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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 13471 | . . . . 5 ⊢ (0[,]+∞) ⊆ ℝ* | |
| 2 | ovexd 7467 | . . . . . 6 ⊢ (𝜑 → (1...𝑀) ∈ V) | |
| 3 | elfznn 13594 | . . . . . . . 8 ⊢ (𝑘 ∈ (1...𝑀) → 𝑘 ∈ ℕ) | |
| 4 | icossicc 13477 | . . . . . . . . 9 ⊢ (0[,)+∞) ⊆ (0[,]+∞) | |
| 5 | esumpmono.3 | . . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,)+∞)) | |
| 6 | 4, 5 | sselid 3980 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → 𝐴 ∈ (0[,]+∞)) | 
| 7 | 3, 6 | sylan2 593 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (1...𝑀)) → 𝐴 ∈ (0[,]+∞)) | 
| 8 | 7 | ralrimiva 3145 | . . . . . 6 ⊢ (𝜑 → ∀𝑘 ∈ (1...𝑀)𝐴 ∈ (0[,]+∞)) | 
| 9 | nfcv 2904 | . . . . . . 7 ⊢ Ⅎ𝑘(1...𝑀) | |
| 10 | 9 | esumcl 34032 | . . . . . 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 3980 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ∈ ℝ*) | 
| 13 | 12 | xrleidd 13195 | . . 3 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 ≤ Σ*𝑘 ∈ (1...𝑀)𝐴) | 
| 14 | ovexd 7467 | . . . . 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 12279 | . . . . . . . . . . 11 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ ℕ) | |
| 18 | nnuz 12922 | . . . . . . . . . . 11 ⊢ ℕ = (ℤ≥‘1) | |
| 19 | 17, 18 | eleqtrdi 2850 | . . . . . . . . . 10 ⊢ (𝑀 ∈ ℕ → (𝑀 + 1) ∈ (ℤ≥‘1)) | 
| 20 | fzss1 13604 | . . . . . . . . . 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 3983 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ ((𝑀 + 1)...𝑁)) → 𝑘 ∈ (1...𝑁)) | 
| 24 | elfznn 13594 | . . . . . . . 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 3145 | . . . . 5 ⊢ (𝜑 → ∀𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ (0[,]+∞)) | 
| 28 | nfcv 2904 | . . . . . 6 ⊢ Ⅎ𝑘((𝑀 + 1)...𝑁) | |
| 29 | 28 | esumcl 34032 | . . . . 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 13498 | . . . . 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 11309 | . . . . 5 ⊢ 0 ∈ ℝ* | |
| 35 | 34 | a1i 11 | . . . 4 ⊢ (𝜑 → 0 ∈ ℝ*) | 
| 36 | 1, 30 | sselid 3980 | . . . 4 ⊢ (𝜑 → Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴 ∈ ℝ*) | 
| 37 | xle2add 13302 | . . . 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 13284 | . . . 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 2742 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑀)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 0)) | 
| 43 | 15, 18 | eleqtrdi 2850 | . . . . 5 ⊢ (𝜑 → 𝑀 ∈ (ℤ≥‘1)) | 
| 44 | esumpmono.2 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | |
| 45 | eluzfz 13560 | . . . . 5 ⊢ ((𝑀 ∈ (ℤ≥‘1) ∧ 𝑁 ∈ (ℤ≥‘𝑀)) → 𝑀 ∈ (1...𝑁)) | |
| 46 | 43, 44, 45 | syl2anc 584 | . . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑁)) | 
| 47 | fzsplit 13591 | . . . 4 ⊢ (𝑀 ∈ (1...𝑁) → (1...𝑁) = ((1...𝑀) ∪ ((𝑀 + 1)...𝑁))) | |
| 48 | esumeq1 34036 | . . . 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 1913 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
| 51 | nnre 12274 | . . . . . 6 ⊢ (𝑀 ∈ ℕ → 𝑀 ∈ ℝ) | |
| 52 | 51 | ltp1d 12199 | . . . . 5 ⊢ (𝑀 ∈ ℕ → 𝑀 < (𝑀 + 1)) | 
| 53 | fzdisj 13592 | . . . . 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 34055 | . . 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 2776 | . 2 ⊢ (𝜑 → Σ*𝑘 ∈ (1...𝑁)𝐴 = (Σ*𝑘 ∈ (1...𝑀)𝐴 +𝑒 Σ*𝑘 ∈ ((𝑀 + 1)...𝑁)𝐴)) | 
