![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsumle | Structured version Visualization version GIF version |
Description: If all of the terms of finite sums compare, so do the sums. (Contributed by NM, 11-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumle.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumle.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fsumle.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) |
fsumle.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) |
Ref | Expression |
---|---|
fsumle | ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumle.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | fsumle.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℝ) | |
3 | fsumle.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
4 | 2, 3 | resubcld 11642 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (𝐶 − 𝐵) ∈ ℝ) |
5 | fsumle.4 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ≤ 𝐶) | |
6 | 2, 3 | subge0d 11804 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (0 ≤ (𝐶 − 𝐵) ↔ 𝐵 ≤ 𝐶)) |
7 | 5, 6 | mpbird 257 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ (𝐶 − 𝐵)) |
8 | 1, 4, 7 | fsumge0 15741 | . . 3 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 (𝐶 − 𝐵)) |
9 | 2 | recnd 11242 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ) |
10 | 3 | recnd 11242 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
11 | 1, 9, 10 | fsumsub 15734 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 (𝐶 − 𝐵) = (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵)) |
12 | 8, 11 | breqtrd 5175 | . 2 ⊢ (𝜑 → 0 ≤ (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵)) |
13 | 1, 2 | fsumrecl 15680 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐶 ∈ ℝ) |
14 | 1, 3 | fsumrecl 15680 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ) |
15 | 13, 14 | subge0d 11804 | . 2 ⊢ (𝜑 → (0 ≤ (Σ𝑘 ∈ 𝐴 𝐶 − Σ𝑘 ∈ 𝐴 𝐵) ↔ Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶)) |
16 | 12, 15 | mpbid 231 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 class class class wbr 5149 (class class class)co 7409 Fincfn 8939 ℝcr 11109 0cc0 11110 ≤ cle 11249 − cmin 11444 Σcsu 15632 |
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 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 |
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 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-ico 13330 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 |
This theorem is referenced by: o1fsum 15759 climcndslem1 15795 climcndslem2 15796 mertenslem1 15830 ovoliunlem1 25019 ovolicc2lem4 25037 uniioombllem4 25103 dvfsumle 25538 dvfsumabs 25540 mtest 25916 mtestbdd 25917 abelthlem7 25950 birthdaylem3 26458 fsumharmonic 26516 ftalem1 26577 ftalem5 26581 basellem8 26592 chtleppi 26713 chpub 26723 logfaclbnd 26725 bposlem1 26787 chebbnd1lem1 26972 chtppilimlem1 26976 vmadivsum 26985 rplogsumlem1 26987 rplogsumlem2 26988 rpvmasumlem 26990 dchrisumlem2 26993 dchrmusum2 26997 dchrvmasumlem3 27002 dchrvmasumiflem1 27004 dchrisum0fno1 27014 dchrisum0lem1 27019 dchrisum0lem2a 27020 mudivsum 27033 mulogsumlem 27034 mulog2sumlem2 27038 vmalogdivsum2 27041 2vmadivsumlem 27043 selberglem2 27049 selbergb 27052 selberg2b 27055 chpdifbndlem1 27056 logdivbnd 27059 selberg3lem1 27060 selberg4lem1 27063 pntrlog2bndlem1 27080 pntrlog2bndlem2 27081 pntrlog2bndlem3 27082 pntrlog2bndlem5 27084 pntrlog2bndlem6 27086 pntpbnd2 27090 pntlemj 27106 reprlt 33631 reprgt 33633 hgt750lemf 33665 hgt750lemb 33668 gg-dvfsumle 35182 knoppndvlem11 35398 geomcau 36627 lcmineqlem17 40910 fltnltalem 41404 stoweidlem11 44727 stoweidlem26 44742 stoweidlem38 44754 stirlinglem12 44801 etransclem23 44973 etransclem32 44982 sge0le 45123 hoidmvlelem2 45312 |
Copyright terms: Public domain | W3C validator |