Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
||
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 12813 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
2 | ovexd 7185 | . . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V) | |
3 | elfznn 12930 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ) | |
4 | icossicc 12818 | . . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
5 | esumpmono.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) | |
6 | 4, 5 | sseldi 3965 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) |
7 | 3, 6 | sylan2 594 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞)) |
8 | 7 | ralrimiva 3182 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
9 | nfcv 2977 | . . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀) | |
10 | 9 | esumcl 31284 | . . . . . 6 ⊢ (((1...𝑀) ∈ V ∧ ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
11 | 2, 8, 10 | syl2anc 586 | . . . . 5 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) |
12 | 1, 11 | sseldi 3965 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ*) |
13 | 12 | xrleidd 12539 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑀)𝐴) |
14 | ovexd 7185 | . . . . 5 ⊢ (𝜑 → ((𝑀 + 1)...𝑁) ∈ V) | |
15 | esumpmono.1 | . . . . . . . . . . 11 ⊢ (𝜑 → 𝑀 ∈ ℕ) | |
16 | 15 | adantr 483 | . . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑀 ∈ ℕ) |
17 | peano2nn 11644 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ) | |
18 | nnuz 12275 | . . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1) | |
19 | 17, 18 | eleqtrdi 2923 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘1)) |
20 | fzss1 12940 | . . . . . . . . . 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 487 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ((𝑀 + 1)...𝑁)) | |
23 | 21, 22 | sseldd 3968 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁)) |
24 | elfznn 12930 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑁) → 𝑘 ∈ ℕ) | |
25 | 23, 24 | syl 17 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ ℕ) |
26 | 25, 6 | syldan 593 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝐴 ∈ (0[,]+∞)) |
27 | 26 | ralrimiva 3182 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
28 | nfcv 2977 | . . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁) | |
29 | 28 | esumcl 31284 | . . . . 5 ⊢ ((((𝑀 + 1)...𝑁) ∈ V ∧ ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
30 | 14, 27, 29 | syl2anc 586 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) |
31 | elxrge0 12839 | . . . . 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 499 | . . . 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 10682 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 34 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) |
36 | 1, 30 | sseldi 3965 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ*) |
37 | xle2add 12646 | . . . 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 697 | . 2 ⊢ (𝜑 → (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 0) ≤ (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
40 | xaddid1 12628 | . . . 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 2827 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 0)) |
43 | 15, 18 | eleqtrdi 2923 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) |
44 | esumpmono.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
45 | eluzfz 12897 | . . . . 5 ⊢ ((𝑀 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (1...𝑁)) | |
46 | 43, 44, 45 | syl2anc 586 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) |
47 | fzsplit 12927 | . . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
48 | esumeq1 31288 | . . . 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 1911 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
51 | nnre 11639 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
52 | 51 | ltp1d 11564 | . . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1)) |
53 | fzdisj 12928 | . . . . 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 31307 | . . 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 2856 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑁)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) |
57 | 39, 42, 56 | 3brtr4d 5091 | 1 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑁)𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 Vcvv 3495 ∪ cun 3934 ∩ cin 3935 ⊆ wss 3936 ∅c0 4291 class class class wbr 5059 ‘cfv 6350 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 ℕcn 11632 ℤ≥cuz 12237 +𝑒 cxad 12499 [,)cico 12734 [,]cicc 12735 ...cfz 12886 Σ*cesum 31281 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-int 4870 df-iun 4914 df-iin 4915 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-se 5510 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-isom 6359 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-of 7403 df-om 7575 df-1st 7683 df-2nd 7684 df-supp 7825 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-er 8283 df-map 8402 df-pm 8403 df-ixp 8456 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-fsupp 8828 df-fi 8869 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-q 12343 df-rp 12384 df-xneg 12501 df-xadd 12502 df-xmul 12503 df-ioo 12736 df-ioc 12737 df-ico 12738 df-icc 12739 df-fz 12887 df-fzo 13028 df-fl 13156 df-mod 13232 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 df-sin 15417 df-cos 15418 df-pi 15420 df-struct 16479 df-ndx 16480 df-slot 16481 df-base 16483 df-sets 16484 df-ress 16485 df-plusg 16572 df-mulr 16573 df-starv 16574 df-sca 16575 df-vsca 16576 df-ip 16577 df-tset 16578 df-ple 16579 df-ds 16581 df-unif 16582 df-hom 16583 df-cco 16584 df-rest 16690 df-topn 16691 df-0g 16709 df-gsum 16710 df-topgen 16711 df-pt 16712 df-prds 16715 df-ordt 16768 df-xrs 16769 df-qtop 16774 df-imas 16775 df-xps 16777 df-mre 16851 df-mrc 16852 df-acs 16854 df-ps 17804 df-tsr 17805 df-plusf 17845 df-mgm 17846 df-sgrp 17895 df-mnd 17906 df-mhm 17950 df-submnd 17951 df-grp 18100 df-minusg 18101 df-sbg 18102 df-mulg 18219 df-subg 18270 df-cntz 18441 df-cmn 18902 df-abl 18903 df-mgp 19234 df-ur 19246 df-ring 19293 df-cring 19294 df-subrg 19527 df-abv 19582 df-lmod 19630 df-scaf 19631 df-sra 19938 df-rgmod 19939 df-psmet 20531 df-xmet 20532 df-met 20533 df-bl 20534 df-mopn 20535 df-fbas 20536 df-fg 20537 df-cnfld 20540 df-top 21496 df-topon 21513 df-topsp 21535 df-bases 21548 df-cld 21621 df-ntr 21622 df-cls 21623 df-nei 21700 df-lp 21738 df-perf 21739 df-cn 21829 df-cnp 21830 df-haus 21917 df-tx 22164 df-hmeo 22357 df-fil 22448 df-fm 22540 df-flim 22541 df-flf 22542 df-tmd 22674 df-tgp 22675 df-tsms 22729 df-trg 22762 df-xms 22924 df-ms 22925 df-tms 22926 df-nm 23186 df-ngp 23187 df-nrg 23189 df-nlm 23190 df-ii 23479 df-cncf 23480 df-limc 24458 df-dv 24459 df-log 25134 df-esum 31282 |
This theorem is referenced by: esumcvg 31340 |
Copyright terms: Public domain | W3C validator |