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Mirrors > Home > ILE Home > Th. List > fprodge0 | GIF version |
Description: If all the terms of a finite product are nonnegative, so is the product. (Contributed by Glauco Siliprandi, 5-Apr-2020.) |
Ref | Expression |
---|---|
fprodge0.kph | ⊢ Ⅎ𝑘𝜑 |
fprodge0.a | ⊢ (𝜑 → 𝐴 ∈ Fin) |
fprodge0.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) |
fprodge0.0leb | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) |
Ref | Expression |
---|---|
fprodge0 | ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0xr 7918 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfxr 7924 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge0.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | rge0ssre 9874 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
5 | ax-resscn 7818 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
6 | 4, 5 | sstri 3137 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
7 | 6 | a1i 9 | . . 3 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
8 | ge0mulcl 9879 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 · 𝑦) ∈ (0[,)+∞)) | |
9 | 8 | adantl 275 | . . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞))) → (𝑥 · 𝑦) ∈ (0[,)+∞)) |
10 | fprodge0.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ Fin) | |
11 | fprodge0.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
12 | fprodge0.0leb | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 0 ≤ 𝐵) | |
13 | elrege0 9873 | . . . 4 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
14 | 11, 12, 13 | sylanbrc 414 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
15 | 1re 7871 | . . . . 5 ⊢ 1 ∈ ℝ | |
16 | 0le1 8350 | . . . . 5 ⊢ 0 ≤ 1 | |
17 | ltpnf 9680 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
19 | 0re 7872 | . . . . . 6 ⊢ 0 ∈ ℝ | |
20 | elico2 9834 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞))) | |
21 | 19, 2, 20 | mp2an 423 | . . . . 5 ⊢ (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞)) |
22 | 15, 16, 18, 21 | mpbir3an 1164 | . . . 4 ⊢ 1 ∈ (0[,)+∞) |
23 | 22 | a1i 9 | . . 3 ⊢ (𝜑 → 1 ∈ (0[,)+∞)) |
24 | 3, 7, 9, 10, 14, 23 | fprodcllemf 11503 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
25 | icogelb 10158 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
26 | 1, 2, 24, 25 | mp3an12i 1323 | 1 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 Ⅎwnf 1440 ∈ wcel 2128 ⊆ wss 3102 class class class wbr 3965 (class class class)co 5821 Fincfn 6682 ℂcc 7724 ℝcr 7725 0cc0 7726 1c1 7727 · cmul 7731 +∞cpnf 7903 ℝ*cxr 7905 < clt 7906 ≤ cle 7907 [,)cico 9787 ∏cprod 11440 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4079 ax-sep 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4495 ax-iinf 4546 ax-cnex 7817 ax-resscn 7818 ax-1cn 7819 ax-1re 7820 ax-icn 7821 ax-addcl 7822 ax-addrcl 7823 ax-mulcl 7824 ax-mulrcl 7825 ax-addcom 7826 ax-mulcom 7827 ax-addass 7828 ax-mulass 7829 ax-distr 7830 ax-i2m1 7831 ax-0lt1 7832 ax-1rid 7833 ax-0id 7834 ax-rnegex 7835 ax-precex 7836 ax-cnre 7837 ax-pre-ltirr 7838 ax-pre-ltwlin 7839 ax-pre-lttrn 7840 ax-pre-apti 7841 ax-pre-ltadd 7842 ax-pre-mulgt0 7843 ax-pre-mulext 7844 ax-arch 7845 ax-caucvg 7846 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-if 3506 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3773 df-int 3808 df-iun 3851 df-br 3966 df-opab 4026 df-mpt 4027 df-tr 4063 df-id 4253 df-po 4256 df-iso 4257 df-iord 4326 df-on 4328 df-ilim 4329 df-suc 4331 df-iom 4549 df-xp 4591 df-rel 4592 df-cnv 4593 df-co 4594 df-dm 4595 df-rn 4596 df-res 4597 df-ima 4598 df-iota 5134 df-fun 5171 df-fn 5172 df-f 5173 df-f1 5174 df-fo 5175 df-f1o 5176 df-fv 5177 df-isom 5178 df-riota 5777 df-ov 5824 df-oprab 5825 df-mpo 5826 df-1st 6085 df-2nd 6086 df-recs 6249 df-irdg 6314 df-frec 6335 df-1o 6360 df-oadd 6364 df-er 6477 df-en 6683 df-dom 6684 df-fin 6685 df-pnf 7908 df-mnf 7909 df-xr 7910 df-ltxr 7911 df-le 7912 df-sub 8042 df-neg 8043 df-reap 8444 df-ap 8451 df-div 8540 df-inn 8828 df-2 8886 df-3 8887 df-4 8888 df-n0 9085 df-z 9162 df-uz 9434 df-q 9522 df-rp 9554 df-ico 9791 df-fz 9906 df-fzo 10035 df-seqfrec 10338 df-exp 10412 df-ihash 10643 df-cj 10735 df-re 10736 df-im 10737 df-rsqrt 10891 df-abs 10892 df-clim 11169 df-proddc 11441 |
This theorem is referenced by: fprodle 11530 |
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