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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 13276 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | ovexd 7385 | . . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V) | |
3 | elfznn 13399 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ) | |
4 | icossicc 13282 | . . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
5 | esumpmono.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) | |
6 | 4, 5 | sselid 3941 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
7 | 3, 6 | sylan2 594 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞)) |
8 | 7 | ralrimiva 3142 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
9 | nfcv 2906 | . . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀) | |
10 | 9 | esumcl 32390 | . . . . . 6 ⊢ (((1...𝑀) ∈ V ∧ ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
11 | 2, 8, 10 | syl2anc 585 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
12 | 1, 11 | sselid 3941 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ*) |
13 | 12 | xrleidd 13000 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑀)𝐴) |
14 | ovexd 7385 | . . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ V) | |
15 | esumpmono.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
16 | 15 | adantr 482 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ) |
17 | peano2nn 12099 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ) | |
18 | nnuz 12735 | . . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1) | |
19 | 17, 18 | eleqtrdi 2849 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘1)) |
20 | fzss1 13409 | . . . . . . . . . 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 486 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ((𝑀 + 1)...𝑁)) | |
23 | 21, 22 | sseldd 3944 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁)) |
24 | elfznn 13399 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ) |
26 | 25, 6 | syldan 592 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ (0[,]+∞)) |
27 | 26 | ralrimiva 3142 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
28 | nfcv 2906 | . . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁) | |
29 | 28 | esumcl 32390 | . . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ V ∧ ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
30 | 14, 27, 29 | syl2anc 585 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
31 | elxrge0 13303 | . . . . 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 498 | . . . 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 11136 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
36 | 1, 30 | sselid 3941 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ*) |
37 | xle2add 13107 | . . . 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 698 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 0) ≤ (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
40 | xaddid1 13089 | . . . 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 2744 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 0)) |
43 | 15, 18 | eleqtrdi 2849 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
44 | esumpmono.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
45 | eluzfz 13365 | . . . . 5 ⊢ ((𝑀 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (1...𝑁)) | |
46 | 43, 44, 45 | syl2anc 585 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
47 | fzsplit 13396 | . . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
48 | esumeq1 32394 | . . . 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 1918 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
51 | nnre 12094 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
52 | 51 | ltp1d 12019 | . . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1)) |
53 | fzdisj 13397 | . . . . 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 32413 | . . 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 2778 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑁)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
