Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > isumge0 | Structured version Visualization version GIF version |
Description: An infinite sum of nonnegative terms is nonnegative. (Contributed by Mario Carneiro, 28-Apr-2014.) |
Ref | Expression |
---|---|
isumrecl.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
isumrecl.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
isumrecl.3 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) |
isumrecl.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) |
isumrecl.5 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) |
isumge0.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) |
Ref | Expression |
---|---|
isumge0 | ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝑍 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isumrecl.1 | . . 3 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | isumrecl.2 | . . 3 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | isumrecl.3 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = 𝐴) | |
4 | isumrecl.4 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℝ) | |
5 | 4 | recnd 10650 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) |
6 | isumrecl.5 | . . . . 5 ⊢ (𝜑 → seq𝑀( + , 𝐹) ∈ dom ⇝ ) | |
7 | 1, 2, 3, 5, 6 | isumclim2 15093 | . . . 4 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑘 ∈ 𝑍 𝐴) |
8 | fveq2 6651 | . . . . . 6 ⊢ (𝑗 = 𝑘 → (𝐹‘𝑗) = (𝐹‘𝑘)) | |
9 | 8 | cbvsumv 15033 | . . . . 5 ⊢ Σ𝑗 ∈ 𝑍 (𝐹‘𝑗) = Σ𝑘 ∈ 𝑍 (𝐹‘𝑘) |
10 | 3 | sumeq2dv 15040 | . . . . 5 ⊢ (𝜑 → Σ𝑘 ∈ 𝑍 (𝐹‘𝑘) = Σ𝑘 ∈ 𝑍 𝐴) |
11 | 9, 10 | syl5eq 2867 | . . . 4 ⊢ (𝜑 → Σ𝑗 ∈ 𝑍 (𝐹‘𝑗) = Σ𝑘 ∈ 𝑍 𝐴) |
12 | 7, 11 | breqtrrd 5075 | . . 3 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ Σ𝑗 ∈ 𝑍 (𝐹‘𝑗)) |
13 | 3, 4 | eqeltrd 2911 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
14 | isumge0.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ 𝐴) | |
15 | 14, 3 | breqtrrd 5075 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
16 | 1, 2, 12, 13, 15 | iserge0 14997 | . 2 ⊢ (𝜑 → 0 ≤ Σ𝑗 ∈ 𝑍 (𝐹‘𝑗)) |
17 | 16, 11 | breqtrd 5073 | 1 ⊢ (𝜑 → 0 ≤ Σ𝑘 ∈ 𝑍 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 class class class wbr 5047 dom cdm 5536 ‘cfv 6336 ℝcr 10517 0cc0 10518 + caddc 10521 ≤ cle 10657 ℤcz 11963 ℤ≥cuz 12225 seqcseq 13354 ⇝ cli 14821 Σcsu 15022 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442 ax-inf2 9085 ax-cnex 10574 ax-resscn 10575 ax-1cn 10576 ax-icn 10577 ax-addcl 10578 ax-addrcl 10579 ax-mulcl 10580 ax-mulrcl 10581 ax-mulcom 10582 ax-addass 10583 ax-mulass 10584 ax-distr 10585 ax-i2m1 10586 ax-1ne0 10587 ax-1rid 10588 ax-rnegex 10589 ax-rrecex 10590 ax-cnre 10591 ax-pre-lttri 10592 ax-pre-lttrn 10593 ax-pre-ltadd 10594 ax-pre-mulgt0 10595 ax-pre-sup 10596 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-nel 3119 df-ral 3138 df-rex 3139 df-reu 3140 df-rmo 3141 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-pss 3937 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-tp 4553 df-op 4555 df-uni 4820 df-int 4858 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-tr 5154 df-id 5441 df-eprel 5446 df-po 5455 df-so 5456 df-fr 5495 df-se 5496 df-we 5497 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6129 df-ord 6175 df-on 6176 df-lim 6177 df-suc 6178 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-isom 6345 df-riota 7095 df-ov 7140 df-oprab 7141 df-mpo 7142 df-om 7562 df-1st 7670 df-2nd 7671 df-wrecs 7928 df-recs 7989 df-rdg 8027 df-1o 8083 df-oadd 8087 df-er 8270 df-pm 8390 df-en 8491 df-dom 8492 df-sdom 8493 df-fin 8494 df-sup 8887 df-inf 8888 df-oi 8955 df-card 9349 df-pnf 10658 df-mnf 10659 df-xr 10660 df-ltxr 10661 df-le 10662 df-sub 10853 df-neg 10854 df-div 11279 df-nn 11620 df-2 11682 df-3 11683 df-n0 11880 df-z 11964 df-uz 12226 df-rp 12372 df-fz 12878 df-fzo 13019 df-fl 13147 df-seq 13355 df-exp 13415 df-hash 13676 df-cj 14438 df-re 14439 df-im 14440 df-sqrt 14574 df-abs 14575 df-clim 14825 df-rlim 14826 df-sum 15023 |
This theorem is referenced by: mertenslem1 15220 mertenslem2 15221 rpnnen2lem12 15558 log2tlbnd 25504 esumcvg 31347 stirlinglem12 42455 |
Copyright terms: Public domain | W3C validator |