esumpmono - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpmono		Structured version Visualization version GIF version

Theorem esumpmono 34164

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 13337	. . . . 5 ⊢ (0[,]+∞) ⊆ ℝ^*
2		ovexd 7390	. . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V)
3		elfznn 13460	. . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ)
4		icossicc 13343	. . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞)
5		esumpmono.3	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞))
6	4, 5	sselid 3928	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞))
7	3, 6	sylan2 593	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞))
8	7	ralrimiva 3125	. . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞))
9		nfcv 2895	. . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀)
10	9	esumcl 34115	. . . . . 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 3928	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ^)
13	12	xrleidd 13057	. . 3 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑀)𝐴 ≤ Σ^𝑘 ∈ (1...𝑀)𝐴)
14		ovexd 7390	. . . . 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 12148	. . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ)
18		nnuz 12781	. . . . . . . . . . 11 ⊢ ℕ = (ℤ_≥‘1)
19	17, 18	eleqtrdi 2843	. . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ_≥‘1))
20		fzss1 13470	. . . . . . . . . 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 3931	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁))
24		elfznn 13460	. . . . . . . 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 3125	. . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞))
28		nfcv 2895	. . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁)
29	28	esumcl 34115	. . . . 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 13364	. . . . 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 11170	. . . . 5 ⊢ 0 ∈ ℝ^*
35	34	a1i 11	. . . 4 ⊢ (𝜑 → 0 ∈ ℝ^*)
36	1, 30	sselid 3928	. . . 4 ⊢ (𝜑 → Σ^𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ^)
37		xle2add 13165	. . . 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 13147	. . . 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 2739	. 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 13426	. . . . 5 ⊢ ((𝑀 ∈ (ℤ_≥‘1) ∧ 𝑁 ∈ (ℤ_≥‘𝑀)) → 𝑀 ∈ (1...𝑁))
46	43, 44, 45	syl2anc 584	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁))
47		fzsplit 13457	. . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁)))
48		esumeq1 34119	. . . 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 12143	. . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ)
52	51	ltp1d 12063	. . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1))
53		fzdisj 13458	. . . . 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 34138	. . 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 2768	. 2 ⊢ (𝜑 → Σ^𝑘 ∈ (1...𝑁)𝐴 = (Σ^𝑘 ∈ (1...𝑀)𝐴 +_𝑒 Σ^*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴))
57	39, 42, 56	3brtr4d 5127	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 3048 Vcvv 3437 ∪ cun 3896 ∩ cin 3897 ⊆ wss 3898 ∅c0 4282 class class class wbr 5095 ‘cfv 6489 (class class class)co 7355 0cc0 11017 1c1 11018 + caddc 11020 +∞cpnf 11154 ℝ^*cxr 11156 < clt 11157 ≤ cle 11158 ℕcn 12136 ℤ_≥cuz 12742 +_𝑒 cxad 13015 [,)cico 13254 [,]cicc 13255 ...cfz 13414 Σ^*cesum 34112

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 2182 ax-ext 2705 ax-rep 5221 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-inf2 9542 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094 ax-pre-sup 11095 ax-addf 11096 ax-mulf 11097

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 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-tp 4582 df-op 4584 df-uni 4861 df-int 4900 df-iun 4945 df-iin 4946 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-of 7619 df-om 7806 df-1st 7930 df-2nd 7931 df-supp 8100 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-1o 8394 df-2o 8395 df-er 8631 df-map 8761 df-pm 8762 df-ixp 8832 df-en 8880 df-dom 8881 df-sdom 8882 df-fin 8883 df-fsupp 9257 df-fi 9306 df-sup 9337 df-inf 9338 df-oi 9407 df-card 9843 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-2 12199 df-3 12200 df-4 12201 df-5 12202 df-6 12203 df-7 12204 df-8 12205 df-9 12206 df-n0 12393 df-z 12480 df-dec 12599 df-uz 12743 df-q 12853 df-rp 12897 df-xneg 13017 df-xadd 13018 df-xmul 13019 df-ioo 13256 df-ioc 13257 df-ico 13258 df-icc 13259 df-fz 13415 df-fzo 13562 df-fl 13703 df-mod 13781 df-seq 13916 df-exp 13976 df-fac 14188 df-bc 14217 df-hash 14245 df-shft 14981 df-cj 15013 df-re 15014 df-im 15015 df-sqrt 15149 df-abs 15150 df-limsup 15385 df-clim 15402 df-rlim 15403 df-sum 15601 df-ef 15981 df-sin 15983 df-cos 15984 df-pi 15986 df-struct 17065 df-sets 17082 df-slot 17100 df-ndx 17112 df-base 17128 df-ress 17149 df-plusg 17181 df-mulr 17182 df-starv 17183 df-sca 17184 df-vsca 17185 df-ip 17186 df-tset 17187 df-ple 17188 df-ds 17190 df-unif 17191 df-hom 17192 df-cco 17193 df-rest 17333 df-topn 17334 df-0g 17352 df-gsum 17353 df-topgen 17354 df-pt 17355 df-prds 17358 df-ordt 17413 df-xrs 17414 df-qtop 17419 df-imas 17420 df-xps 17422 df-mre 17496 df-mrc 17497 df-acs 17499 df-ps 18480 df-tsr 18481 df-plusf 18555 df-mgm 18556 df-sgrp 18635 df-mnd 18651 df-mhm 18699 df-submnd 18700 df-grp 18857 df-minusg 18858 df-sbg 18859 df-mulg 18989 df-subg 19044 df-cntz 19237 df-cmn 19702 df-abl 19703 df-mgp 20067 df-rng 20079 df-ur 20108 df-ring 20161 df-cring 20162 df-subrng 20470 df-subrg 20494 df-abv 20733 df-lmod 20804 df-scaf 20805 df-sra 21116 df-rgmod 21117 df-psmet 21292 df-xmet 21293 df-met 21294 df-bl 21295 df-mopn 21296 df-fbas 21297 df-fg 21298 df-cnfld 21301 df-top 22829 df-topon 22846 df-topsp 22868 df-bases 22881 df-cld 22954 df-ntr 22955 df-cls 22956 df-nei 23033 df-lp 23071 df-perf 23072 df-cn 23162 df-cnp 23163 df-haus 23250 df-tx 23497 df-hmeo 23690 df-fil 23781 df-fm 23873 df-flim 23874 df-flf 23875 df-tmd 24007 df-tgp 24008 df-tsms 24062 df-trg 24095 df-xms 24255 df-ms 24256 df-tms 24257 df-nm 24517 df-ngp 24518 df-nrg 24520 df-nlm 24521 df-ii 24817 df-cncf 24818 df-limc 25814 df-dv 25815 df-log 26512 df-esum 34113

This theorem is referenced by: esumcvg 34171

W3C validator