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Mirrors > Home > MPE Home > Th. List > fprodge0 | Structured version Visualization version 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 10682 | . 2 ⊢ 0 ∈ ℝ* | |
2 | pnfxr 10689 | . 2 ⊢ +∞ ∈ ℝ* | |
3 | fprodge0.kph | . . 3 ⊢ Ⅎ𝑘𝜑 | |
4 | rge0ssre 12838 | . . . . 5 ⊢ (0[,)+∞) ⊆ ℝ | |
5 | ax-resscn 10588 | . . . . 5 ⊢ ℝ ⊆ ℂ | |
6 | 4, 5 | sstri 3975 | . . . 4 ⊢ (0[,)+∞) ⊆ ℂ |
7 | 6 | a1i 11 | . . 3 ⊢ (𝜑 → (0[,)+∞) ⊆ ℂ) |
8 | ge0mulcl 12843 | . . . 4 ⊢ ((𝑥 ∈ (0[,)+∞) ∧ 𝑦 ∈ (0[,)+∞)) → (𝑥 · 𝑦) ∈ (0[,)+∞)) | |
9 | 8 | adantl 484 | . . 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 12836 | . . . 4 ⊢ (𝐵 ∈ (0[,)+∞) ↔ (𝐵 ∈ ℝ ∧ 0 ≤ 𝐵)) | |
14 | 11, 12, 13 | sylanbrc 585 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐵 ∈ (0[,)+∞)) |
15 | 1re 10635 | . . . . 5 ⊢ 1 ∈ ℝ | |
16 | 0le1 11157 | . . . . 5 ⊢ 0 ≤ 1 | |
17 | ltpnf 12509 | . . . . . 6 ⊢ (1 ∈ ℝ → 1 < +∞) | |
18 | 15, 17 | ax-mp 5 | . . . . 5 ⊢ 1 < +∞ |
19 | 0re 10637 | . . . . . 6 ⊢ 0 ∈ ℝ | |
20 | elico2 12794 | . . . . . 6 ⊢ ((0 ∈ ℝ ∧ +∞ ∈ ℝ*) → (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞))) | |
21 | 19, 2, 20 | mp2an 690 | . . . . 5 ⊢ (1 ∈ (0[,)+∞) ↔ (1 ∈ ℝ ∧ 0 ≤ 1 ∧ 1 < +∞)) |
22 | 15, 16, 18, 21 | mpbir3an 1337 | . . . 4 ⊢ 1 ∈ (0[,)+∞) |
23 | 22 | a1i 11 | . . 3 ⊢ (𝜑 → 1 ∈ (0[,)+∞)) |
24 | 3, 7, 9, 10, 14, 23 | fprodcllemf 15306 | . 2 ⊢ (𝜑 → ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) |
25 | icogelb 12782 | . 2 ⊢ ((0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ∏𝑘 ∈ 𝐴 𝐵 ∈ (0[,)+∞)) → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) | |
26 | 1, 2, 24, 25 | mp3an12i 1461 | 1 ⊢ (𝜑 → 0 ≤ ∏𝑘 ∈ 𝐴 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 Ⅎwnf 1780 ∈ wcel 2110 ⊆ wss 3935 class class class wbr 5058 (class class class)co 7150 Fincfn 8503 ℂcc 10529 ℝcr 10530 0cc0 10531 1c1 10532 · cmul 10536 +∞cpnf 10666 ℝ*cxr 10668 < clt 10669 ≤ cle 10670 [,)cico 12734 ∏cprod 15253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-n0 11892 df-z 11976 df-uz 12238 df-rp 12384 df-ico 12738 df-fz 12887 df-fzo 13028 df-seq 13364 df-exp 13424 df-hash 13685 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-clim 14839 df-prod 15254 |
This theorem is referenced by: fprodle 15344 hoiprodcl 42823 hoiprodcl3 42856 hoidmvcl 42858 hsphoidmvle2 42861 hsphoidmvle 42862 |
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