
47		fzsplit 13466	. . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
48		esumeq1 34191	. . . 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 1915	. . . 4 ⊢ Ⅎ𝑘𝜑
51		nnre 12152	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
52	51	ltp1d 12072	. . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1))
53		fzdisj 13467	. . . . 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 34210	. . 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 2771	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
57	39, 42, 56	3brtr4d 5130	1 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑁)𝐴)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1541 ∈ wcel 2113 ∀wral 3051 Vcvv 3440 ∪ cun 3899 ∩ cin 3900 ⊆ wss 3901 ∅c0 4285 class class class wbr 5098 ‘cfv 6492 (class class class)co 7358 0cc0 11026 1c1 11027 + caddc 11029 +∞cpnf 11163 ℝ^*cxr 11165 < clt 11166 ≤ cle 11167 ℕcn 12145 ℤ_≥cuz 12751 +_𝑒 cxad 13024 [,)cico 13263 [,]cicc 13264 ...cfz 13423 Σ^*cesum 34184

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2184 ax-ext 2708 ax-rep 5224 ax-sep 5241 ax-nul 5251 ax-pow 5310 ax-pr 5377 ax-un 7680 ax-inf2 9550 ax-cnex 11082 ax-resscn 11083 ax-1cn 11084 ax-icn 11085 ax-addcl 11086 ax-addrcl 11087 ax-mulcl 11088 ax-mulrcl 11089 ax-mulcom 11090 ax-addass 11091 ax-mulass 11092 ax-distr 11093 ax-i2m1 11094 ax-1ne0 11095 ax-1rid 11096 ax-rnegex 11097 ax-rrecex 11098 ax-cnre 11099 ax-pre-lttri 11100 ax-pre-lttrn 11101 ax-pre-ltadd 11102 ax-pre-mulgt0 11103 ax-pre-sup 11104 ax-addf 11105 ax-mulf 11106

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3350 df-reu 3351 df-rab 3400 df-v 3442 df-sbc 3741 df-csb 3850 df-dif 3904 df-un 3906 df-in 3908 df-ss 3918 df-pss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4581 df-pr 4583 df-tp 4585 df-op 4587 df-uni 4864 df-int 4903 df-iun 4948 df-iin 4949 df-br 5099 df-opab 5161 df-mpt 5180 df-tr 5206 df-id 5519 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5577 df-se 5578 df-we 5579 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-pred 6259 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-isom 6501 df-riota 7315 df-ov 7361 df-oprab 7362 df-mpo 7363 df-of 7622 df-om 7809 df-1st 7933 df-2nd 7934 df-supp 8103 df-frecs 8223 df-wrecs 8254 df-recs 8303 df-rdg 8341 df-1o 8397 df-2o 8398 df-er 8635 df-map 8765 df-pm 8766 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9265 df-fi 9314 df-sup 9345 df-inf 9346 df-oi 9415 df-card 9851 df-pnf 11168 df-mnf 11169 df-xr 11170 df-ltxr 11171 df-le 11172 df-sub 11366 df-neg 11367 df-div 11795 df-nn 12146 df-2 12208 df-3 12209 df-4 12210 df-5 12211 df-6 12212 df-7 12213 df-8 12214 df-9 12215 df-n0 12402 df-z 12489 df-dec 12608 df-uz 12752 df-q 12862 df-rp 12906 df-xneg 13026 df-xadd 13027 df-xmul 13028 df-ioo 13265 df-ioc 13266 df-ico 13267 df-icc 13268 df-fz 13424 df-fzo 13571 df-fl 13712 df-mod 13790 df-seq 13925 df-exp 13985 df-fac 14197 df-bc 14226 df-hash 14254 df-shft 14990 df-cj 15022 df-re 15023 df-im 15024 df-sqrt 15158 df-abs 15159 df-limsup 15394 df-clim 15411 df-rlim 15412 df-sum 15610 df-ef 15990 df-sin 15992 df-cos 15993 df-pi 15995 df-struct 17074 df-sets 17091 df-slot 17109 df-ndx 17121 df-base 17137 df-ress 17158 df-plusg 17190 df-mulr 17191 df-starv 17192 df-sca 17193 df-vsca 17194 df-ip 17195 df-tset 17196 df-ple 17197 df-ds 17199 df-unif 17200 df-hom 17201 df-cco 17202 df-rest 17342 df-topn 17343 df-0g 17361 df-gsum 17362 df-topgen 17363 df-pt 17364 df-prds 17367 df-ordt 17422 df-xrs 17423 df-qtop 17428 df-imas 17429 df-xps 17431 df-mre 17505 df-mrc 17506 df-acs 17508 df-ps 18489 df-tsr 18490 df-plusf 18564 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-mhm 18708 df-submnd 18709 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18998 df-subg 19053 df-cntz 19246 df-cmn 19711 df-abl 19712 df-mgp 20076 df-rng 20088 df-ur 20117 df-ring 20170 df-cring 20171 df-subrng 20479 df-subrg 20503 df-abv 20742 df-lmod 20813 df-scaf 20814 df-sra 21125 df-rgmod 21126 df-psmet 21301 df-xmet 21302 df-met 21303 df-bl 21304 df-mopn 21305 df-fbas 21306 df-fg 21307 df-cnfld 21310 df-top 22838 df-topon 22855 df-topsp 22877 df-bases 22890 df-cld 22963 df-ntr 22964 df-cls 22965 df-nei 23042 df-lp 23080 df-perf 23081 df-cn 23171 df-cnp 23172 df-haus 23259 df-tx 23506 df-hmeo 23699 df-fil 23790 df-fm 23882 df-flim 23883 df-flf 23884 df-tmd 24016 df-tgp 24017 df-tsms 24071 df-trg 24104 df-xms 24264 df-ms 24265 df-tms 24266 df-nm 24526 df-ngp 24527 df-nrg 24529 df-nlm 24530 df-ii 24826 df-cncf 24827 df-limc 25823 df-dv 25824 df-log 26521 df-esum 34185

This theorem is referenced by: esumcvg 34243

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