Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > iserge0 | GIF version |
Description: The limit of an infinite series of nonnegative reals is nonnegative. (Contributed by Paul Chapman, 9-Feb-2008.) (Revised by Mario Carneiro, 3-Feb-2014.) |
Ref | Expression |
---|---|
clim2iser.1 | ⊢ 𝑍 = (ℤ≥‘𝑀) |
iserge0.2 | ⊢ (𝜑 → 𝑀 ∈ ℤ) |
iserge0.3 | ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) |
iserge0.4 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) |
iserge0.5 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) |
Ref | Expression |
---|---|
iserge0 | ⊢ (𝜑 → 0 ≤ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clim2iser.1 | . 2 ⊢ 𝑍 = (ℤ≥‘𝑀) | |
2 | iserge0.2 | . 2 ⊢ (𝜑 → 𝑀 ∈ ℤ) | |
3 | serclim0 11029 | . . 3 ⊢ (𝑀 ∈ ℤ → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) | |
4 | 2, 3 | syl 14 | . 2 ⊢ (𝜑 → seq𝑀( + , ((ℤ≥‘𝑀) × {0})) ⇝ 0) |
5 | iserge0.3 | . 2 ⊢ (𝜑 → seq𝑀( + , 𝐹) ⇝ 𝐴) | |
6 | simpr 109 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ 𝑍) | |
7 | 6, 1 | eleqtrdi 2210 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ (ℤ≥‘𝑀)) |
8 | c0ex 7728 | . . . . 5 ⊢ 0 ∈ V | |
9 | 8 | fvconst2 5604 | . . . 4 ⊢ (𝑘 ∈ (ℤ≥‘𝑀) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) |
10 | 7, 9 | syl 14 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((ℤ≥‘𝑀) × {0})‘𝑘) = 0) |
11 | 0re 7734 | . . 3 ⊢ 0 ∈ ℝ | |
12 | 10, 11 | syl6eqel 2208 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((ℤ≥‘𝑀) × {0})‘𝑘) ∈ ℝ) |
13 | iserge0.4 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℝ) | |
14 | iserge0.5 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 0 ≤ (𝐹‘𝑘)) | |
15 | 10, 14 | eqbrtrd 3920 | . 2 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((ℤ≥‘𝑀) × {0})‘𝑘) ≤ (𝐹‘𝑘)) |
16 | 1, 2, 4, 5, 12, 13, 15 | iserle 11066 | 1 ⊢ (𝜑 → 0 ≤ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 = wceq 1316 ∈ wcel 1465 {csn 3497 class class class wbr 3899 × cxp 4507 ‘cfv 5093 ℝcr 7587 0cc0 7588 + caddc 7591 ≤ cle 7769 ℤcz 9012 ℤ≥cuz 9282 seqcseq 10173 ⇝ cli 11002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 588 ax-in2 589 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-13 1476 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-coll 4013 ax-sep 4016 ax-nul 4024 ax-pow 4068 ax-pr 4101 ax-un 4325 ax-setind 4422 ax-iinf 4472 ax-cnex 7679 ax-resscn 7680 ax-1cn 7681 ax-1re 7682 ax-icn 7683 ax-addcl 7684 ax-addrcl 7685 ax-mulcl 7686 ax-mulrcl 7687 ax-addcom 7688 ax-mulcom 7689 ax-addass 7690 ax-mulass 7691 ax-distr 7692 ax-i2m1 7693 ax-0lt1 7694 ax-1rid 7695 ax-0id 7696 ax-rnegex 7697 ax-precex 7698 ax-cnre 7699 ax-pre-ltirr 7700 ax-pre-ltwlin 7701 ax-pre-lttrn 7702 ax-pre-apti 7703 ax-pre-ltadd 7704 ax-pre-mulgt0 7705 ax-pre-mulext 7706 ax-arch 7707 ax-caucvg 7708 |
This theorem depends on definitions: df-bi 116 df-dc 805 df-3or 948 df-3an 949 df-tru 1319 df-fal 1322 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ne 2286 df-nel 2381 df-ral 2398 df-rex 2399 df-reu 2400 df-rmo 2401 df-rab 2402 df-v 2662 df-sbc 2883 df-csb 2976 df-dif 3043 df-un 3045 df-in 3047 df-ss 3054 df-nul 3334 df-if 3445 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-int 3742 df-iun 3785 df-br 3900 df-opab 3960 df-mpt 3961 df-tr 3997 df-id 4185 df-po 4188 df-iso 4189 df-iord 4258 df-on 4260 df-ilim 4261 df-suc 4263 df-iom 4475 df-xp 4515 df-rel 4516 df-cnv 4517 df-co 4518 df-dm 4519 df-rn 4520 df-res 4521 df-ima 4522 df-iota 5058 df-fun 5095 df-fn 5096 df-f 5097 df-f1 5098 df-fo 5099 df-f1o 5100 df-fv 5101 df-riota 5698 df-ov 5745 df-oprab 5746 df-mpo 5747 df-1st 6006 df-2nd 6007 df-recs 6170 df-frec 6256 df-pnf 7770 df-mnf 7771 df-xr 7772 df-ltxr 7773 df-le 7774 df-sub 7903 df-neg 7904 df-reap 8304 df-ap 8311 df-div 8400 df-inn 8685 df-2 8743 df-3 8744 df-4 8745 df-n0 8936 df-z 9013 df-uz 9283 df-rp 9398 df-fz 9746 df-fzo 9875 df-seqfrec 10174 df-exp 10248 df-cj 10569 df-re 10570 df-im 10571 df-rsqrt 10725 df-abs 10726 df-clim 11003 |
This theorem is referenced by: isumge0 11154 |
Copyright terms: Public domain | W3C validator |