![]() |
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 13439 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
2 | ax-resscn 11169 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
3 | 1, 2 | sstri 3986 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
4 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
5 | ge0addcl 13443 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 + 𝑦) ∈ (0[,)+∞)) | |
6 | 5 | adantl 481 | . . 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 13437 | . . . 4 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
11 | 8, 9, 10 | sylanbrc 582 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
12 | 0e0icopnf 13441 | . . . 4 ⊢ 0 ∈ (0[,)+∞) | |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → 0 ∈ (0[,)+∞)) |
14 | 4, 6, 7, 11, 13 | fsumcllem 15684 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
15 | elrege0 13437 | . . 3 ⊢ (Σ𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞) ↔ (Σ𝑘 ∈ 𝐴 𝐵 ∈ ℝ ∧ 0 ≤ Σ𝑘 ∈ 𝐴 𝐵)) | |
16 | 15 | simprbi 496 | . 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 395 ∈ wcel 2098 ⊆ wss 3943 class class class wbr 5141 (class class class)co 7405 Fincfn 8941 ℂcc 11110 ℝcr 11111 0cc0 11112 + caddc 11115 +∞cpnf 11249 ≤ cle 11253 [,)cico 13332 Σcsu 15638 |
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 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rmo 3370 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-pss 3962 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-int 4944 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-tr 5259 df-id 5567 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-se 5625 df-we 5626 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-ord 6361 df-on 6362 df-lim 6363 df-suc 6364 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7853 df-1st 7974 df-2nd 7975 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-1o 8467 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-oi 9507 df-card 9936 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-rp 12981 df-ico 13336 df-fz 13491 df-fzo 13634 df-seq 13973 df-exp 14033 df-hash 14296 df-cj 15052 df-re 15053 df-im 15054 df-sqrt 15188 df-abs 15189 df-clim 15438 df-sum 15639 |
This theorem is referenced by: fsumless 15748 fsumle 15751 o1fsum 15765 rrxcph 25275 csbren 25282 trirn 25283 rrxmet 25291 rrxdstprj1 25292 itg1ge0 25570 itg1ge0a 25596 mtest 26295 abelthlem7 26330 abelthlem8 26331 ftalem4 26963 ftalem5 26964 chtge0 26999 vmadivsum 27370 vmadivsumb 27371 rpvmasumlem 27375 dchrvmasumlem2 27386 dchrisum0re 27401 rplogsum 27415 dirith2 27416 mulog2sumlem2 27423 vmalogdivsum2 27426 2vmadivsumlem 27428 selbergb 27437 selberg2b 27440 logdivbnd 27444 selberg3lem2 27446 selberg4lem1 27448 pntrlog2bndlem1 27465 pntrlog2bndlem2 27466 pntrlog2bnd 27472 pntpbnd1 27474 pntlemf 27493 axsegconlem3 28685 ax5seglem3 28697 sibfof 33869 eulerpartlemgc 33891 eulerpartlemb 33897 hgt750leme 34199 rrnmet 37210 rrndstprj1 37211 rrndstprj2 37212 sticksstones6 41525 fsumge0cl 44858 stoweidlem26 45311 stoweidlem38 45323 stoweidlem44 45329 etransclem35 45554 rrndistlt 45575 hoiqssbllem2 45908 |
Copyright terms: Public domain | W3C validator |