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Mirrors > Home > MPE Home > Th. List > fsumless | Structured version Visualization version GIF version |
Description: A shorter sum of nonnegative terms is smaller than a longer one. (Contributed by NM, 26-Dec-2005.) (Proof shortened by Mario Carneiro, 24-Apr-2014.) |
Ref | Expression |
---|---|
fsumge0.1 | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fsumge0.2 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fsumge0.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
fsumless.4 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
Ref | Expression |
---|---|
fsumless | ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fsumge0.1 | . . . . 5 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
2 | difss 4145 | . . . . 5 ⊢ (𝐴 ∖ 𝐶) ⊆ 𝐴 | |
3 | ssfi 9211 | . . . . 5 ⊢ ((𝐴 ∈ Fin ∧ (𝐴 ∖ 𝐶) ⊆ 𝐴) → (𝐴 ∖ 𝐶) ∈ Fin) | |
4 | 1, 2, 3 | sylancl 586 | . . . 4 ⊢ (𝜑 → (𝐴 ∖ 𝐶) ∈ Fin) |
5 | eldifi 4140 | . . . . 5 ⊢ (𝑘 ∈ (𝐴 ∖ 𝐶) → 𝑘 ∈ 𝐴) | |
6 | fsumge0.2 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
7 | 5, 6 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → 𝐵 ∈ ℝ) |
8 | fsumge0.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) | |
9 | 5, 8 | sylan2 593 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐶)) → 0 ≤ 𝐵) |
10 | 4, 7, 9 | fsumge0 15827 | . . 3 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵) |
11 | fsumless.4 | . . . . . 6 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
12 | 1, 11 | ssfid 9298 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ Fin) |
13 | 11 | sselda 3994 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝑘 ∈ 𝐴) |
14 | 13, 6 | syldan 591 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐶) → 𝐵 ∈ ℝ) |
15 | 12, 14 | fsumrecl 15766 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ∈ ℝ) |
16 | 4, 7 | fsumrecl 15766 | . . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵 ∈ ℝ) |
17 | 15, 16 | addge01d 11848 | . . 3 ⊢ (𝜑 → (0 ≤ Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵 ↔ Σ𝑘 ∈ 𝐶 𝐵 ≤ (Σ𝑘 ∈ 𝐶 𝐵 + Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵))) |
18 | 10, 17 | mpbid 232 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ (Σ𝑘 ∈ 𝐶 𝐵 + Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵)) |
19 | disjdif 4477 | . . . 4 ⊢ (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅ | |
20 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → (𝐶 ∩ (𝐴 ∖ 𝐶)) = ∅) |
21 | undif 4487 | . . . . 5 ⊢ (𝐶 ⊆ 𝐴 ↔ (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴) | |
22 | 11, 21 | sylib 218 | . . . 4 ⊢ (𝜑 → (𝐶 ∪ (𝐴 ∖ 𝐶)) = 𝐴) |
23 | 22 | eqcomd 2740 | . . 3 ⊢ (𝜑 → 𝐴 = (𝐶 ∪ (𝐴 ∖ 𝐶))) |
24 | 6 | recnd 11286 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℂ) |
25 | 20, 23, 1, 24 | fsumsplit 15773 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 = (Σ𝑘 ∈ 𝐶 𝐵 + Σ𝑘 ∈ (𝐴 ∖ 𝐶)𝐵)) |
26 | 18, 25 | breqtrrd 5175 | 1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐶 𝐵 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∖ cdif 3959 ∪ cun 3960 ∩ cin 3961 ⊆ wss 3962 ∅c0 4338 class class class wbr 5147 (class class class)co 7430 Fincfn 8983 ℝcr 11151 0cc0 11152 + caddc 11155 ≤ cle 11293 Σcsu 15718 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-n0 12524 df-z 12611 df-uz 12876 df-rp 13032 df-ico 13389 df-fz 13544 df-fzo 13691 df-seq 14039 df-exp 14099 df-hash 14366 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-clim 15520 df-sum 15719 |
This theorem is referenced by: fsumge1 15829 fsum00 15830 ovolicc2lem4 25568 fsumharmonic 27069 chtwordi 27213 chpwordi 27214 chtlepsi 27264 chtublem 27269 perfectlem2 27288 chtppilimlem1 27531 vmadivsumb 27541 rplogsumlem2 27543 rpvmasumlem 27545 dchrvmasumiflem1 27559 rplogsum 27585 dirith2 27586 mulog2sumlem2 27593 selbergb 27607 selberg2b 27610 chpdifbndlem1 27611 logdivbnd 27614 selberg3lem2 27616 pntrsumbnd 27624 pntlemf 27663 fsumiunle 32835 esumpcvgval 34058 eulerpartlemgc 34343 reprinfz1 34615 hgt750lemb 34649 fsumlessf 45532 sge0fsum 46342 sge0xaddlem1 46388 sge0seq 46401 carageniuncllem2 46477 perfectALTVlem2 47646 |
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