57 | 39, 42, 56 | 3brtr4d 5136 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑁)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3063 Vcvv 3444 ∪ cun 3907 ∩ cin 3908 ⊆ wss 3909 ∅c0 4281 class class class wbr 5104 ‘cfv 6492 (class class class)co 7350 0cc0 10985 1c1 10986 + caddc 10988 +∞cpnf 11120 ℝ*cxr 11122 < clt 11123 ≤ cle 11124 ℕcn 12087 ℤ≥cuz 12696 +𝑒 cxad 12960 [,)cico 13195 [,]cicc 13196 ...cfz 13353 Σ*cesum 32387 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2709 ax-rep 5241 ax-sep 5255 ax-nul 5262 ax-pow 5319 ax-pr 5383 ax-un 7663 ax-inf2 9511 ax-cnex 11041 ax-resscn 11042 ax-1cn 11043 ax-icn 11044 ax-addcl 11045 ax-addrcl 11046 ax-mulcl 11047 ax-mulrcl 11048 ax-mulcom 11049 ax-addass 11050 ax-mulass 11051 ax-distr 11052 ax-i2m1 11053 ax-1ne0 11054 ax-1rid 11055 ax-rnegex 11056 ax-rrecex 11057 ax-cnre 11058 ax-pre-lttri 11059 ax-pre-lttrn 11060 ax-pre-ltadd 11061 ax-pre-mulgt0 11062 ax-pre-sup 11063 ax-addf 11064 ax-mulf 11065 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3064 df-rex 3073 df-rmo 3352 df-reu 3353 df-rab 3407 df-v 3446 df-sbc 3739 df-csb 3855 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-tp 4590 df-op 4592 df-uni 4865 df-int 4907 df-iun 4955 df-iin 4956 df-br 5105 df-opab 5167 df-mpt 5188 df-tr 5222 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-se 5587 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6250 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6444 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 7306 df-ov 7353 df-oprab 7354 df-mpo 7355 df-of 7608 df-om 7794 df-1st 7912 df-2nd 7913 df-supp 8061 df-frecs 8180 df-wrecs 8211 df-recs 8285 df-rdg 8324 df-1o 8380 df-2o 8381 df-er 8582 df-map 8701 df-pm 8702 df-ixp 8770 df-en 8818 df-dom 8819 df-sdom 8820 df-fin 8821 df-fsupp 9240 df-fi 9281 df-sup 9312 df-inf 9313 df-oi 9380 df-card 9809 df-pnf 11125 df-mnf 11126 df-xr 11127 df-ltxr 11128 df-le 11129 df-sub 11321 df-neg 11322 df-div 11747 df-nn 12088 df-2 12150 df-3 12151 df-4 12152 df-5 12153 df-6 12154 df-7 12155 df-8 12156 df-9 12157 df-n0 12348 df-z 12434 df-dec 12552 df-uz 12697 df-q 12803 df-rp 12845 df-xneg 12962 df-xadd 12963 df-xmul 12964 df-ioo 13197 df-ioc 13198 df-ico 13199 df-icc 13200 df-fz 13354 df-fzo 13497 df-fl 13626 df-mod 13704 df-seq 13836 df-exp 13897 df-fac 14102 df-bc 14131 df-hash 14159 df-shft 14886 df-cj 14918 df-re 14919 df-im 14920 df-sqrt 15054 df-abs 15055 df-limsup 15288 df-clim 15305 df-rlim 15306 df-sum 15506 df-ef 15885 df-sin 15887 df-cos 15888 df-pi 15890 df-struct 16954 df-sets 16971 df-slot 16989 df-ndx 17001 df-base 17019 df-ress 17048 df-plusg 17081 df-mulr 17082 df-starv 17083 df-sca 17084 df-vsca 17085 df-ip 17086 df-tset 17087 df-ple 17088 df-ds 17090 df-unif 17091 df-hom 17092 df-cco 17093 df-rest 17239 df-topn 17240 df-0g 17258 df-gsum 17259 df-topgen 17260 df-pt 17261 df-prds 17264 df-ordt 17318 df-xrs 17319 df-qtop 17324 df-imas 17325 df-xps 17327 df-mre 17401 df-mrc 17402 df-acs 17404 df-ps 18390 df-tsr 18391 df-plusf 18431 df-mgm 18432 df-sgrp 18481 df-mnd 18492 df-mhm 18536 df-submnd 18537 df-grp 18686 df-minusg 18687 df-sbg 18688 df-mulg 18807 df-subg 18858 df-cntz 19029 df-cmn 19493 df-abl 19494 df-mgp 19826 df-ur 19843 df-ring 19890 df-cring 19891 df-subrg 20143 df-abv 20199 df-lmod 20247 df-scaf 20248 df-sra 20556 df-rgmod 20557 df-psmet 20711 df-xmet 20712 df-met 20713 df-bl 20714 df-mopn 20715 df-fbas 20716 df-fg 20717 df-cnfld 20720 df-top 22165 df-topon 22182 df-topsp 22204 df-bases 22218 df-cld 22292 df-ntr 22293 df-cls 22294 df-nei 22371 df-lp 22409 df-perf 22410 df-cn 22500 df-cnp 22501 df-haus 22588 df-tx 22835 df-hmeo 23028 df-fil 23119 df-fm 23211 df-flim 23212 df-flf 23213 df-tmd 23345 df-tgp 23346 df-tsms 23400 df-trg 23433 df-xms 23595 df-ms 23596 df-tms 23597 df-nm 23860 df-ngp 23861 df-nrg 23863 df-nlm 23864 df-ii 24162 df-cncf 24163 df-limc 25152 df-dv 25153 df-log 25834 df-esum 32388 |
This theorem is referenced by: esumcvg 32446 |
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