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Mirrors > Home > MPE Home > Th. List > Mathboxes > esumpmono | Structured version Visualization version GIF version |
Description: The partial sums in an extended sum form a monotonic sequence. (Contributed by Thierry Arnoux, 31-Aug-2017.) |
Ref | Expression |
---|---|
esumpmono.1 | ⊢ (𝜑 → 𝑀 ∈ ℕ) |
esumpmono.2 | ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
esumpmono.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) |
Ref | Expression |
---|---|
esumpmono | ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑁)𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | iccssxr 13439 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | ovexd 7451 | . . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V) | |
3 | elfznn 13562 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ) | |
4 | icossicc 13445 | . . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
5 | esumpmono.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) | |
6 | 4, 5 | sselid 3970 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
7 | 3, 6 | sylan2 591 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞)) |
8 | 7 | ralrimiva 3136 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
9 | nfcv 2892 | . . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀) | |
10 | 9 | esumcl 33706 | . . . . . 6 ⊢ (((1...𝑀) ∈ V ∧ ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
11 | 2, 8, 10 | syl2anc 582 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
12 | 1, 11 | sselid 3970 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ*) |
13 | 12 | xrleidd 13163 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑀)𝐴) |
14 | ovexd 7451 | . . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ V) | |
15 | esumpmono.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
16 | 15 | adantr 479 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ) |
17 | peano2nn 12254 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ) | |
18 | nnuz 12895 | . . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1) | |
19 | 17, 18 | eleqtrdi 2835 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘1)) |
20 | fzss1 13572 | . . . . . . . . . 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 483 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ((𝑀 + 1)...𝑁)) | |
23 | 21, 22 | sseldd 3973 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁)) |
24 | elfznn 13562 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ) |
26 | 25, 6 | syldan 589 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ (0[,]+∞)) |
27 | 26 | ralrimiva 3136 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
28 | nfcv 2892 | . . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁) | |
29 | 28 | esumcl 33706 | . . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ V ∧ ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
30 | 14, 27, 29 | syl2anc 582 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
31 | elxrge0 13466 | . . . . 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 495 | . . . 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 11291 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
36 | 1, 30 | sselid 3970 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ*) |
37 | xle2add 13270 | . . . 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 13252 | . . . 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 2731 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 0)) |
43 | 15, 18 | eleqtrdi 2835 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
44 | esumpmono.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
45 | eluzfz 13528 | . . . . 5 ⊢ ((𝑀 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (1...𝑁)) | |
46 | 43, 44, 45 | syl2anc 582 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
47 | fzsplit 13559 | . . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
48 | esumeq1 33710 | . . . 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 12249 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
52 | 51 | ltp1d 12174 | . . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1)) |
53 | fzdisj 13560 | . . . . 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 33729 | . . 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 2765 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑁)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
57 | 39, 42, 56 | 3brtr4d 5175 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑁)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3051 Vcvv 3463 ∪ cun 3937 ∩ cin 3938 ⊆ wss 3939 ∅c0 4318 class class class wbr 5143 ‘cfv 6543 (class class class)co 7416 0cc0 11138 1c1 11139 + caddc 11141 +∞cpnf 11275 ℝ*cxr 11277 < clt 11278 ≤ cle 11279 ℕcn 12242 ℤ≥cuz 12852 +𝑒 cxad 13122 [,)cico 13358 [,]cicc 13359 ...cfz 13516 Σ*cesum 33703 |
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 2166 ax-ext 2696 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7738 ax-inf2 9664 ax-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216 ax-addf 11217 ax-mulf 11218 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3465 df-sbc 3769 df-csb 3885 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-pss 3959 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-tr 5261 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-of 7682 df-om 7869 df-1st 7991 df-2nd 7992 df-supp 8164 df-frecs 8285 df-wrecs 8316 df-recs 8390 df-rdg 8429 df-1o 8485 df-2o 8486 df-er 8723 df-map 8845 df-pm 8846 df-ixp 8915 df-en 8963 df-dom 8964 df-sdom 8965 df-fin 8966 df-fsupp 9386 df-fi 9434 df-sup 9465 df-inf 9466 df-oi 9533 df-card 9962 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477 df-div 11902 df-nn 12243 df-2 12305 df-3 12306 df-4 12307 df-5 12308 df-6 12309 df-7 12310 df-8 12311 df-9 12312 df-n0 12503 df-z 12589 df-dec 12708 df-uz 12853 df-q 12963 df-rp 13007 df-xneg 13124 df-xadd 13125 df-xmul 13126 df-ioo 13360 df-ioc 13361 df-ico 13362 df-icc 13363 df-fz 13517 df-fzo 13660 df-fl 13789 df-mod 13867 df-seq 13999 df-exp 14059 df-fac 14265 df-bc 14294 df-hash 14322 df-shft 15046 df-cj 15078 df-re 15079 df-im 15080 df-sqrt 15214 df-abs 15215 df-limsup 15447 df-clim 15464 df-rlim 15465 df-sum 15665 df-ef 16043 df-sin 16045 df-cos 16046 df-pi 16048 df-struct 17115 df-sets 17132 df-slot 17150 df-ndx 17162 df-base 17180 df-ress 17209 df-plusg 17245 df-mulr 17246 df-starv 17247 df-sca 17248 df-vsca 17249 df-ip 17250 df-tset 17251 df-ple 17252 df-ds 17254 df-unif 17255 df-hom 17256 df-cco 17257 df-rest 17403 df-topn 17404 df-0g 17422 df-gsum 17423 df-topgen 17424 df-pt 17425 df-prds 17428 df-ordt 17482 df-xrs 17483 df-qtop 17488 df-imas 17489 df-xps 17491 df-mre 17565 df-mrc 17566 df-acs 17568 df-ps 18557 df-tsr 18558 df-plusf 18598 df-mgm 18599 df-sgrp 18678 df-mnd 18694 df-mhm 18739 df-submnd 18740 df-grp 18897 df-minusg 18898 df-sbg 18899 df-mulg 19028 df-subg 19082 df-cntz 19272 df-cmn 19741 df-abl 19742 df-mgp 20079 df-rng 20097 df-ur 20126 df-ring 20179 df-cring 20180 df-subrng 20487 df-subrg 20512 df-abv 20701 df-lmod 20749 df-scaf 20750 df-sra 21062 df-rgmod 21063 df-psmet 21275 df-xmet 21276 df-met 21277 df-bl 21278 df-mopn 21279 df-fbas 21280 df-fg 21281 df-cnfld 21284 df-top 22814 df-topon 22831 df-topsp 22853 df-bases 22867 df-cld 22941 df-ntr 22942 df-cls 22943 df-nei 23020 df-lp 23058 df-perf 23059 df-cn 23149 df-cnp 23150 df-haus 23237 df-tx 23484 df-hmeo 23677 df-fil 23768 df-fm 23860 df-flim 23861 df-flf 23862 df-tmd 23994 df-tgp 23995 df-tsms 24049 df-trg 24082 df-xms 24244 df-ms 24245 df-tms 24246 df-nm 24509 df-ngp 24510 df-nrg 24512 df-nlm 24513 df-ii 24815 df-cncf 24816 df-limc 25813 df-dv 25814 df-log 26508 df-esum 33704 |
This theorem is referenced by: esumcvg 33762 |
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