esumpmono - Mathbox for Thierry Arnoux

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Theorem esumpmono 33607

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 13413	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
2		ovexd 7440	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V)
3		elfznn 13536	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
4		icossicc 13419	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
5		esumpmono.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6	4, 5	sselid 3975	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
7	3, 6	sylan2 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞))
8	7	ralrimiva 3140	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
9		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀)
10	9	esumcl 33558	. . . . . 6 ⊢ (((1...𝑀) ∈ V ∧ ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
11	2, 8, 10	syl2anc 583	. . . . 5 ⊢ (𝜑 → Σ^*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
12	1, 11	sselid 3975	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^)
13	12	xrleidd 13137	. . 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 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ)
17		peano2nn 12228	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
18		nnuz 12869	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
19	17, 18	eleqtrdi 2837	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ_≥‘1))
20		fzss1 13546	. . . . . . . . . 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 3978	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁))
24		elfznn 13536	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ)
25	23, 24	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ)
26	25, 6	syldan 590	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ (0[,]+∞))
27	26	ralrimiva 3140	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
28		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁)
29	28	esumcl 33558	. . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ V ∧ ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) → Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
30	14, 27, 29	syl2anc 583	. . . 4 ⊢ (𝜑 → Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
31		elxrge0 13440	. . . . 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 11265	. . . . 5 ⊢ 0 ∈ ℝ^*
35	34	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ^*)
36	1, 30	sselid 3975	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^)
37		xle2add 13244	. . . 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 836	. . 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 696	. 2 ⊢ (𝜑 → (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0) ≤ (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
40		xaddrid 13226	. . . 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 2732	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 0))
43	15, 18	eleqtrdi 2837	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ_≥‘1))
44		esumpmono.2	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
45		eluzfz 13502	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ (1...𝑁))
46	43, 44, 45	syl2anc 583	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
47		fzsplit 13533	. . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
48		esumeq1 33562	. . . 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 1909	. . . 4 ⊢ Ⅎ𝑘𝜑
51		nnre 12223	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
52	51	ltp1d 12148	. . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1))
53		fzdisj 13534	. . . . 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 33581	. . 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 2766	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
57	39, 42, 56	3brtr4d 5173	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 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ∪ cun 3941 ∩ cin 3942 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 ‘cfv 6537 (class class class)co 7405 0cc0 11112 1c1 11113 + caddc 11115 +∞cpnf 11249 ℝ^*cxr 11251 < clt 11252 ≤ cle 11253 ℕcn 12216 ℤ_≥cuz 12826 +_𝑒 cxad 13096 [,)cico 13332 [,]cicc 13333 ...cfz 13490 Σ^*cesum 33555

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-tp 4628 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-iin 4993 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-of 7667 df-om 7853 df-1st 7974 df-2nd 7975 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-2o 8468 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-4 12281 df-5 12282 df-6 12283 df-7 12284 df-8 12285 df-9 12286 df-n0 12477 df-z 12563 df-dec 12682 df-uz 12827 df-q 12937 df-rp 12981 df-xneg 13098 df-xadd 13099 df-xmul 13100 df-ioo 13334 df-ioc 13335 df-ico 13336 df-icc 13337 df-fz 13491 df-fzo 13634 df-fl 13763 df-mod 13841 df-seq 13973 df-exp 14033 df-fac 14239 df-bc 14268 df-hash 14296 df-shft 15020 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-limsup 15421 df-clim 15438 df-rlim 15439 df-sum 15639 df-ef 16017 df-sin 16019 df-cos 16020 df-pi 16022 df-struct 17089 df-sets 17106 df-slot 17124 df-ndx 17136 df-base 17154 df-ress 17183 df-plusg 17219 df-mulr 17220 df-starv 17221 df-sca 17222 df-vsca 17223 df-ip 17224 df-tset 17225 df-ple 17226 df-ds 17228 df-unif 17229 df-hom 17230 df-cco 17231 df-rest 17377 df-topn 17378 df-0g 17396 df-gsum 17397 df-topgen 17398 df-pt 17399 df-prds 17402 df-ordt 17456 df-xrs 17457 df-qtop 17462 df-imas 17463 df-xps 17465 df-mre 17539 df-mrc 17540 df-acs 17542 df-ps 18531 df-tsr 18532 df-plusf 18572 df-mgm 18573 df-sgrp 18652 df-mnd 18668 df-mhm 18713 df-submnd 18714 df-grp 18866 df-minusg 18867 df-sbg 18868 df-mulg 18996 df-subg 19050 df-cntz 19233 df-cmn 19702 df-abl 19703 df-mgp 20040 df-rng 20058 df-ur 20087 df-ring 20140 df-cring 20141 df-subrng 20446 df-subrg 20471 df-abv 20660 df-lmod 20708 df-scaf 20709 df-sra 21021 df-rgmod 21022 df-psmet 21232 df-xmet 21233 df-met 21234 df-bl 21235 df-mopn 21236 df-fbas 21237 df-fg 21238 df-cnfld 21241 df-top 22751 df-topon 22768 df-topsp 22790 df-bases 22804 df-cld 22878 df-ntr 22879 df-cls 22880 df-nei 22957 df-lp 22995 df-perf 22996 df-cn 23086 df-cnp 23087 df-haus 23174 df-tx 23421 df-hmeo 23614 df-fil 23705 df-fm 23797 df-flim 23798 df-flf 23799 df-tmd 23931 df-tgp 23932 df-tsms 23986 df-trg 24019 df-xms 24181 df-ms 24182 df-tms 24183 df-nm 24446 df-ngp 24447 df-nrg 24449 df-nlm 24450 df-ii 24752 df-cncf 24753 df-limc 25750 df-dv 25751 df-log 26445 df-esum 33556

This theorem is referenced by: esumcvg 33614

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