| 57 | 39, 42, 56 | 3brtr4d 5174 | 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 1539 ∈ wcel 2107 ∀wral 3060 Vcvv 3479 ∪ cun 3948 ∩ cin 3949 ⊆ wss 3950 ∅c0 4332 class class class wbr 5142 ‘cfv 6560 (class class class)co 7432 0cc0 11156 1c1 11157 + caddc 11159 +∞cpnf 11293 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 ℕcn 12267 ℤ≥cuz 12879 +𝑒 cxad 13153 [,)cico 13390 [,]cicc 13391 ...cfz 13548 Σ*cesum 34029 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 ax-addf 11235 ax-mulf 11236 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-tp 4630 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-of 7698 df-om 7889 df-1st 8015 df-2nd 8016 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-1o 8507 df-2o 8508 df-er 8746 df-map 8869 df-pm 8870 df-ixp 8939 df-en 8987 df-dom 8988 df-sdom 8989 df-fin 8990 df-fsupp 9403 df-fi 9452 df-sup 9483 df-inf 9484 df-oi 9551 df-card 9980 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-2 12330 df-3 12331 df-4 12332 df-5 12333 df-6 12334 df-7 12335 df-8 12336 df-9 12337 df-n0 12529 df-z 12616 df-dec 12736 df-uz 12880 df-q 12992 df-rp 13036 df-xneg 13155 df-xadd 13156 df-xmul 13157 df-ioo 13392 df-ioc 13393 df-ico 13394 df-icc 13395 df-fz 13549 df-fzo 13696 df-fl 13833 df-mod 13911 df-seq 14044 df-exp 14104 df-fac 14314 df-bc 14343 df-hash 14371 df-shft 15107 df-cj 15139 df-re 15140 df-im 15141 df-sqrt 15275 df-abs 15276 df-limsup 15508 df-clim 15525 df-rlim 15526 df-sum 15724 df-ef 16104 df-sin 16106 df-cos 16107 df-pi 16109 df-struct 17185 df-sets 17202 df-slot 17220 df-ndx 17232 df-base 17249 df-ress 17276 df-plusg 17311 df-mulr 17312 df-starv 17313 df-sca 17314 df-vsca 17315 df-ip 17316 df-tset 17317 df-ple 17318 df-ds 17320 df-unif 17321 df-hom 17322 df-cco 17323 df-rest 17468 df-topn 17469 df-0g 17487 df-gsum 17488 df-topgen 17489 df-pt 17490 df-prds 17493 df-ordt 17547 df-xrs 17548 df-qtop 17553 df-imas 17554 df-xps 17556 df-mre 17630 df-mrc 17631 df-acs 17633 df-ps 18612 df-tsr 18613 df-plusf 18653 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-mhm 18797 df-submnd 18798 df-grp 18955 df-minusg 18956 df-sbg 18957 df-mulg 19087 df-subg 19142 df-cntz 19336 df-cmn 19801 df-abl 19802 df-mgp 20139 df-rng 20151 df-ur 20180 df-ring 20233 df-cring 20234 df-subrng 20547 df-subrg 20571 df-abv 20811 df-lmod 20861 df-scaf 20862 df-sra 21173 df-rgmod 21174 df-psmet 21357 df-xmet 21358 df-met 21359 df-bl 21360 df-mopn 21361 df-fbas 21362 df-fg 21363 df-cnfld 21366 df-top 22901 df-topon 22918 df-topsp 22940 df-bases 22954 df-cld 23028 df-ntr 23029 df-cls 23030 df-nei 23107 df-lp 23145 df-perf 23146 df-cn 23236 df-cnp 23237 df-haus 23324 df-tx 23571 df-hmeo 23764 df-fil 23855 df-fm 23947 df-flim 23948 df-flf 23949 df-tmd 24081 df-tgp 24082 df-tsms 24136 df-trg 24169 df-xms 24331 df-ms 24332 df-tms 24333 df-nm 24596 df-ngp 24597 df-nrg 24599 df-nlm 24600 df-ii 24904 df-cncf 24905 df-limc 25902 df-dv 25903 df-log 26599 df-esum 34030 | 
| This theorem is referenced by: esumcvg 34088 | 
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