![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > fsumge0 | Structured version Visualization version GIF version |
Description: If all of the terms of a finite sum are nonnegative, so is the sum. (Contributed by NM, 26-Dec-2005.) (Revised 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) |
Ref | Expression |
---|---|
fsumge0 | ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rge0ssre 13473 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | ax-resscn 11203 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3991 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
5 | ge0addcl 13477 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞)) | |
6 | 5 | adantl 480 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 + 𝑦) ∈ (0[,)+∞)) |
7 | fsumge0.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
8 | fsumge0.2 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
9 | fsumge0.3 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) | |
10 | elrege0 13471 | . . . 4 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
11 | 8, 9, 10 | sylanbrc 581 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
12 | 0e0icopnf 13475 | . . . 4 ⊢ 0 ∈ (0[,)+∞) | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
14 | 4, 6, 7, 11, 13 | fsumcllem 15718 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
15 | elrege0 13471 | . . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) ↔ (Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ 𝐴 𝐵)) | |
16 | 15 | simprbi 495 | . 2 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) → 0 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
17 | 14, 16 | syl 17 | 1 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ⊆ wss 3949 class class class wbr 5152 (class class class)co 7426 Fincfn 8970 ℂcc 11144 ℝcr 11145 0cc0 11146 + caddc 11149 +∞cpnf 11283 ≤ cle 11287 [,)cico 13366 Σcsu 15672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-ico 13370 df-fz 13525 df-fzo 13668 df-seq 14007 df-exp 14067 df-hash 14330 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-clim 15472 df-sum 15673 |
This theorem is referenced by: fsumless 15782 fsumle 15785 o1fsum 15799 rrxcph 25340 csbren 25347 trirn 25348 rrxmet 25356 rrxdstprj1 25357 itg1ge0 25635 itg1ge0a 25661 mtest 26360 abelthlem7 26395 abelthlem8 26396 ftalem4 27028 ftalem5 27029 chtge0 27064 vmadivsum 27435 vmadivsumb 27436 rpvmasumlem 27440 dchrvmasumlem2 27451 dchrisum0re 27466 rplogsum 27480 dirith2 27481 mulog2sumlem2 27488 vmalogdivsum2 27491 2vmadivsumlem 27493 selbergb 27502 selberg2b 27505 logdivbnd 27509 selberg3lem2 27511 selberg4lem1 27513 pntrlog2bndlem1 27530 pntrlog2bndlem2 27531 pntrlog2bnd 27537 pntpbnd1 27539 pntlemf 27558 axsegconlem3 28750 ax5seglem3 28762 sibfof 33993 eulerpartlemgc 34015 eulerpartlemb 34021 hgt750leme 34323 rrnmet 37335 rrndstprj1 37336 rrndstprj2 37337 sticksstones6 41655 fsumge0cl 44990 stoweidlem26 45443 stoweidlem38 45455 stoweidlem44 45461 etransclem35 45686 rrndistlt 45707 hoiqssbllem2 46040 |
Copyright terms: Public domain | W3C validator |